Fixed Income Analysis: Securities, Pricing, and Risk Management

Fixed Income Analysis: Securiies, Pricing, and Risk Managemen Claus Munk This version: January 23, 2003 Deparmen of Accouning and Finance, Universiy of Souhern Denmark, Campusvej 55, DK-5230 Odense M, Denmark. Phone: 6550 3257. Fax: 6593 0726. E-mail: cmu@sam.sdu.dk. Inerne homepage: hp://www.sam.sdu.dk/ cmu

Conens Preface viii 1 Basic ineres rae markes, conceps, and relaions 1 1.1 Inroducion........................................ 1 1.2 Markes for bonds and ineres raes.......................... 1 1.3 Discoun facors and zero-coupon bonds........................ 5 1.4 Zero-coupon raes and forward raes.......................... 6 1.4.1 Annual compounding.............................. 7 1.4.2 Compounding over oher discree periods LIBOR raes.......... 8 1.4.3 Coninuous compounding............................ 8 1.4.4 Differen ways o represen he erm srucure of ineres raes....... 10 1.5 Deermining he zero-coupon yield curve: Boosrapping............... 10 1.6 Deermining he zero-coupon yield curve: Parameerized forms........... 14 1.6.1 Cubic splines................................... 14 1.6.2 The Nelson-Siegel parameerizaion....................... 18 1.6.3 Addiional remarks on yield curve esimaion................. 19 1.7 Exercises......................................... 20 2 Fixed income securiies 22 2.1 Inroducion........................................ 22 2.2 Floaing rae bonds.................................... 22 2.3 Forwards on bonds.................................... 23 2.4 Ineres rae forwards forward rae agreemens................... 25 2.5 Fuures on bonds..................................... 26 2.6 Ineres rae fuures Eurodollar fuures........................ 26 2.7 Opions on bonds..................................... 28 2.7.1 Opions on zero-coupon bonds......................... 29 2.7.2 Opions on coupon bonds............................ 30 2.8 Caps, floors, and collars................................. 31 2.8.1 Caps........................................ 31 2.8.2 Floors....................................... 33 2.8.3 Collars....................................... 33 2.8.4 Exoic caps and floors.............................. 34 2.9 Swaps and swapions................................... 35 i

Conens ii 2.9.1 Swaps....................................... 35 2.9.2 Swapions..................................... 39 2.9.3 Exoic swap insrumens............................. 40 2.10 Exercises......................................... 41 3 Sochasic processes and sochasic calculus 43 3.1 Probabiliy spaces.................................... 43 3.2 Sochasic processes................................... 44 3.2.1 Differen ypes of sochasic processes..................... 44 3.2.2 Basic conceps.................................. 45 3.2.3 Markov processes and maringales....................... 46 3.2.4 Coninuous or disconinuous pahs....................... 46 3.3 Brownian moions.................................... 47 3.4 Diffusion processes.................................... 51 3.5 Iô processes....................................... 52 3.6 Sochasic inegrals.................................... 53 3.6.1 Definiion and properies of sochasic inegrals................ 53 3.6.2 Leibniz rule for sochasic inegrals...................... 54 3.7 Iô s Lemma........................................ 56 3.8 Imporan diffusion processes.............................. 57 3.8.1 Geomeric Brownian moions.......................... 57 3.8.2 Ornsein-Uhlenbeck processes.......................... 59 3.8.3 Square roo processes.............................. 62 3.9 Muli-dimensional processes............................... 64 3.10 Change of probabiliy measure............................. 67 3.11 Exercises......................................... 70 4 Asse pricing and erm srucures: discree-ime models 71 4.1 Inroducion........................................ 71 4.2 A one-period model................................... 72 4.2.1 Asses, porfolios, and arbirage........................ 72 4.2.2 Invesors..................................... 73 4.2.3 Sae-price vecors and deflaors........................ 75 4.2.4 Risk-neural probabiliies............................ 77 4.2.5 Redundan asses................................. 79 4.2.6 Complee markes................................ 80 4.2.7 Equilibrium and represenaive agens in complee markes......... 82 4.3 A muli-period, discree-ime model.......................... 84 4.3.1 Asses, rading sraegies, and arbirage.................... 85 4.3.2 Invesors..................................... 86 4.3.3 Sae-price vecors and deflaors........................ 87 4.3.4 Risk-neural probabiliy measures....................... 89 4.3.5 Redundan asses................................. 91 4.3.6 Complee markes................................ 92

Conens iii 4.3.7 Equilibrium and represenaive agens in complee markes......... 93 4.4 Discree-ime, finie-sae models of he erm srucure................ 94 4.5 Concluding remarks................................... 95 4.6 Exercises......................................... 95 5 Asse pricing and erm srucures: an inroducion o coninuous-ime models 97 5.1 Inroducion........................................ 97 5.2 Asse pricing in coninuous-ime models........................ 98 5.2.1 Asses, rading sraegies, and arbirage.................... 99 5.2.2 Invesors..................................... 101 5.2.3 Sae-price deflaors............................... 102 5.2.4 Risk-neural probabiliy measures....................... 103 5.2.5 From no arbirage o sae-price deflaors and risk-neural measures.... 105 5.2.6 Marke prices of risk............................... 105 5.2.7 Complee vs. incomplee markes........................ 108 5.2.8 Exension o inermediae dividends...................... 109 5.2.9 Equilibrium and represenaive agens in complee markes......... 110 5.3 Oher probabiliy measures convenien for pricing................... 111 5.3.1 A zero-coupon bond as he numeraire forward maringale measures.... 113 5.3.2 An annuiy as he numeraire swap maringale measures.......... 114 5.3.3 A general pricing formula for European opions................ 114 5.4 Forward prices and fuures prices............................ 116 5.4.1 Forward prices.................................. 116 5.4.2 Fuures prices................................... 117 5.4.3 A comparison of forward prices and fuures prices.............. 118 5.5 American-syle derivaives................................ 119 5.6 Diffusion models and he fundamenal parial differenial equaion......... 119 5.6.1 One-facor diffusion models........................... 120 5.6.2 Muli-facor diffusion models.......................... 125 5.7 The Black-Scholes-Meron model and Black s varian................. 127 5.7.1 The Black-Scholes-Meron model........................ 127 5.7.2 Black s model................................... 129 5.7.3 Problems in applying Black s model o fixed income securiies........ 133 5.8 An overview of coninuous-ime erm srucure models................ 134 5.8.1 Overall caegorizaion.............................. 135 5.8.2 Some frequenly applied models......................... 136 5.9 Exercises......................................... 138 6 The Economics of he Term Srucure of Ineres Raes 139 6.1 Inroducion........................................ 139 6.2 Real ineres raes and aggregae consumpion.................... 140 6.3 Real ineres raes and aggregae producion..................... 142 6.4 Equilibrium ineres rae models............................ 145 6.4.1 Producion-based models............................ 145

Conens iv 6.4.2 Consumpion-based models........................... 146 6.5 Real and nominal ineres raes and erm srucures................. 147 6.5.1 Real and nominal asse pricing......................... 148 6.5.2 No real effecs of inflaion............................ 151 6.5.3 A model wih real effecs of money....................... 152 6.6 The expecaion hypohesis............................... 157 6.6.1 Versions of he pure expecaion hypohesis.................. 157 6.6.2 The pure expecaion hypohesis and equilibrium............... 158 6.6.3 The weak expecaion hypohesis........................ 159 6.7 Liquidiy preference, marke segmenaion, and preferred habias......... 160 6.8 Concluding remarks................................... 161 6.9 Exercises......................................... 161 7 One-facor diffusion models 163 7.1 Inroducion........................................ 163 7.2 Affine models....................................... 164 7.2.1 Bond prices, zero-coupon raes, and forward raes.............. 165 7.2.2 Forwards and fuures............................... 167 7.2.3 European opions on coupon bonds: Jamshidian s rick........... 169 7.3 Meron s model...................................... 172 7.3.1 The shor rae process.............................. 172 7.3.2 Bond pricing................................... 173 7.3.3 The yield curve.................................. 173 7.3.4 Forwards and fuures............................... 174 7.3.5 Opion pricing.................................. 174 7.4 Vasicek s model...................................... 176 7.4.1 The shor rae process.............................. 176 7.4.2 Bond pricing................................... 178 7.4.3 The yield curve.................................. 181 7.4.4 Forwards and fuures............................... 184 7.4.5 Opion pricing.................................. 185 7.5 The Cox-Ingersoll-Ross model.............................. 188 7.5.1 The shor rae process.............................. 188 7.5.2 Bond pricing................................... 189 7.5.3 The yield curve.................................. 190 7.5.4 Forwards and fuures............................... 191 7.5.5 Opion pricing.................................. 192 7.6 Non-affine models..................................... 193 7.7 Parameer esimaion and empirical ess....................... 196 7.8 Concluding remarks................................... 199 7.9 Exercises......................................... 199

Conens v 8 Muli-facor diffusion models 201 8.1 Wha is wrong wih one-facor models?........................ 201 8.2 Muli-facor diffusion models of he erm srucure.................. 203 8.3 Muli-facor affine diffusion models........................... 205 8.3.1 Two-facor affine diffusion models....................... 205 8.3.2 n-facor affine diffusion models......................... 208 8.3.3 European opions on coupon bonds....................... 209 8.4 Muli-facor Gaussian diffusion models......................... 209 8.4.1 General analysis................................. 209 8.4.2 A specific example: he wo-facor Vasicek model............... 211 8.5 Muli-facor CIR models................................. 212 8.5.1 General analysis................................. 212 8.5.2 A specific example: he Longsaff-Schwarz model.............. 214 8.6 Oher muli-facor diffusion models........................... 219 8.6.1 Models wih sochasic consumer prices.................... 219 8.6.2 Models wih sochasic long-erm level and volailiy............. 220 8.6.3 A model wih a shor and a long rae..................... 222 8.6.4 Key rae models................................. 222 8.6.5 Quadraic models................................. 223 8.7 Final remarks....................................... 223 9 Calibraion of diffusion models 225 9.1 Inroducion........................................ 225 9.2 Time inhomogeneous affine models........................... 226 9.3 The Ho-Lee model exended Meron)......................... 228 9.4 The Hull-Whie model exended Vasicek)....................... 230 9.5 The exended CIR model................................ 232 9.6 Calibraion o oher marke daa............................ 233 9.7 Iniial and fuure erm srucures in calibraed models................ 234 9.8 Calibraed non-affine models.............................. 236 9.9 Is a calibraed one-facor model jus as good as a muli-facor model?....... 237 9.10 Final remarks....................................... 238 9.11 Exercises......................................... 239 10 Heah-Jarrow-Moron models 240 10.1 Inroducion........................................ 240 10.2 Basic assumpions.................................... 240 10.3 Bond price dynamics and he drif resricion..................... 242 10.4 Three well-known special cases............................. 244 10.4.1 The Ho-Lee exended Meron) model..................... 244 10.4.2 The Hull-Whie exended Vasicek) model................... 245 10.4.3 The exended CIR model............................ 246 10.5 Gaussian HJM models.................................. 247 10.6 Diffusion represenaions of HJM models........................ 248

Conens vi 10.6.1 On he use of numerical echniques for diffusion and non-diffusion models. 249 10.6.2 In which HJM models does he shor rae follow a diffusion process?.... 249 10.6.3 A wo-facor diffusion represenaion of a one-facor HJM model...... 252 10.7 HJM-models wih forward-rae dependen volailiies................. 253 10.8 Concluding remarks................................... 254 11 Marke models 256 11.1 Inroducion........................................ 256 11.2 General LIBOR marke models............................. 257 11.2.1 Model descripion................................ 257 11.2.2 The dynamics of all forward raes under he same probabiliy measure... 258 11.2.3 Consisen pricing................................ 263 11.3 The lognormal LIBOR marke model.......................... 263 11.3.1 Model descripion................................ 263 11.3.2 The pricing of oher securiies.......................... 265 11.4 Alernaive LIBOR marke models........................... 267 11.5 Swap marke models................................... 268 11.6 Furher remarks..................................... 270 11.7 Exercises......................................... 271 12 The measuremen and managemen of ineres rae risk 272 12.1 Inroducion........................................ 272 12.2 Tradiional measures of ineres rae risk........................ 272 12.2.1 Macaulay duraion and convexiy........................ 272 12.2.2 The Fisher-Weil duraion and convexiy.................... 274 12.2.3 The no-arbirage principle and parallel shifs of he yield curve....... 275 12.3 Risk measures in one-facor diffusion models...................... 276 12.3.1 Definiions and relaions............................. 276 12.3.2 Compuaion of he risk measures in affine models.............. 279 12.3.3 A comparison wih radiional duraions.................... 281 12.4 Immunizaion....................................... 282 12.4.1 Consrucion of immunizaion sraegies.................... 282 12.4.2 An experimenal comparison of immunizaion sraegies........... 284 12.5 Risk measures in muli-facor diffusion models..................... 288 12.5.1 Facor duraions, convexiies, and ime value................. 288 12.5.2 One-dimensional risk measures in muli-facor models............ 290 12.6 Duraion-based pricing of opions on bonds...................... 292 12.6.1 The general idea................................. 292 12.6.2 A mahemaical analysis of he approximaion................ 293 12.6.3 The accuracy of he approximaion in he Longsaff-Schwarz model.... 294 12.7 Alernaive measures of ineres rae risk........................ 296 13 Morgage-backed securiies 299

Conens vii 14 Credi risky securiies 300 15 Sochasic ineres raes and he pricing of sock and currency derivaives 301 15.1 Inroducion........................................ 301 15.2 Sock opions....................................... 301 15.2.1 General analysis................................. 301 15.2.2 Deerminisic volailiies............................. 304 15.3 Opions on forwards and fuures............................ 306 15.3.1 Forward and fuures prices........................... 306 15.3.2 Opions on forwards............................... 307 15.3.3 Opions on fuures................................ 308 15.4 Currency derivaives................................... 309 15.4.1 Currency forwards................................ 309 15.4.2 A model for he exchange rae......................... 310 15.4.3 Currency fuures................................. 311 15.4.4 Currency opions................................. 311 15.4.5 Alernaive exchange rae models........................ 313 15.5 Final remarks....................................... 313 16 Numerical echniques 315 A Resuls on he lognormal disribuion 316 References 319

Preface This book provides an inroducion o he markes for fixed-income securiies and he models and mehods ha are used o analyze such securiies. The class of fixed-income securiies covers securiies where he issuer promises one or several fixed, predeermined paymens a given poins in ime. This is he case for sandard deposi arrangemens and bonds. However, several relaed securiies wih paymens ha are ied o he developmen in some paricular index, ineres rae, or asse price are ypically also ermed fixed-income securiies. In he broades sense of he erm, he many differen ineres rae and bond derivaives are also considered fixed-income producs. Maybe a more descripive erm for his broad class of securiies is ineres rae securiies, since he values of hese financial conracs are derived from curren ineres raes and expecaions and uncerainy abou fuure ineres raes. The key concep in he analysis of fixed-income securiies is he erm srucure of ineres raes, which is loosely defined as he dependence beween ineres raes and mauriies. The ouline of his book is as follows. The firs wo chapers deal wih he mos common fixedincome securiies, fix much of he noaion and erminology, and discuss basic relaions beween key conceps. The main par of he book discusses models of he evoluion of he erm srucure of ineres raes over ime. Chaper 3 inroduces much of he mahemaics needed for developing and analyzing modern dynamic models of ineres raes. In Chapers 4 and 5 we review some of he imporan general resuls on asse pricing. In paricular, we define and relae he key conceps of arbirage, sae-price deflaors, and risk-neural probabiliy measures. The connecion o marke compleeness and individual invesors behavior is also addressed, jus as he implicaions of he general asse pricing heory for he modeling of he erm srucure are discussed. Chaper 6 applies he general asse pricing ools o explore he economics of he erm srucure of ineres raes. For example we discuss he relaion beween he erm srucure of ineres raes and macro-economic variables such as aggregae consumpion, producion, and inflaion. We will also review some of he radiional hypoheses on he shape of he yield curve, e.g. he expecaion hypoheses. Chapers 7 o 12 develops models for he pricing of fixed income securiies and he managemen of ineres rae risk. Chaper 7 goes hrough so-called one-facor models. This ype of models was he firs o be applied in he lieraure and daes back a leas o 1970. The one-facor models of Vasicek and Cox, Ingersoll, and Ross are sill frequenly applied boh in pracice and in academic research. Chaper 8 explores muli-facor models which have several advanages over one-facor models, bu are also more complicaed o analyze and apply. In Chaper 9 we discuss how oneand muli-facor models can be exended o be consisen wih curren marke informaion, such as bond prices and volailiies. Chaper 10 inroduces and analyzes so-called Heah-Jarrow-Moron viii

Preface ix models, which are characerized by aking he curren marke erm srucure of ineres raes as given and hen modeling he evoluion of he enire erm srucure in an arbirage-free way. We will explore he relaion beween hese models and he facor models sudied in earlier chapers. Ye anoher class of models is he subjec of Chaper 11. These marke models are designed for he pricing and hedging of specific producs ha are raded on a large scale in he inernaional markes, namely caps, floors, and swapions. Chaper 12 discusses how he differen ineres rae models can be applied for ineres rae risk managemen. The subjec of Chaper 13 only some references are lised in he curren version) is how o consruc models for he pricing and risk managemen of morgage-backed securiies. The main concern is how o adjus he models sudied in earlier chapers o ake he prepaymen opions involved in morgages ino accoun. In Chaper 14 only some references are lised in he curren version) we discuss he pricing of corporae bonds and oher fixed-income securiies where he defaul risk of he issuer canno be ignored. Chaper 15 focuses on he consequences which sochasic variaions in ineres raes have for he valuaion of securiies wih paymens ha are no direcly relaed o ineres raes, such as sock opions and currency opions. Finally, Chaper 16 only some references are lised in he curren version) describes several numerical echniques ha can be applied in cases where explici pricing and hedging formulas are no available. Syle... Prerequisies... There are several oher books ha cover much of he same maerial or focus on paricular elemens discussed in his book. Books emphasizing descripions of markes and producs: Fabozzi 2000), van Horne 2001). Books emphasizing modern ineres rae modeling: James and Webber 2000), Pelsser 2000), Rebonao 1996),... I appreciae commens and correcions from Lennar Damgaard, Hans Frimor, Mee Hansen, Frank Emil Jensen, Kasper Larsen, Moren Mosegaard, Per Plonikoff, and oher people. I also appreciae he excellen secrearial assisance of Lene Holbæk.

Chaper 1 Basic ineres rae markes, conceps, and relaions 1.1 Inroducion The value of an asse equals he value of is fuure cash flow. Tradiionally, he value of a bond is compued by discouning all is fuure paymens using he same ineres rae, namely he yield of he bond. If we have wo bonds wih he same paymen daes bu differen yields, we will herefore discoun he paymens from he bonds wih differen ineres raes. This is clearly illogical. The presen value of a given paymen a a given fuure poin in ime is independen of which asse he paymen sems from. All sure paymens a he same dae should be discouned wih he same rae. On he oher hand, here is no reason o discoun paymens a differen daes wih he same discoun rae. The ineres rae on a loan will normally depend on he mauriy of he loan, and on he bond markes here will ofen be differences beween he yields on shor-erm bonds and long-erm bonds. The erm srucure of ineres raes is he relaion beween ineres raes and heir mauriy. In his chaper we firs provide a brief inroducion o he mos basic markes for borrowing and lending and discuss differen ypes of ineres raes and discoun facors as well as he relaion beween hem. We will briefly inroduce some heories of he level and shape of he erm srucure of ineres raes. Finally, we will discuss how informaion abou he erm srucure can be exraced from marke daa. 1.2 Markes for bonds and ineres raes This secion gives a brief descripion of markes for bonds and oher deb conracs. A bond is simply a radable loan agreemen. Mos bonds ha are issued and subsequenly raded a eiher organized exchanges or over-he-couner OTC) represen medium- or long-erm loans wih mauriies in excess of one year and ofen beween 10 and 30 years. Some shor-erm bonds are also issued and raded, bu much of he shor-erm borrowing aciviy akes place in he so-called money marke, where large financial insiuions including he cenral bank) and large corporaions form several ypes of loan agreemens wih mauriies ranging from a few hours up o around one year. These agreemens are usually no raded afer he original conracing. The ineres raes se in he money marke direcly affec he ineres raes ha banks offer and charge heir commercial 1

1.2 Markes for bonds and ineres raes 2 and household cusomers. Small invesors may paricipae in he money markes hrough money marke funds. Below, we inroduce he mos imporan ypes of bonds ha are raded. More deails on bond markes can be found in e.g. Fabozzi 2000). We also look a some of he deb conracs available in he money marke. In Chaper 2 we will discuss many oher ineres rae relaed securiies, such as fuures and opions on bonds and ineres raes. The issuer of he bond he debor or borrower) issues a conrac in which he is obligaed o pay cerain paymens a cerain fuure poins in ime. Typically, a bond issue consiss of a series of idenical bonds. The simples possible bond is a zero-coupon bond, which is a bond promising a single paymen a a given fuure dae, he mauriy dae of he bond, and no oher paymens. Bonds which promise more han one paymen when issued are referred o as coupon bonds. Typically, he paymens of coupon bonds follow a regular schedule so ha he paymens occur a regular inervals quarerly, semi-annually, or annually) and he size of each of he paymens is deermined by he face value, he coupon rae, and he amorizaion principle of he bond. The face value is also known as he par value or principal of he bond, and he coupon rae is also called he nominal rae or saed ineres rae. Mos coupon bonds are so-called bulle bonds or sraigh-coupon bonds where all he paymens before he final paymen are equal o he produc of he coupon rae and he face value. The final paymen a he mauriy dae is he sum of he same ineres rae paymen and he face value. Oher bonds are so-called annuiy bonds, which are consruced so ha he oal paymen is equal for all paymen daes. Each paymen is he sum of an ineres paymen and a parial repaymen of he face value. The ousanding deb and he ineres paymen are gradually decreasing over he life of an annuiy, so ha he repaymen increases over ime. Some bonds are so-called serial bonds where he face value is paid back in equal insalmens. The paymen a a given paymen dae is hen he sum of he insalmen and he ineres rae on he ousanding deb. The ineres rae paymens, and hence he oal paymens, will herefore decrease over he life of he bond. Finally, few bonds are perpeuiies or consols ha las forever and only pay ineres. The face value of a perpeuiy is never repaid. Mos coupon bonds have a fixed coupon rae, bu a small minoriy of bonds have coupon raes ha are rese periodically over he life of he bond. Such bonds are called floaing rae bonds. Typically, he coupon rae effecive for he paymen a he end of one period is se a he beginning of he period a he curren marke ineres rae for ha period, e.g. o he 6-monh ineres rae for a floaing rae bond wih semi-annual paymens. Bond markes can be divided ino he naional markes of differen counries and he inernaional marke also known as he Eurobond marke). The larges naional bond markes are hose of he U.S., Japan, Germany, Ialy, and France followed by oher Wesern European counries and Ausralia. In he naional marke of a counry, primarily bonds issued by domesic issuers are raded, bu ofen some bonds issued by cerain foreign governmens or corporaions or inernaional associaions are also raded. The bonds issued in a given naional marke mus comply wih he regulaion of ha paricular counry. Bonds issued in he less regulaed Eurobond marke are usually underwrien by an inernaional syndicae and offered o invesors in several counries simulaneously. Many Eurobonds are lised on one naional exchange, ofen in Luxembourg or London, bu mos of he rading in hese bonds akes place OTC over-he-couner). Oher Eurobonds are issued as a privae placemen wih financial insiuions. Mos Eurobonds are issued

1.2 Markes for bonds and ineres raes 3 in U.S. dollars Eurodollar bonds ), he common European currency Euro, Pound Serling, Swiss francs, or Japanese yen. Divided according o he ype of issuer, naional bond markes have wo or hree major caegories: governmen-relaed) bonds, corporae bonds, and in some counries morgage-backed bonds. In addiion, some bonds issued by inernaional insiuions or foreign governmens are ofen raded. Eurobonds are ypically issued by inernaional insiuions, governmens, or large corporaions. In mos naional bond markes, he major par of bond rading is in governmen bonds, which are simply bonds issued by he governmen o finance and refinance he public deb. In mos counries, such bonds can be considered o be free of defaul risk, and ineres raes in he governmen bond marke are hen a benchmark agains which he ineres raes on oher bonds are measured. However, in some economically and poliically unsable counries, he defaul risk on governmen bonds canno be ignored. In he U.S., governmen bonds are issued by he Deparmen of he Treasury and called Treasury securiies. These securiies are divided ino hree caegories: bills, noes, and bonds. Treasury bills or simply T-bills) are shor-erm securiies ha maure in one year or less from heir issue dae. T-bills are zero-coupon bonds since hey have a single paymen equal o he face value. Treasury noes and bonds are coupon-bearing bulle bonds wih semi-annual paymens. The only difference beween noes and bonds is he ime-omauriy when firs issued. Treasury noes are issued wih a ime-o-mauriy of 1-10 years, while Treasury bonds maure in more han 10 years and up o 30 years from heir issue dae. Treasury sells wo ypes of noes and bonds, fixed-principal and inflaion-indexed. The fixed-principal ype promises given dollar paymens in he fuure, whereas he dollar paymens of he inflaion-indexed ype are adjused o reflec inflaion in consumer prices. 1 Finally, he U.S. Treasury also issue socalled savings bonds o individuals and cerain organizaions, bu hese bonds are no subsequenly radable. While Treasury noes and bonds are issued as coupon bonds, he Treasury Deparmen inroduced he so-called STRIPS program in 1985 ha les invesors hold and rade he individual ineres and principal componens of mos Treasury noes and bonds as separae securiies. 2 These separae securiies, which are usually referred o as STRIPs, are zero-coupon bonds. Marke paricipans creae STRIPs by separaing he ineres and principal pars of a Treasury noe or bond. For example, a 10-year Treasury noe consiss of 20 semi-annual ineres paymens and a principal paymen payable a mauriy. When his securiy is sripped, each of he 20 ineres paymens and he principal paymen become separae securiies and can be held and ransferred separaely. 3 In some counries including he U.S., bonds issued by various public insiuions, e.g. uiliy companies, railway companies, expor suppor funds, ec., are backed by he governmen, so ha he defaul risk on such bonds is he risk ha he governmen defauls. In addiion, some bonds are issued by governmen-sponsored eniies creaed o faciliae borrowing and reduce borrowing 1 The principal value of an inflaion-indexed noe or bond is adjused before each paymen dae according o he change in he consumer price index. Since he semi-annual ineres paymens are compued as he produc of he fixed coupon rae and he curren principal, all he paymens of an inflaion-indexed noe or bond are inflaionadjused. 2 STRIPS is shor for Separae Trading of Regisered Ineres and Principal of Securiies. 3 More informaion on Treasury securiies can be found on he homepage of he Bureau of he Public Deb a he Deparmen of he Treasury, see www.publicdeb.reas.gov.

1.2 Markes for bonds and ineres raes 4 coss for e.g. farmers, homeowners, and sudens. However, hese bonds are ypically no backed by he governmen and are herefore exposed o he risk of defaul of he issuing organizaion. Bonds may also be issued by local governmens. In he U.S. such bonds are known as municipal bonds. In some counries, corporaions will radiionally borrow funds by issuing bonds, so-called corporae bonds. This is he case in he U.S., where here is a large marke for such bonds. In oher counries, e.g. Germany, corporaions end o borrow funds hrough bank loans, so ha he marke for corporae bonds is very limied. For corporae bonds, invesors canno ignore he possibiliy ha he issuer defauls and canno mee he obligaions represened by he bonds. Bond invesors can eiher perform heir own analysis of he crediworhiness of he issuer or rely on he analysis of professional raing agencies such as Moody s Invesors Service or Sandard & Poor s Corporaion. These agencies designae leer codes o bond issuers boh in he U.S. and in oher counries. Invesors will ypically rea bonds wih he same raing as having nearly) he same defaul risk. Due o he defaul risk, corporae bonds are raded a lower prices han similar defaul-free) governmen bonds. The managemen of he issuing corporaion can effecively ransfer wealh from bond-holders o equiy-holders, e.g. by increasing dividends, aking on more risky invesmen projecs, or issuing new bonds wih he same or even higher prioriy in case of defaul. Corporae bonds are ofen issued wih bond covenans or bond indenures ha resric managemen from implemening such acions. U.S. corporae bonds are ypically issued wih mauriies of 10-30 years and are ofen callable bonds, so ha he issuer has he righ o buy back he bonds on cerain erms a given poins in ime and for a given price). Some corporae bonds are converible bonds meaning ha he bondholders may conver he bonds ino socks of he issuing corporaion on predeermined erms. Alhough mos corporae bonds are lised on a naional exchange, much of he rading in hese bonds is in he OTC marke. Morgage-backed bonds consiue a large par of some bond markes, e.g. in he U.S., Swizerland, and Denmark. A morgage is a loan ha can parly) finance he borrower s purchase of a given real esae propery, which is hen used as collaeral for he loan. Morgages can be residenial family houses, aparmens, ec.) or non-residenial corporaions, farms, ec.). The issuer of he loan he lender) is a financial insiuion. Typical morgages have a mauriy beween 15 and 30 years and are annuiies in he sense ha he oal scheduled paymen ineres plus repaymen) a all paymen daes are idenical. Fixed-rae morgages have a fixed ineres rae, while adjusable-rae morgages have an ineres rae which is rese periodically according o some reference rae. A characerisic feaure of mos morgages is he prepaymen opion. A any paymen dae in he life of he loan, he borrower has he righ o pay off all or par of he ousanding deb. This can occur due o a sale of he underlying real esae propery, bu can also occur afer a drop in marke ineres raes, since he borrower hen have he chance o ge a cheaper loan. Morgages are pooled eiher by he issuers or oher insiuions, who hen issue morgage-backed securiies ha have an ownership ineres in a given pool of morgage loans. The mos common ype of morgage-backed securiies is he so-called pass-hrough, where he pooling insiuion simply collecs he paymens from borrowers wih loans in a given pool and passes hrough he cash flow o invesors less some servicing and guaraneeing fees. Many pass-hroughs have

1.3 Discoun facors and zero-coupon bonds 5 paymen schemes equal o he paymen schemes of bonds, e.g. pass-hroughs issued on he basis of a pool of fixed-rae annuiy morgage loans have a paymen schedule equal o ha of annuiy bond. However, when borrowers in he pool prepay heir morgage, hese prepaymens are also passed hrough o he securiy-holders, so ha heir paymens will be differen from annuiies. In general, owners of pass-hrough securiies mus ake ino accoun he risk ha he morgage borrowers in he pool defaul on heir loans. In he U.S. mos pass-hroughs are issued by hree organizaions ha guaranee he paymens o he securiies even if borrowers defaul. These organizaions are he Governmen Naional Morgage Associaion called Ginnie Mae ), he Federal Home Loan Morgage Corporaion Freddie Mac ), and he Federal Naional Morgage Associaion Fannie Mae ). Ginnie Mae pass-hroughs are even guaraneed by he U.S. governmen, bu he securiies issued by he wo oher insiuions are also considered virually free of defaul risk. Finally, le us ake a brief look a some of he deb conracs made in he money marke. While we focus on he U.S. marke, similar conracs exis in many oher counries and many of he conracs are also made in he Euromarke. The deb conracs in he money marke are mainly zero-coupon loans, which have a single repaymen dae. Financial insiuions borrow large amouns over shor periods from each oher by issuing cerificaes of deposi, also known in he marke as CDs. In he Euromarke deposis are negoiaed for various erms and currencies, bu mos deposis are in U.S. dollars and for a period of one, hree, or six monhs. Ineres raes se on deposis a he London inerbank marke are called LIBOR raes LIBOR is shor for London Inerbank offered rae). To manage very shor-erm liquidiy, financial insiuions ofen agree on overnigh loans, socalled federal funds. The ineres rae charged on such loans is called he Fed funds rae. The Federal Reserve has a arge Fed funds rae and buys and sells securiies in open marke operaions o manage he liquidiy in he marke, hereby also affecing he Fed funds rae. Banks may obain emporary credi direcly from he Federal Reserve a he so-called discoun window. The ineres rae charged by he Fed on such credi is called he federal discoun rae, bu since such borrowing is quie uncommon nowadays, he federal discoun rae serves more as a signaling device for he arges of he Federal Reserve. Large corporaions, boh financial corporaions and ohers, ofen borrow shor-erm by issuing so-called commercial papers. Anoher sandard money marke conrac is a repurchase agreemen or simply repo. One pary of his conrac sells a cerain asse, e.g. a shor-erm Treasury bill, o he oher pary and promises o buy back ha asse a a given fuure dae a he marke price a ha dae. A repo is effecively a collaeralized loan, where he underlying asse serves as collaeral. As cenral banks in oher counries, he Federal Reserve in he U.S. paricipaes acively in he repo marke o implemen heir moneary policy. The ineres rae on repos is called he repo rae. 1.3 Discoun facors and zero-coupon bonds We will assume hroughou ha he face value is equal o 1 dollar) unless saed oherwise. Suppose ha a some dae a zero-coupon bond wih mauriy T is raded in he financial markes a a price of B T. This price reflecs he marke discoun facor for sure ime T paymens. If many zero-coupon bonds wih differen mauriies are raded, we can form he funcion T B T, which we call he marke) discoun funcion prevailing a ime. Noe ha B = 1, since

1.4 Zero-coupon raes and forward raes 6 he value of geing 1 dollar righ away is 1 dollar, of course. Presumably, all invesors will prefer geing 1 dollar a some ime T raher han a a laer ime S. Therefore, he discoun funcion mus be decreasing, i.e. 1.1) 1 B T B S 0, T < S. An example of an esimaed marke discoun funcion is shown in Figure 1.1 on page 17. Nex, consider a coupon bond wih paymen daes 1, 2,..., n, where we assume wihou loss of generaliy ha 1 < 2 < < n. The paymen a dae i is denoed by Y i. Such a coupon bond can be seen as a porfolio of zero-coupon bonds, namely a porfolio of Y 1 zero-coupon bonds mauring a 1, Y 2 zero-coupon bonds mauring a 2, ec. If all hese zero-coupon bonds are raded in he marke, he price of he coupon bond a any ime mus be 1.2) B = i> Y i B i, where he sum is over all fuure paymen daes of he coupon bond. If his relaion does no hold, here will be a clear arbirage opporuniy in he marke. Example 1.1 Consider a bulle bond wih a face value of 100, a coupon rae of 7%, annual paymens, and exacly hree years o mauriy. Suppose zero-coupon bonds are raded wih face values of 1 dollar and ime-o-mauriy of 1, 2, and 3 years, respecively. Assume ha he prices of hese zero-coupon bonds are B +1 = 0.94, B +2 = 0.90, and B +3 = 0.87. According o 1.2), he price of he bulle bond mus hen be B = 7 0.94 + 7 0.90 + 107 0.87 = 105.97. If he price is lower han 105.97, riskfree profis can be locked in by buying he bulle bond and selling 7 one-year, 7 wo-year, and 107 hree-year zero-coupon bonds. If he price of he bulle bond is higher han 105.97, sell he bulle bond and buy 7 one-year, 7 wo-year, and 107 hree-year zero-coupon bonds. If no all he relevan zero-coupon bonds are raded, we canno jusify he relaion 1.2) as a resul of he no-arbirage principle. Sill, i is a valuable relaion. Suppose ha an invesor has deermined from privae or macro economic informaion) a discoun funcion showing he value she aribues o paymens a differen fuure poins in ime. Then she can value all sure cash flows in a consisen way by subsiuing ha discoun funcion ino 1.2). The marke prices of all bonds reflec a marke discoun funcion, which is he resul of he supply and demand for he bonds of all marke paricipans. We can hink of he marke discoun funcion as a very complex average of he individual discoun funcions of he marke paricipans. In mos markes only few zero-coupon bonds are raded, so ha informaion abou he discoun funcion mus be inferred from marke prices of coupon bonds. We discuss ways of doing ha in Secions 1.5 1.6. 1.4 Zero-coupon raes and forward raes Alhough discoun facors provide full informaion abou how o discoun amouns back and forh, i is prey hard o relae o a 5-year discoun facor of 0.7835. I is far easier o relae o he

1.4 Zero-coupon raes and forward raes 7 informaion ha he five-year ineres rae is 5%. Ineres raes are always quoed on an annual basis, i.e. as some percenage per year. However, o apply and assess he magniude of an ineres rae, we also need o know he compounding frequency of ha rae. More frequen compounding of a given ineres rae per year resuls in higher effecive ineres raes. Furhermore, we need o know a which ime he ineres rae is se or observed and for which period of ime he ineres rae applies. Spo raes applies o a period beginning a he ime he rae is se, whereas forward raes applies o a fuure period of ime. The precise definiions follow below. 1.4.1 Annual compounding Given he price B T a ime on a zero-coupon bond mauring a ime T, he relevan discoun rae beween ime and ime T is he yield on he zero-coupon bond, he so-called zero-coupon rae or spo rae for dae T. Le ŷ T hen have he following relaionship: 1.3) B T = 1 + ŷ T T ) ) or 1.4) ŷ T = B T ) 1/T ) 1. denoe his rae compued using annual compounding. We The zero-coupon raes as a funcion of mauriy is called he zero-coupon yield curve or simply he yield curve. I is one way o express he erm srucure of ineres raes. An example of a zero-coupon yield curve is shown in Figure 1.2 on page 17. While a zero-coupon or spo rae reflecs he price on a loan beween oday and a given fuure dae, a forward rae reflecs he price on a loan beween wo fuure daes. The annually compounded relevan forward rae a ime for he period beween ime T and ime S is denoed by ˆf T,S. Here, we have T < S. This is he rae, which is appropriae a ime for discouning beween ime T and S. We can hink of discouning from ime S back o ime by firs discouning from ime S o ime T and hen discouning from ime T o ime. We mus herefore have ha 1.5) 1 + ŷ S ) S ) = 1 + ŷ T ) T ) 1 + ˆf T,S ) S T ), from which we find ha ˆf T,S T )/S T ) = 1 + ŷt ) 1 + ŷ S 1. ) S )/S T ) We can also wrie 1.5) in erms of zero-coupon bond prices as ) S T ) 1.6) B S = B T T,S 1 + ˆf, so ha he forward rae is given by 1.7) ˆf T,S = Noe ha since B = 1, we have B T B S ) 1/S T ) 1. ˆf,S = B B S ) 1/S ) 1 = B S ) 1/S ) 1 = ŷ S, i.e. he forward rae for a period saring oday equals he zero-coupon rae or spo rae for he same period.

1.4 Zero-coupon raes and forward raes 8 1.4.2 Compounding over oher discree periods LIBOR raes In pracice, many ineres raes are quoed using semi-annually, quarerly, or monhly compounding. An ineres rae or R per year compounded m imes a year, corresponds o a discoun facor of 1 + R/m) m over a year. The annually compounded ineres rae ha corresponds o an ineres rae of R compounded m imes a year is 1 + R/m) m 1. This is someimes called he effecive ineres rae corresponding o he nominal ineres rae R. As discussed earlier, ineres raes are se for loans wih various mauriies and currencies a he inernaional money markes, he mos commonly used being he LIBOR raes ha are fixed in London. Tradiionally, hese raes are quoed using a compounding period equal o he mauriy of he ineres rae. If, for example, he hree-monh ineres rae is l +0.25 he presen value of one dollar paid hree monhs from now is Conversely, he hree-monh rae is More generally, he relaions are B +0.25 = l +0.25 = 1 0.25 1.8) B T = and 1 1 + 0.25 l +0.25 ) 1 B +0.25 1. 1 1 + l T T ) l T = 1 1 T B T ) 1.. per year, i means ha T,T +0.5 Similarly, a six-monh forward rae of L valid for he period [T, T + 0.5] means ha so ha B T +0.5 = B T 1 + 0.5 L T,T +0.5 ) 1, T,T +0.5 L = 1 ) B T 0.5 B T +0.5 1. More generally, 1.9) L T,S = 1 B T S T B S ) 1. Alhough such spo and forward raes are quoed on many differen money markes, we shall use he erm spo/forward) LIBOR rae for all such money marke ineres raes compued wih discree compounding. 1.4.3 Coninuous compounding Increasing he compounding frequency m, he effecive annual reurn of one dollar invesed a he ineres rae R per year increases o e R, due o he mahemaical resul saying ha 1 + m) R m = e R. lim m

1.4 Zero-coupon raes and forward raes 9 A nominal, coninuously compounded ineres rae R is equivalen o an annually compounded ineres rae of e R 1 which is bigger han R). Similarly, he zero-coupon bond price B T is relaed o he coninuously compounded zerocoupon rae y T by 1.10) B T = e yt T ), so ha 1.11) y T = 1 T ln BT. The funcion T y T is also a zero-coupon yield curve ha conains exacly he same informaion as he discoun funcion T B T and also he same informaion as he annually compounded yield curve T ŷ T. We have he following relaion beween he coninuously compounded and he annually compounded zero-coupon raes: y T = ln1 + ŷ T ). If f T,S denoes he coninuously compounded forward rae prevailing a ime for he period beween T and S, we mus have ha B S = B T T,S f e S T ), in analogy wih 1.6). Consequenly, 1.12) f T,S = ln BS ln B T S T Using 1.10), we ge he following relaion beween zero-coupon raes and forward raes under coninuous compounding: 1.13) f T,S = ys S ) y T T ). S T In he following chapers, we shall ofen focus on forward raes for fuure periods of infiniesimal lengh. The forward rae for an infiniesimal period saring a ime T is simply referred o as he forward rae for ime T and is defined as f T = lim S T f T,S. The funcion T f T is called he erm srucure of forward raes. Leing S T in he expression 1.12), we ge. 1.14) f T = ln BT T = BT / T B T, assuming ha he discoun funcion T B T is differeniable. Conversely, 1.15) B T = e T f u du. Applying 1.13), he relaion beween he infiniesimal forward rae and he spo raes can be wrien as 1.16) f T = [yt T )] T = y T + yt T T )

1.5 Deermining he zero-coupon yield curve: Boosrapping 10 under he assumpion of a differeniable erm srucure of spo raes T y T. The forward rae reflecs he slope of he zero-coupon yield curve. In paricular, he forward rae f T and he zerocoupon rae y T will coincide if and only if he zero-coupon yield curve has a horizonal angen a T. Conversely, we see from 1.15) and 1.10) ha 1.17) y T = 1 T f u du, T i.e. he zero-coupon rae is an average of he forward raes. 1.4.4 Differen ways o represen he erm srucure of ineres raes I is imporan o realize ha discoun facors, spo raes, and forward raes wih any compounding frequency) are perfecly equivalen ways of expressing he same informaion. If a complee yield curve of, say, quarerly compounded spo raes is given, we can compue he discoun funcion and spo raes and forward raes for any given period and wih any given compounding frequency. If a complee erm srucure of forward raes is known, we can compue discoun funcions and spo raes, ec. Academics frequenly apply coninuous compounding since he mahemaics involved in many relevan compuaions is more elegan when exponenials are used. There are even more ways of represening he erm srucure of ineres raes. Since mos bonds are bulle bonds, many raders and analyss are used o hinking in erms of yields of bulle bonds raher han in erms of discoun facors or zero-coupon raes. The par yield for a given mauriy is he coupon rae ha causes a bulle bond of he given mauriy o have a price equal o is face value. Again we have o fix he coupon period of he bond. U.S. reasury bonds ypically have semi-annual coupons which are herefore ofen used when compuing par yields. Given a discoun funcion T B T, he n-year par yield is he value of c ha solves he equaion 2n i=1 c B 2) +0.5i + B +n = 1. I reflecs he curren marke ineres rae for an n-year bulle bond. The par yield is closely relaed o he so-called swap rae, which is a key concep in he swap markes, cf. Secion 2.9. 1.5 Deermining he zero-coupon yield curve: Boosrapping In many bond markes only very few zero-coupon bonds are issued and raded. All bonds issued as coupon bonds will evenually become a zero-coupon bond afer heir nex-o-las paymen dae.) Usually, such zero-coupon bonds have a very shor mauriy. To obain knowledge of he marke zero-coupon yields for longer mauriies,we have o exrac informaion from he prices of raded coupon bonds. In some markes i is possible o consruc some longer-erm zero-coupon bonds by forming porfolios of raded coupon bonds. Marke prices of hese synheical zero-coupon bonds and he associaed zero-coupon yields can hen be derived. Example 1.2 Consider a marke where wo bulle bonds are raded, a 10% bond expiring in one year and a 5% bond expiring in wo years. Boh have annual paymens and a face value of 100. The one-year bond has he paymen srucure of a zero-coupon bond: 110 dollars in one year and nohing a all oher poins in ime. A share of 1/110 of his bond corresponds exacly o a

1.5 Deermining he zero-coupon yield curve: Boosrapping 11 zero-coupon bond paying one dollar in a year. If he price of he one-year bulle bond is 100, he one-year discoun facor is given by B +1 = 1 100 0.9091. 110 The wo-year bond provides paymens of 5 dollars in one year and 105 dollars in wo years. Hence, i can be seen as a porfolio of five one-year zero-coupon bonds and 105 wo-year zero-coupon bonds, all wih a face value of one dollar. The price of he wo-year bulle bond is herefore cf. 1.2). Isolaing B +2, we ge B 2 = 5B +1 + 105B +2, 1.18) B +2 = 1 105 B2 5 105 B+1. If for example he price of he wo-year bulle bond is 90, he wo-year discoun facor will be B +2 = 1 105 90 5 0.9091 0.8139. 105 From 1.18) we see ha we can consruc a wo-year zero-coupon bond as a porfolio of 1/105 unis of he wo-year bulle bond and 5/105 unis of he one-year zero-coupon bond. This is equivalen o a porfolio of 1/105 unis of he wo-year bulle bond and 5/105 110) unis of he one-year bulle bond. Given he discoun facors, zero-coupon raes and forward raes can be calculaed as shown in Secion 1.4.1 1.4.3. The example above can easily be generalized o more periods. Suppose we have M bonds wih mauriies of 1, 2,..., M periods, respecively, one paymen dae each period and idenical paymen dae. Then we can consruc successively zero-coupon bonds for each of hese mauriies and hence compue he marke discoun facors B +1, B +2,..., B +M. Firs, B +1 is compued using he shores bond. Then, B +2 is compued using he nex-o-shores bond and he already compued value of B +1, ec. Given he discoun facors B +1, B +2,..., B +M, we can compue he zero-coupon ineres raes and hence he zero-coupon yield curve up o ime + M for he M seleced mauriies). This approach is called boosrapping or yield curve sripping. Boosrapping also applies o he case where he mauriies of he M bonds are no all differen and regularly increasing as above. As long as he M bonds ogeher have a mos M differen paymen daes and each bond has a mos one paymen dae, where none of he bonds provide paymens, hen we can consruc zero-coupon bonds for each of hese paymen daes and compue he associaed discoun facors and raes. Le us denoe he paymen of bond i i = 1,..., M) a ime +j j = 1,..., M) by Y ij. Some of hese paymens may well be zero, e.g. if he bond maures before ime + M. Le B i denoe he price of bond i. From 1.2) we have ha he discoun facors, B +2,..., B +M B +1 1.19) mus saisfy he sysem of equaions B 1 Y 11 Y 12... Y 1M B 2 Y 21 Y 22... Y 2M. =...... Y M1 Y M2... Y MM B M B +1 B +2.. B +M.

1.5 Deermining he zero-coupon yield curve: Boosrapping 12 The condiions in he bonds ensure ha he paymen marix of his equaion sysem is non-singular so ha a unique soluion will exis. For each of he paymen daes + j, we can consruc a porfolio of he M bonds, which is equivalen o a zero-coupon bond wih a paymen of 1 a ime + j. Denoe by x i j) he number of unis of bond i which eners he porfolio replicaing he zero-coupon bond mauring a + j. Then we mus have ha 0 Y 11 Y 21......... Y M1 x 1 j) 0 Y 12 Y 22......... Y M2 x 2 j)........ 1.20) =, 1 Y 1j Y 2j......... Y Mj x j j).......... 0 Y 1M Y 2M......... Y MM x M j) where he 1 on he lef-hand side of he equaion is a he j h enry of he vecor. Of course, here will be he following relaion beween he soluion B +1,..., B +M ) o 1.19) and he soluion x 1 j),..., x M j)) o 1.20): 4 1.21) M i=1 x i j)b i = B +j. Thus, firs he zero-coupon bonds can be consruced, i.e. 1.20) is solved for each j = 1,..., M, and nex 1.21) can be applied o compue he discoun facors. Example 1.3 In Example 1.2 we considered a wo-year 5% bulle bond. Assume now ha a wo-year 8% serial bond wih he same paymen daes is raded. The paymens from his bond are 58 dollars in one year and 54 dollars in wo years. Assume ha he price of he serial bond is 98 dollars. From hese wo bonds we can se up he following equaion sysem o solve for he discoun facors B +1 and B +2 : ) ) ) 90 5 105 B +1 =. 98 58 54 B +2 The soluion is B +1 0.9330 and B +2 0.8127. More generally, if here are M raded bonds having in oal N differen paymen daes, he sysem 1.19) becomes one of M equaions in N unknowns. If M > N, he sysem may no have any soluion, since i may be impossible o find discoun facors consisen wih he prices of all M bonds. If no such soluion can be found, here will be an arbirage opporuniy. Example 1.4 In he Examples 1.2 and 1.3 we have considered hree bonds: a one-year bulle bond, a wo-year bulle bond, and a wo-year serial bond. In oal, hese hree bonds have wo 4 In marix noaion, B = YP and e j = Y xj), where e j is he vecor on he lef hand side of 1.20), and he oher symbols are self-explanaory he symbol indicaes ransposiion). Hence, which is equivalen o 1.21). xj) B = xj) YP = e j P = B+j,

1.5 Deermining he zero-coupon yield curve: Boosrapping 13 differen paymen daes. According o he prices and paymens of hese hree bonds, he discoun facors B +1 and B +2 mus saisfy he following hree equaions: 100 = 110B +1, 90 = 5B +1 + 105B +2, 98 = 58B +1 + 54B +2. No soluion exiss. In Example 1.2 we found ha he soluion o he firs wo equaions is B +1 0.9091 and B +2 0.8139. In conras, we found in Example 1.3 ha he soluion o he las wo equaions is B +1 0.9330 and B +2 0.8127. If he firs soluion is correc, he price on he serial bond should be 1.22) 58 0.9091 + 54 0.8139 96.68, bu i is no. The serial bond is mispriced relaive o he wo bulle bonds. More precisely, he serial bond is oo expensive. We can exploi his by selling he serial bond and buying a porfolio of he wo bulle bonds ha replicaes he serial bond, i.e. provides he same cash flow. We know ha he serial bond is equivalen o a porfolio of 58 one-year zero-coupon bonds and 54 wo-year zero-coupon bonds, all wih a face value of 1 dollar. In Example 1.2 we found ha he one-year zero-coupon bond is equivalen o 1/110 unis of he one-year bulle bond, and ha he wo-year zero-coupon bond is equivalen o a porfolio of 5/105 110) unis of he one-year bulle bond and 1/105 unis of he wo-year bulle bond. I follows ha he serial bond is equivalen o a porfolio consising of 58 unis of he one-year bulle bond and 1 110 54 5 105 110 0.5039 54 1 105 0.5143 unis of he wo-year bulle bond. This porfolio will give exacly he same cash flow as he serial bond, i.e. 58 dollars in one year and 54 dollars in wo years. The price of he porfolio is 0.5039 100 + 0.5143 90 96.68, which is exacly he price found in 1.22). In some markes, he governmen bonds are issued wih many differen paymen daes. The sysem 1.19) will hen ypically have fewer equaions han unknowns. In ha case here are many soluions o he equaion sysem, i.e. many ses of discoun facors can be consisen boh wih observed prices and he no-arbirage pricing principle.

1.6 Deermining he zero-coupon yield curve: Parameerized forms 14 1.6 Deermining he zero-coupon yield curve: Parameerized forms Boosrapping can only provide knowledge of he discoun facors for some of) he paymen daes of he raded bonds. In many siuaions informaion abou marke discoun facors for oher fuure daes will be valuable. We will nex consider mehods o esimae he enire discoun funcion T B T a leas up o some large T ). The basic assumpion is ha he discoun funcion is of a given funcional form wih some unknown parameers. The value of hese parameers are hen esimaed o obain he bes possible agreemen beween observed bond prices and heoreical bond prices compued using he funcional form. Typically, he assumed funcional forms are eiher polynomials or exponenial funcions of mauriy or some combinaion. This is consisen wih he usual percepion ha discoun funcions and yield curves are coninuous and smooh. If he yield for a given mauriy was much higher han he yield for anoher mauriy very close o he firs, mos bond owners would probably shif from bonds wih he low-yield mauriy o bonds wih he high-yield mauriy. Conversely, bond issuers borrowers) would shif o he low-yield mauriy. These changes in supply and demand will cause he gap beween he yields for he wo mauriies o shrink. Hence, he equilibrium yield curve should be coninuous and smooh. The unknown parameers can be esimaed by leas-squares mehods. I can be quie hard o fi a relaively simple funcional form o prices of a large number of bonds wih very differen mauriies. To enhance flexibiliy, some of he procedures suggesed in he lieraure divide he mauriy axis ino subinervals and use separae funcions of he same ype) in each subinerval. To ensure a coninuous and smooh erm srucure of ineres raes, one mus impose cerain condiions for he mauriies separaing he subinervals. Procedures of his ype are called spline mehods. In his secion we will focus on wo of he mos frequenly applied parameerizaion echniques, namely cubic splines and he Nelson-Siegel parameerizaion. An overview of some of he many oher approaches suggesed in he lieraure can be seen in Anderson, Breedon, Deacon, Derry, and Murphy 1996, Ch. 2). For some recen procedures, see Jaschke 1998) and Linon, Mammen, Nielsen, and Tanggaard 2001). The purpose of he procedures is o esimae he erm srucure of ineres rae a a given dae from he marke prices of bonds a ha dae. To simplify he noaion in wha follows, le P τ) denoe he discoun facor for he nex τ periods, i.e. P τ) = B +τ. Hence, he funcion P τ) for τ [0, ) represens he ime marke discoun funcion. In paricular, P 0) = 1. We will use a similar noaion for zero-coupon raes and forward raes: ȳτ) = y +τ and fτ) = f +τ. 1.6.1 Cubic splines In his subsecion we will discuss a version of he cubic splines approach inroduced by Mc- Culloch 1971) and laer modified by McCulloch 1975) and Lizenberger and Rolfo 1984). Given prices for M bonds wih ime-o-mauriies of T 1 T 2 T M. Divide he mauriy axis ino subinervals defined by he kno poins 0 = τ 0 < τ 1 < < τ k = T M. A spline approximaion of he discoun funcion P τ) is based on an expression like k 1 P τ) = G j τ)i j τ), j=0

1.6 Deermining he zero-coupon yield curve: Parameerized forms 15 where he G j s are basis funcions, and he I j s are he sep funcions 1, if τ τ j, I j τ) = 0, oherwise. We demand ha he G j s are coninuous and differeniable and ensure a smooh ransiion in he kno poins τ j. A polynomial spline is a spline where he basis funcions are polynomials. Le us consider a cubic spline, where G j τ) = α j + β j τ τ j ) + γ j τ τ j ) 2 + δ j τ τ j ) 3, and α j, β j, γ j, and δ j are consans. For τ [0, τ 1 ), we have 1.23) P τ) = α0 + β 0 τ + γ 0 τ 2 + δ 0 τ 3. Since P 0) = 1, we mus have α 0 = 1. For τ [τ 1, τ 2 ), we have 1.24) P τ) = 1 + β0 τ + γ 0 τ 2 + δ 0 τ 3) + α 1 + β 1 τ τ 1 ) + γ 1 τ τ 1 ) 2 + δ 1 τ τ 1 ) 3). To ge a smooh ransiion beween 1.23) and 1.24) in he poin τ = τ 1, we demand ha 1.25) 1.26) 1.27) P τ 1 ) = P τ 1 +), P τ 1 ) = P τ 1 +), P τ 1 ) = P τ 1 +), where P τ 1 ) = lim τ τ1,τ<τ 1 P τ), P τ1 +) = lim τ τ1,τ>τ 1 P τ), ec. The condiion 1.25) implies α 1 = 0. Differeniaing 1.23) and 1.24), we find P τ) = β 0 + 2γ 0 τ + 3δ 0 τ 2, 0 τ < τ 1, and P τ) = β 0 + 2γ 0 τ + 3δ 0 τ 2 + β 1 + 2γ 1 τ τ 1 ) + 3δ 1 τ τ 1 ) 2, τ 1 τ < τ 2. The condiion 1.26) now implies β 1 = 0. Differeniaing again, we ge P τ) = 2γ 0 + 6δ 0 τ, 0 τ < τ 1, and P τ) = 2γ 0 + 6δ 0 τ + 2γ 1 + 6δ 1 τ τ 1 ), τ 1 τ < τ 2. Consequenly, he condiion 1.27) implies γ 1 = 0. Similarly, i can be shown ha α j = β j = γ j = 0 for all j = 1,..., k 1. The cubic spline is herefore reduced o k 1 1.28) P τ) = 1 + β0 τ + γ 0 τ 2 + δ 0 τ 3 + δ j τ τ j ) 3 I j τ). Le 1, 2,..., N denoe he ime disance from oday dae ) o he each of he paymen daes in he se of all paymen daes of he bonds in he daa se. Le Y in denoe he paymen of bond i in n periods. From he no-arbirage pricing relaion 1.2), we should have ha B i = j=1 N Y in P n ), n=1

1.6 Deermining he zero-coupon yield curve: Parameerized forms 16 where B i is he curren marke price of bond i. Since no all he zero-coupon bonds involved in his equaion are raded, we will allow for a deviaion ε i so ha 1.29) B i = N Y in P n ) + ε i. n=1 We assume ha ε i is normally disribued wih mean zero and variance σ 2 assumed o be he same for all bonds) and ha he deviaions for differen bonds are muually independen. We wan o pick parameer values ha minimize he sum of squared deviaions M i=1 ε2 i. Subsiuing 1.28) ino 1.29) yields B i N Y in = β 0 n=1 N n=1 Y in n + γ 0 N n=1 Y in 2 n + δ 0 N n=1 k 1 Y in 3 n + N δ j j=1 n=1 Y in n τ j ) 3 I j n ) + ε i. Given he prices and paymen schemes of he M bonds, he k +2 parameers β 0, γ 0, δ 0, δ 1,..., δ k 1 can now be esimaed using ordinary leas squares. 5 Subsiuing he esimaed parameers ino 1.28), we ge an esimaed discoun funcion, from which esimaed zero-coupon yield curves and forward rae curves can be derived as explained earlier in he chaper. I remains o describe how he number of subinervals k and he kno poins τ j are o be chosen. McCulloch suggesed o le k be he neares ineger o M and o define he kno poins by τ j = T hj + θ j T hj+1 T hj ), where h j = [j M/k] here he square brackes mean he ineger par) and θ j = j M/k h j. In paricular, τ k = T M. Alernaively, he kno poins can be placed a for example 1 year, 5 years, and 10 years, so ha he inervals broadly correspond o he shor-erm, inermediae-erm, and long-erm segmens of he marke, cf. he preferred habias hypohesis. Figure 1.1 shows he discoun funcion on he Danish governmen bond markes on February 14, 2000 esimaed using cubic splines and daa from 14 bonds wih mauriies up o 25 years. Figure 1.2 shows he associaed zero-coupon yield curve and he erm srucure of forward raes. Discoun funcions esimaed using cubic splines will usually have a credible form for mauriies less han he longes mauriy in he daa se. Alhough here is nohing in he approach ha ensures ha he resuling discoun funcion is posiive and decreasing, as i should be according o 1.1), his will almos always be he case. As he mauriy approaches infiniy, he cubic spline discoun funcion will approach eiher plus or minus infiniy depending on he sign of he coefficien of he hird order erm. Of course, boh properies are unaccepable, and he mehod canno be expeced o provide reasonable values beyond he longes mauriy T M, since none of he bonds are affeced by ha very long end of he erm srucure. Two oher properies of he cubic splines approach are more disurbing. Firs, he derived zero-coupon raes will ofen increase or decrease significanly for mauriies approaching T M, cf. Shea 1984, 1985). Second, he derived forward rae curve will ypically be quie rugged especially near he kno poins, and he curve ends o be very sensiive o he bond prices and he precise locaion of he kno poins. Therefore, forward rae curves esimaed using cubic splines should only be applied wih grea cauion. 5 See, for example, Johnson 1984).

1.6 Deermining he zero-coupon yield curve: Parameerized forms 17 6.5% 6.0% ineres raes 5.5% 5.0% zero-cpn raes forward raes 4.5% 4.0% 0 5 10 15 20 25 mauriy, years Figure 1.1: The discoun funcion, τ P τ), esimaed using cubic splines and prices of Danish governmen bonds February 14, 2000. 1 0.9 0.8 discoun facors 0.7 0.6 0.5 0.4 0.3 0.2 0 5 10 15 20 25 mauriy, years Figure 1.2: The zero-coupon yield curve, τ ȳτ), and he erm srucure of forward raes, τ fτ), esimaed using cubic splines and prices of Danish governmen bonds February 14, 2000.

1.6 Deermining he zero-coupon yield curve: Parameerized forms 18 1.2 1 0.8 0.6 0.4 long medium shor 0.2 0 0 2 4 6 8 10 scaled mauriy Figure 1.3: The hree curves which he Nelson-Siegel parameerizaion combines. 1.6.2 The Nelson-Siegel parameerizaion Nelson and Siegel 1987) proposed a simple parameerizaion of he erm srucure of ineres raes, which has become quie popular. The approach is based on he following parameerizaion of he forward raes: 1.30) fτ) = β0 + β 1 e τ/θ + β 2 τ θ e τ/θ, where β 0, β 1, β 2, and θ are consans o be esimaed. The same consans are assumed o apply for all mauriies, so no splines are involved. The simple funcional form ensures a smooh and ye quie flexible curve. Figure 1.3 shows he graphs of he hree funcions ha consiues 1.30). The fla curve corresponding o he consan erm β 0 ) will by iself deermine he long-erm forward raes, he erm β 1 e τ/θ is mosly affecing he shor-erm forward raes, while he erm β 2 τ/θe τ/θ is imporan for medium-erm forward raes. The value of he parameer θ deermines how large a mauriy inerval he non-consan erms will affec. The value of he parameers β 0, β 1, and β 2 deermine he relaive weighing of he hree curves. According o 1.17) on page 10, he erm srucure of zero-coupon raes is given by ȳτ) = β 0 + β 1 + β 2 ) 1 e τ/θ τ/θ β 2 e τ/θ, which we will rewrie as 1.31) ȳτ) = a + b 1 e τ/θ τ/θ + ce τ/θ. Figure 1.4 depics he possible forms of he zero-coupon yield curve for differen values of a, b, and c. By varying he parameer θ, he curves can be sreched or compressed in he horizonal dimension.

1.6 Deermining he zero-coupon yield curve: Parameerized forms 19 zero-coupon raes 0 2 4 6 8 10 scaled mauriy Figure 1.4: Possible forms of he zero-coupon yield curve using he Nelson-Siegel parameerizaion. If we could direcly observe zero-coupon raes ȳt i ) for differen mauriies T i, i = 1,..., M, we could, given θ, esimae he parameers a, b, and c using simple linear regression on he model ȳτ) = a + b 1 e τ/θ τ/θ + ce τ/θ + ε i, where ε i N0, σ 2 ), i = 1,..., M, are independen error erms. Doing his for various choices of θ, we could pick he θ and he corresponding regression esimaes of a, b and c ha resul in he highes R 2, i.e. ha bes explain he daa. This is exacly he procedure used by Nelson and Siegel on daa on shor-erm zero-coupon bonds in he U.S. marke. When he daa se involves coupon bonds, he esimaion procedure is slighly more complicaed. The discoun funcion associaed wih he forward rae srucure in 1.31) is given by { P τ) = exp aτ bθ 1 e τ/θ) cτe τ/θ}. Subsiuing his ino 1.29), we ge 1.32) B i = N n=1 { Y in exp a n bθ 1 e n/θ) c n e n/θ} + ε i. Since his is a non-linear expression in he unknown parameers, he esimaion mus be based on generalized leas squares, i.e. non-linear regression echniques. See e.g. Gallan 1987). 1.6.3 Addiional remarks on yield curve esimaion Above we looked a wo of he many esimaion procedures based on a given parameerized form of eiher he discoun funcion, he zero-coupon yield curve, or he forward rae curve. A clear

1.7 Exercises 20 disadvanage of boh mehods is ha he esimaed discoun funcion is no necessarily consisen wih hose probably few) discoun facors ha can be derived from marke prices assuming only no-arbirage. The procedures do no punish deviaions from no-arbirage values. A more essenial disadvanage of all such esimaion procedures is ha hey only consider he erm srucure of ineres raes a one paricular poin in ime. Esimaions a wo differen daes are compleely independen and do no ake ino accoun he possible dynamics of he erm srucure over ime. As we shall see in Chapers 7 and 8, here are many dynamic erm srucure models which also provide a parameerized form for he erm srucure a any given dae. Applying such models, he esimaion can and should) be based on bond price observaions a differen daes. Typically, he possible forms of he erm srucure in such models resemble hose of he Nelson-Siegel approach. We will reurn o his discussion in Chaper 7. Finally, we will emphasize ha he esimaed erm srucure of ineres raes should be used wih cauion. An obvious use of he esimaed yield curve is o value fixed-income securiies. In paricular, he coupon bonds in he daa se used in he esimaion can be priced using he esimaed discoun funcion. For some of he bonds he price according o he esimaed curve will be lower higher) han he marke price. Therefore, one migh hink such bonds are overvalued undervalued) by he marke. In an esimaion like 1.29) his can be seen direcly from he residual ε i.) I would seem a good sraegy o sell he overvalued and buy he undervalued bonds. However, such a sraegy is no a riskless arbirage, bu a risky sraegy, since he applied discoun funcion is no derived from he no-arbirage principle only, bu depends on he assumed parameric form and he oher bonds in he daa se. Wih anoher parameerized form or a differen se of bonds he esimaed discoun funcion and, hence, he assessmen of over- and undervaluaion can be differen. 1.7 Exercises EXERCISE 1.1 Find a lis of curren price quoes on governmen bonds a an exchange in your counry. Derive as many discoun facors and zero-coupon yields as possible using only he no-arbirage pricing principle. EXERCISE 1.2 Consider wo annuiy bonds wih idenical ime-o-mauriy and paymen daes. Le n be he number of remaining paymens on each bond, and le R 1 and R 2 be he coupon raes of he wo bonds. In addiion, le B 1 and B 2 denoe he prices quoed price plus accrued ineress) of he bonds. Show ha in he absence of arbirage, he following relaion mus hold: B 1 B 2 = R1 R 2 1 1 + R 2) n 1 1 + R 1) n. In some markes callable morgage-backed annuiy bonds wih differen coupon raes, bu he same mauriy and paymen daes, are raded. Should you expec he relaion above o hold for such bonds? Explain! EXERCISE 1.3 Show ha if he discoun funcion does no saisfy he condiion B T B S, T < S, hen negaive forward raes will exis. Are non-negaive forward raes likely o exis? Explain! EXERCISE 1.4 Consider wo bulle bonds, boh wih annual paymens and exacly four years o ma-

1.7 Exercises 21 uriy. The firs bond has a coupon rae of 6% and is raded a a price of 101.00. The oher bond has a coupon rae of 4% and is raded a a price of 93.20. Wha is he four-year discoun facor? Wha is he four-year zero-coupon ineres rae?

Chaper 2 Fixed income securiies 2.1 Inroducion This chaper provides a descripion of he mos imporan financial securiies whose cash flows and values only depend on he erm srucure of ineres raes. In Chaper 1 we already considered bonds and he relaion beween bond prices and he erm srucure of ineres raes. In his chaper we will describe and discuss floaing-rae bonds, forwards, fuures and opions wrien on bonds, ineres rae fuures, caps, floors, ineres rae swaps and swapions. We will also derive numerous relaions beween he prices of hese securiies ha will hold in he absence of arbirage and hence independenly of he preferences of invesors and he fuure dynamics of he erm srucure. For example we will show a pu-call pariy for opions on bonds, boh zero-coupon bonds and coupon bonds, ha prices of caps and floors follow from prices of porfolios on cerain European opions on zero-coupon bonds ha prices of European swapions follow from prices of cerain European opions on coupon bonds. Consequenly, we can price many frequenly raded securiies as long as we price bonds and European opions on bonds. In laer chapers we can herefore focus on he pricing of hese basic securiies. 2.2 Floaing rae bonds Floaing rae bonds have coupon raes ha are rese periodically over he life of he bond. We will consider he mos common floaing rae bond, which is a bulle bond, where he coupon rae effecive for he paymen a he end of one period is se a he beginning of he period a he curren marke ineres rae for ha period. Suppose ha he paymen daes of he bond are T 1 < < T n, where T i T i 1 = δ for all i. In pracice, δ will ypically equal 0.25, 0.5, or 1 year, corresponding o quarerly, semi-annual, or annual paymens. The annualized coupon rae valid for he period [T i 1, T i ] is he δ-period marke rae a dae T i 1 compued wih a compounding frequency of δ. We will denoe his ineres rae by l Ti T i 1, alhough he rae is no necessarily a LIBOR rae, bu can also be a Treasury rae. If he 22

2.3 Forwards on bonds 23 face value of he bond is H, he paymen a ime T i i = 1, 2,..., n 1) equals Hδl Ti T i 1, while he final paymen a ime T n equals H1 + δl Ti T i 1 ). If we define T 0 = T 1 δ, he daes T 0, T 1,..., T n 1 are ofen referred o as he rese daes of he bond. Le us look a he valuaion of a floaing rae bond. We will argue ha immediaely afer each rese dae, he value of he bond will equal is face value. To see his, firs noe ha immediaely afer he las rese dae T n 1, he bond is equivalen o a zero-coupon bond wih a coupon rae equal o he marke ineres rae for he las coupon period. By definiion of ha marke ineres rae, he ime T n 1 value of he bond will be exacly equal o he face value H. In mahemaical erms, he marke discoun facor o apply for he discouning of ime T n paymens back o ime T n 1 is 1 + δl Tn T n 1 ) 1, so he ime T n 1 value of a paymen of H1 + δl Tn T n 1 ) a ime T n is precisely H. Immediaely afer he nex-o-las rese dae T n 2, we know ha we will receive a paymen of Hδl Tn 1 T n 2 a ime T n 1 and ha he ime T n 1 value of he following paymen received a T n ) equals H. We herefore have o discoun he sum Hδl Tn 1 T n 2 + H = H1 + δl Tn 1 T n 2 ) from T n 1 back o T n 2. The discouned value is exacly H. Coninuing his procedure, we ge ha immediaely afer a rese of he coupon rae, he floaing rae bond is valued a par. We can also derive he value of he floaing rae bond beween wo paymen daes. Suppose we are ineresed in he value a some ime beween T 0 and T n. Inroduce he noaion i) = min {i {1, 2,..., n} : T i > }, so ha T i) is he neares following paymen dae afer ime. We know ha he following paymen a ime T i) equals Hδl T i) T i) 1 and ha he value a ime T i) of all he remaining paymens will equal H. The value of he bond a ime will hen be 2.1) B fl = H1 + δl T i) T i) 1 )B T i), T 0 < T n. This expression also holds a paymen daes = T i, where i resuls in H, which is he value excluding he paymen a ha dae. While few floaing rae bonds are raded, he resuls above are also very useful for he analysis of ineres rae swaps sudied in Secion 2.9. 2.3 Forwards on bonds A forward conrac is an agreemen beween wo paries on a rade of a given asse or securiy a a given fuure poin in ime and a a price ha is already fixed when he agreemen is made. We will refer o his price as he delivery price. Usually, he delivery price is se so ha he ne presen value of he fuure ransacion compued a he agreemen dae) is equal o zero. This value of he delivery price is called he forward price. Noe ha he forward price will depend on he ime of delivery and he underlying asse o be ransaced. In he following we will ake a closer look a forwards on zero-coupon bonds and coupon bonds. For a general inroducion o forwards, see Hull 2003). Consider an agreemen on a delivery a ime T of a zero-coupon bond mauring a ime S where S > T ) for a price of K. We assume ha he face value of zero-coupon bonds is 1 dollar). The following heorem gives formulas for he ime value V T,S of a long posiion in such a forward conrac and for he forward price of he zero-coupon bond. Here and hroughou he ex, B T denoes he ime price of a zero-coupon bond ha pays 1 dollar) a ime T.

2.3 Forwards on bonds 24 Theorem 2.1 The unique no-arbirage value a ime of a forward wih delivery a ime T of a zero-coupon bond mauring a ime S a he delivery price K is given by 2.2) V T,S = B S KB T. The unique no-arbirage forward price on he zero-coupon bond is 2.3) F T,S Proof: Consider wo porfolios, namely = BS B T A) a long posiion in he forward conrac and K zero-coupon bonds mauring a ime T, B) a zero-coupon bond mauring a ime S i.e. he underlying bond). A ime T he forward yields a payoff of BT S K, so ha he oal ime T value of porfolio A is BT S K + K = BS T, which is idenical o he value of porfolio B. Since none of he porfolios yield paymens before ime T, he no-arbirage principle implies ha hey mus have he same curren price, i.e. V T,S + KB T = B S, from which 2.2) follows. Since he forward price F T,S equaion 2.3) follows direcly from 2.2).. is ha value of K ha makes V T,S = 0, A he delivery ime T he gain or loss from he forward posiion will be known. The gain from a long posiion in a forward wrien on a zero-coupon wih face value H and mauriy a S is equal o HBT S F T,S ). If we wrie he spo bond price BT S in erms of he spo LIBOR rae ls T forward bond price F T,S ha he gain is equal o 2.4) and he in erms of he forward LIBOR rae L T,S, i follows from 1.8) and 1.9) ) H HBT S F T,S 1 = H 1 + S T )lt S 1 1 + S T )L T,S S T )L T,S lt S = ) )H ). 1 + S T )l S T 1 + S T )L T,S An invesor wih a long posiion in he forward will realize a gain if he spo bond price a delivery urns ou o be above he forward price, i.e. if he spo ineres rae a delivery urns ou o be below he forward ineres rae when he forward posiion was aken. We can hink of a shor posiion in a forward on a zero-coupon bond as a way o lock in he borrowing rae for he period beween he delivery dae of he forward and he mauriy dae of he bond. Nex, le us consider a forward on a coupon bond. As before, le T be he delivery dae and K be he delivery price. The underlying coupon bond is assumed o yield paymens a he poins in ime T 1 < T 2 < < T n, where T < T n. The ime T i paymen is denoed by Y i, i = 1, 2,..., n. The ime value of he bond is herefore given by B = T i> Y i B Ti, ) where he sum is over all fuure paymen daes; cf. 1.2) on page 6. Le V T,kup value of his forward conrac. We hen have he following resul: denoe he ime

2.4 Ineres rae forwards forward rae agreemens 25 Theorem 2.2 The unique no-arbirage value of a forward on a coupon bond is given by 2.5) V T,cpn = T i>t Y i B Ti KB T = B <T i<t Y i B Ti KB T, and he unique no-arbirage forward price of a coupon bond is 2.6) F T,cpn T = Y i>t ib Ti B T = B <T Y i<t ib Ti B T = T i>t Y i F T,Ti. Proof: Consider he porfolios A) a forward conrac, K zero-coupon bonds mauring a T and, for each T i wih < T i < T, Y i zero-coupon bonds mauring a T i, B) he underlying coupon bond. Beween ime and he delivery dae T, he wo porfolios have exacly he same paymens. A ime T he value of porfolio A equals B T K + K = B T, which is idenical o he value of porfolio B. Absence of arbirage implies ha V T,cpn + KB T + <T i<t Y i B Ti = B, from which 2.5) follows. The forward price follows immediaely. 2.4 Ineres rae forwards forward rae agreemens As discussed in Secion 1.4 forward ineres raes are raes for a fuure period relaive o he ime where he rae is se. Many paricipans in he financial markes may on occasion be ineresed in locking in an ineres rae for a fuure period, eiher in order o hedge risk involved wih varying ineres raes or o speculae in specific changes in ineres raes. In he money markes he agens can lock in an ineres rae by enering a so-called forward rae agreemen FRA). Suppose he relevan fuure period is he ime inerval beween T and S, where S > T. In principle, a forward rae agreemen wih a face value H and a conrac rae of K involves wo paymens: a paymen of H a ime T and a paymen of H[1 + S T )K] a ime S. Of course, he paymens o he oher par of he agreemen are H a ime T and H[1 + S T )K] a ime S.) In pracice, he conrac is ypically seled a ime T, so ha he wo paymens are replaced by a single paymen of BT S H[1 + S T )K] H a ime T. Usually he conrac rae K is se so ha he presen value of he fuure paymens) is zero a he ime he conrac is made. Suppose he conrac is made a ime < T. Then he ime value of he wo fuure paymens of he conrac is equal o HB T if and only if K = 1 B T S T B S ) 1 = L T,S, + H[1 + S T )K]B S. This is zero cf. 1.9), i.e. when he conrac rae equals he forward rae prevailing a ime for he period beween T and S. For his conrac rae, we can hink of he forward rae agreemen having a

2.5 Fuures on bonds 26 single paymen a ime T, which is given by ) 2.7) BT S 1 + S T )L T,S S T )LT,S lt S H[1 + S T )K] H = H 1 + lt S 1 = )H 1 + S T )lt S. The numeraor is exacly he ineres los by lending ou H from ime T o ime S a he forward rae given by he FRA raher han he realized spo rae. Of course, his amoun may be negaive, so ha a gain is realized. The division by 1 + S T )lt S corresponds o discouning he gain/loss from ime S back o ime T. The ime T value saed in 2.7) is closely relaed, bu no idenical, o he gain/loss on a forward on a zero-coupon bond, cf. 2.4). 2.5 Fuures on bonds As a forward conrac, a fuures conrac is also an agreemen of a fuure ransacion of a given asse or securiy. The disinc characerisic of a fuure is ha changes in is value are seled coninuously hroughou he life of he conrac usually once every rading day). This so-called marking-o-marke ensures ha he value of he conrac i.e. he value of he fuure paymens) reurns o zero immediaely afer each selemen. This procedure makes i pracically possible o rade fuures a organized exchanges, since here is no need o keep rack of when he fuures posiion was originally aken. Fuures on governmen bonds are raded a many leading exchanges. The marking-o-marke a a given dae involves he paymen of he change in he so-called fuures price of he conrac relaive o he previous selemen dae. A mauriy of he conrac he fuures gives a payoff equal o he difference beween he price of he underlying asse a ha dae and he fuures price a he previous selemen dae. Afer he las selemen before mauriy, he fuures is herefore indisinguishable from he corresponding forward conrac, so he values of he fuures and he forward a ha selemen dae mus be idenical. A he nex-olas selemen dae before mauriy, he fuures price is se o ha value ha ensures ha he ne presen value of he upcoming selemen a he las selemen dae before mauriy which depends on his fuures price) and he final payoff is equal o zero. Similarly a earlier selemen daes. Due o he marking-o-marke selemen procedure, i is far more difficul o price fuures han forwards. In paricular, he no-arbirage principle is no sufficien o derive a unique fuures price. As shown by Cox, Ingersoll, and Ross 1981b), he fuures price and he forward price will be idenical a all poins in ime if here is no uncerainy abou fuure ineres raes, since in ha case he iming of he payoffs does no maer. 1 Such an assumpion is of course unaccepable in he case of fuures on asses ha depend on he erm srucure of ineres raes, such as fuures on bonds. We will reurn o he valuaion of fuures and he relaion beween forward prices and fuures prices in Chaper 5. 2.6 Ineres rae fuures Eurodollar fuures Ineres rae fuures is a class of fixed income insrumens ha rade wih a very high volume a several inernaional exchanges, e.g. CME Chicago Mercanile Exchange), LIFFE London Inernaional Financial Fuures & Opions Exchange), and MATIF Marché à Terme Inernaional 1 A proof of his resul can also be found in Appendix 3A in Hull 2003).

2.6 Ineres rae fuures Eurodollar fuures 27 de France). The CME ineres rae fuures involve he hree-monh Eurodollar deposi rae and are called Eurodollar fuures. The ineres rae involved in he fuures conracs raded a LIFFE and MATIF is he hree-monh LIBOR rae on he Euro currency. We shall simply refer o all hese conracs as Eurodollar fuures and refer o he underlying ineres rae as he hree-monh LIBOR rae, whose value a ime we denoe by l +0.25. The price quoaion of Eurodollar fuures is a bi complicaed, since he amouns paid in he marking-o-marke selemens are no exacly he changes in he quoed fuures price. We mus herefore disinguish beween he quoed fuures price, Ẽ T, and he acual fuures price, E T, wih he selemens being equal o changes in he acual fuures price. A he mauriy dae of he conrac, T, he quoed Eurodollar fuures price is defined in erms of he prevailing hree-monh LIBOR rae according o he relaion 2.8) ẼT T = 100 1 l T +0.25 ) T, which using 1.8) on page 8 can be rewrien as )) ẼT T 1 = 100 1 4 B T +0.25 1 = 500 400 T 1 B T +0.25 T Traders and analyss ypically ransform he Eurodollar fuures price o an ineres rae, he socalled LIBOR fuures rae, which is defined by. ϕ T = 1 Ẽ T 100 Ẽ T = 100 1 ϕ T ). I follows from 2.8) ha he LIBOR fuures rae converges o he hree-monh LIBOR spo rae, as he mauriy of he fuures conrac approaches. The acual Eurodollar fuures price is given by E T = 100 0.25100 Ẽ T ) = 100 25ϕ T per 100 dollars of nominal value. I is he change in he acual fuures price which is exchanged in he marking-o-marke selemens. A he CME he nominal value of he Eurodollar fuures is 1 million dollars. A quoed fuures price of Ẽ T = 94.47 corresponds o a LIBOR fuures rae of 5.53% and an acual fuures price of 1 000 000 100 [100 25 0.0553] = 986 175. If he quoed fuures price increases o 94.48 he nex day, corresponding o a drop in he LIBOR fuures rae of one basis poin 0.01 percenage poins), he acual fuures price becomes 1 000 000 100 [100 25 0.0552] = 986 200. An invesor wih a long posiion will herefore receive 986 200 986 175 = 25 dollars a he selemen a he end of ha day. If we simply sum up he individual selemens wihou discouning hem o he erminal dae, he oal gain on a long posiion in a Eurodollar fuures conrac from o expiraion a T is given by ET T E T = 100 25ϕ T ) ) T 100 25ϕ T = 25 ϕ T T ϕ T )

2.7 Opions on bonds 28 per 100 dollars of nominal value, i.e. he oal gain on a conrac wih nominal value H is equal o 0.25 ϕ T T ) ϕt H. The gain will be posiive if he hree-monh spo rae a expiraion urns ou o be below he fuures rae when he posiion was aken. Conversely for a shor posiion. The gain/loss on a Eurodollar fuures conrac is closely relaed o he gain/loss on a forward rae agreemen, as can be seen from subsiuing S = T + 0.25 ino 2.7). Recall ha he raes ϕ T T and l T +0.25 T are idenical. However, i should be emphasized ha in general he fuures rae ϕ T and T,T +0.25 he forward rae L will be differen due o he marking-o-marke of he fuures conrac. 2.7 Opions on bonds Opions on governmen bonds are raded a several exchanges and also on he OTC-markes OTC: Over-he-couner). In addiion, many bonds are issued wih embedded opions. For example, many morgage-backed bonds and corporae bonds are callable, in he sense ha he issuer has he righ o buy back he bond a a pre-specified price. We will firs consider opions on zero-coupon bonds alhough, apparenly, no such opions are raded a any exchange. However, we shall see laer ha oher, frequenly raded, fixed income securiies can be considered as porfolios of European opions on zero-coupon bonds. This is rue for caps and floors, which we urn o in Secion 2.8. We will also show laer ha, under cerain assumpions on he dynamics of ineres raes, any European opion on a coupon bond is equivalen o a porfolio of cerain European opions on zero-coupon bonds; see Chaper 7. For hese reasons, i is imporan o be able o price European opions on zero-coupon bonds. I is well-known ha he no-arbirage principle in iself does no yield a unique price for sock opions, bu only upper and lower boundaries for he price, cf. Meron 1973) or Hull 2003). This is also he case for opions on bonds. The bounds ha can be obained for bond opions are no jus a simple reformulaion of he bounds available for sock opions due o he close relaion beween he appropriae discoun facor and he price of he underlying asse, he exisence of an upper bound on he price of he underlying bond: under he reasonable assumpion ha all forward raes are non-negaive, he price of a bond will be less han or equal o he sum of is remaining paymens. Alhough he obainable bounds for bond opions are igher han hose for sock opions, hey sill leave quie a large inerval in which he price can lie. For proofs and examples see Munk 2002) and Exercise 2.1. Jus as for sock opions, he absence of arbirage is sufficien o derive a precise relaion beween prices on European call and pu opions wih he same underlying asse, exercise price, and mauriy dae. This relaion, he so-called pu-call pariy, is saed below boh for opions on zero-coupon bonds and opions on coupon bonds. For American opions on bonds, i is also possible o find no-arbirage price bounds, and, as a counerpar o he pu-call pariy, relaively igh bounds on he difference beween he prices of an American call and an American pu. Again he reader is referred o Munk 2002).

2.7 Opions on bonds 29 2.7.1 Opions on zero-coupon bonds Le us firs fix some noaion. The ime of mauriy of he opion is denoed by T. The underlying zero-coupon bond gives a paymen of 1 dollar) a ime S, where S T. The exercise price of he opion is denoed by K. A European call opion on his zero-coupon bond gives he owner he righ, bu no he obligaion, o buy he zero-coupon bond a ime T for a price of K. We le C K,T,S denoe he ime price of he call opion. A mauriy he value of he call equals is payoff: C K,T,S T = max B S T K, 0 ). A European pu opion on he zero-coupon bond gives he owner he righ, bu no he obligaion, o sell he zero-coupon bond a ime T for a price of K. We le π K,T,S denoe he ime price of he pu opion. The value a mauriy is equal o π K,T,S T = max K B S T, 0 ). If he owner of he opion makes use of his righ o buy/sell he underlying asse, he opion is said o be exercised. Noe ha only opions wih an exercise price beween 0 and 1 are ineresing, since he price of he underlying zero-coupon bond a expiry of he opion will be in his inerval, assuming non-negaive ineres raes. The pu-call pariy gives a precise relaion beween he prices of European call and pu opions wih he same underlying asse, exercise price and mauriy dae. bonds he relaion is as follows: For opions on zero-coupon Theorem 2.3 In absence of arbirage, he prices of European call and pu opions on zero-coupon bonds saisfy he relaion 2.9) C K,T,S + KB T = π K,T,S + B S. Proof: A porfolio consising of a call opion and K zero-coupon bonds mauring a he same ime as he opion yields a payoff a ime T of max B S T K, 0 ) + K = max B S T, K ) and will have a curren ime price given by he lef-hand side of 2.9). Anoher porfolio consising of a pu opion and one uni of he underlying zero-coupon bond has a ime T value of max K BT S, 0 ) + BT S = max K, BT S ) and a ime price corresponding o he righ-hand side of 2.9). None of he porfolios provide paymens before ime T. Therefore, here will be an obvious arbirage opporuniy unless 2.9) is saisfied. A consequence of he pu-call pariy is ha we can focus on he pricing of European call opions. The prices of European pu opions will hen follow immediaely. American opions can be exercised a he ime of mauriy T or any poin in ime before ime T. I is well-known ha i is never sricly advanageous o exercise an American call opion on a

2.7 Opions on bonds 30 non-dividend paying sock before ime T ; cf. Meron 1973) and Hull 2003). By analogy, his is also rue for American call opions on zero-coupon bonds. A firs glance, i may appear opimal o exercise an American call on a zero-coupon bond immediaely in case he price of he underlying bond is equal o 1, because his will imply a payoff of 1 K, which is he maximum possible payoff under he assumpion of non-negaive ineres raes. However, he price of he underlying bond will only equal 1, if ineres raes are zero and say a zero for sure. Therefore, exercising he opion a ime T will also provide a payoff of 1 K, and since ineres raes are zero, he presen value of he payoff is also equal o 1 K. Hence, here is no sric advanage o early exercise. As for sock opions, premaure exercise of an American pu opion on a zero-coupon bond will be advanageous for sufficienly low prices of he underlying zero-coupon bond, i.e. sufficienly high ineres raes. 2.7.2 Opions on coupon bonds Consider a coupon bond wih paymens Y i a ime T i i = 1, 2,..., n), where T 1 < T 2 < < T n. Le B denoe he ime price of his bond, i.e. B = T i> Y i B Ti. Le C K,T,cpn and π K,T,cpn denoe he ime prices of a European call and a European pu, respecively, expiring a ime T, having an exercise price of K and he coupon bond above as he underlying asse. Of course, we mus have ha T < T n. The ime T value of he opions is given by heir payoffs: C K,T,cpn T π K,T,cpn T = max B T K, 0) = max = max K B T, 0) = max T i>t K Y i B Ti T K, 0 T i>t Y i B Ti T, 0 Such opions are only ineresing, if he exercise price is posiive and less han T i>t Y i, which is he upper bound for B T wih non-negaive forward raes. Noe ha 1) only he paymens of he bonds afer mauriy of he opion are relevan for he payoff and he value of he opion; 2 2) we have assumed ha he payoff of he opion is deermined by he difference beween he exercise price and he rue bond price raher han he quoed bond price. The rue bond price is he sum of he quoed bond price and accrued ineres. 3 Some aspecs of opions on he quoed bond price are discussed by Munk 2002). The pu-call pariy for European opions on coupon bonds is as follows: ) ),. 2 In paricular, we assume ha in he case where he expiry dae of he opion coincides wih a paymen dae of he underlying bond, i is he bond price excluding ha paymen which deermines he payoff of he opion. 3 The quoed price is someimes referred o as he clean price. Similarly, he rue price is someimes called he diry price.

2.8 Caps, floors, and collars 31 Theorem 2.4 Absence of arbirage implies ha 2.10) C K,T,cpn + KB T = π K,T,cpn + B The proof of his resul is lef for he reader in Exercise 2.2. <T i T Y i B Ti. When and under wha circumsances should one consider exercising an American call on a coupon bond? This is equivalen o he quesion of exercising an American call on a dividendpaying sock, which is discussed e.g. in Hull 2003, Chap. 12). herefore be saed. The following conclusions can The only poins in ime when i can be opimal o exercise an American call on a bond is jus before he paymen daes of he bond. Le T l be he las paymen dae before expiraion of he opion. Then i canno be opimal o exercise he call jus before T l if he paymen Y l is less han K1 B T T l ). If he opposie relaion holds, i may be opimal o exercise jus before T l. Similarly, a any earlier paymen dae T i [, T l ], exercise is ruled ou if he paymen a ha dae Y i is less han K1 B Ti+1 T i ). Broadly speaking, early exercise of he call will only be relevan if he shor-erm ineres rae is relaively low and he bond paymen is relaively high. 4 Regarding early exercise of pu opions, i can never be opimal o exercise an American pu on a bond jus before a paymen on he bond. A all oher poins in ime early exercise may be opimal for sufficienly low bond prices, i.e. high ineres raes. 2.8 Caps, floors, and collars 2.8.1 Caps An ineres rae) cap is designed o proec an invesor who has borrowed funds on a floaing ineres rae basis agains he risk of paying very high ineres raes. Suppose he loan has a face value of H and paymen daes T 1 < T 2 < < T n, where T i+1 T i = δ for all i. 5 The ineres rae o be paid a ime T i is deermined by he δ-period money marke ineres rae prevailing a ime T i δ, i.e. he paymen a ime T i is equal o Hδl Ti T i δ. Noe ha he ineres rae is se a he beginning of he period, bu paid a he end. Define T 0 = T 1 δ. The daes T 0, T 1,..., T n 1 where he rae for he coming period is deermined are called he rese daes of he loan. A cap wih a face value of H, paymen daes T i i = 1,..., n) as above, and a so-called cap rae K yields a ime T i payoff of Hδ maxl Ti T i δ K, 0), for i = 1, 2,..., n. If a borrower buys such a cap, he oal paymen a ime T i canno exceed HδK. The period lengh δ is ofen referred o as he frequency or he enor of he cap. 6 In pracice, he frequency is ypically eiher 3, 6, or 12 monhs. Noe ha he ime disance beween paymen daes coincides wih he mauriy of he floaing ineres rae. Also noe ha while a cap is ailored for ineres rae hedging, i can also be used for ineres rae speculaion. A cap can be seen as a porfolio of n caples, namely one for each paymen dae of he cap. 4 Some counries have markes wih rade in morgage-backed bonds where he issuer has an American call opion on he bond. These bonds are annuiy bonds where he paymens are considerably higher han for a sandard bulle bond wih he same face value. Opimaliy of early exercise of such a call is herefore more likely han exercise of a call on a sandard bond. 5 In pracice, here will no be exacly he same number of days beween successive rese daes, and he calculaions below mus be slighly adjused by using he relevan day coun convenion. 6 The word enor is someimes used for he se of paymen daes T 1,..., T n.

2.8 Caps, floors, and collars 32 The i h caple yields a payoff a ime T i of 2.11) C i T i = Hδ max l Ti T i δ K, 0 ) and no oher paymens. A caple is a call opion on he zero-coupon yield prevailing a ime T i δ for a period of lengh δ, bu where he paymen akes place a ime T i alhough i is already fixed a ime T i δ. In he following we will find he value of he i h caple before ime T i. Since he payoff becomes known a ime T i δ, we can obain is value in he inerval beween T i δ and T i by a simple discouning of he payoff, i.e. ) C i = B Ti Hδ max l Ti T K, 0 i δ, T i δ T i. In paricular, C i T i δ = B Ti T i δ Hδ max l Ti T i δ K, 0 ). Applying 1.8) on page 8, we can rewrie his value as ) CT i i δ = B Ti T H max i δ 1 + δl Ti T [1 + δk], 0 i δ ) = B Ti T H max 1 i δ [1 + δk], 0 = H1 + δk) max B Ti T i δ ) 1 1 + δk BTi T, 0 i δ. We can now see ha he value a ime T i δ is idenical o he payoff of a European pu opion expiring a ime T i δ ha has an exercise price of 1/1 + δk) and is wrien on a zero-coupon bond mauring a ime T i. Accordingly, he value of he i h caple a an earlier poin in ime T i δ mus equal he value of ha pu opion. Wih he noaion used earlier we can wrie his as 2.12) C i = H1 + δk)π 1+δK) 1,T i δ,t i. To find he value of he enire cap conrac we simply have o add up he values of all he caples corresponding o he remaining paymen daes of he cap. Before he firs rese dae, T 0, none of he cap paymens are known, so he value of he cap is given by n n 2.13) C = C i = H1 + δk) π 1+δK) 1,T i δ,t i, < T 0. i=1 i=1 A all daes afer he firs rese dae, he nex paymen of he cap will already be known. If we again use he noaion T i) for he neares following paymen dae afer ime, he value of he cap a any ime in [T 0, T n ] exclusive of any paymen received exacly a ime ) can be wrien as 2.14) ) C = HB T i) δ max l T i) T i) δ K, 0 n + 1 + δk)h π 1+δK) 1,T i δ,t i, T 0 T n. i=i)+1 If T n 1 < < T n, we have i) = n, and here will be no erms in he sum, which is hen considered o be equal o zero. In laer chapers we will discuss models for pricing bond opions. From he resuls above, cap prices will follow from prices of European pus on zero-coupon bonds.

2.8 Caps, floors, and collars 33 Noe ha he ineres raes and he discoun facors appearing in he expressions above are aken from he money marke, no from he governmen bond marke. Also noe ha since caps and mos oher conracs relaed o money marke raes rade OTC, one should ake he defaul risk of he wo paries ino accoun when valuing he cap. Here, defaul simply means ha he pary canno pay he amouns promised in he conrac. Official money marke raes and he associaed discoun funcion apply o loan and deposi arrangemens beween large financial insiuions, and hus hey reflec he defaul risk of hese corporaions. If he paries in an OTC ransacion have a defaul risk significanly differen from ha, he discoun raes in he formulas should be adjused accordingly. However, i is quie complicaed o do ha in a heoreically correc manner, so we will no discuss his issue any furher a his poin. 2.8.2 Floors An ineres rae) floor is designed o proec an invesor who has len funds on a floaing rae basis agains receiving very low ineres raes. The conrac is consruced jus as a cap excep ha he payoff a ime T i i = 1,..., n) is given by ) 2.15) FT i i = Hδ max K l Ti T, 0 i δ, where K is called he floor rae. Buying an appropriae floor, an invesor who has provided anoher invesor wih a floaing rae loan will in oal a leas receive he floor rae. Of course, an invesor can also speculae in low fuure ineres raes by buying a floor. The hypoheical) conracs ha only yield one of he paymens in 2.15) are called floorles. Obviously, we can hink of a floorle as a European pu on he floaing ineres rae wih delayed paymen of he payoff. Analogously o he analysis for caps, we can also hink of a floorle as a European call on a zero-coupon bond, and hence a floor is equivalen o a porfolio of European calls on zero-coupon bonds. More precisely, he value of he i h floorle a ime T i δ is ) 2.16) FT i i δ = H1 + δk) max B Ti T 1 i δ 1 + δk, 0. The oal value of he floor conrac a any ime < T 0 is herefore given by 2.17) F = H1 + δk) n i=1 and laer he value is ) F = HB T i) δ max K l T i) T i) δ, 0 2.18) 2.8.3 Collars + 1 + δk)h n i=i)+1 C 1+δK) 1,T i δ,t i, < T 0, C 1+δK) 1,T i δ,t i, T 0 T n. A collar is a conrac designed o ensure ha he ineres rae paymens on a floaing rae borrowing arrangemen says beween wo pre-specified levels. A collar can be seen as a porfolio of a long posiion in a cap wih a cap rae K c and a shor posiion in a floor wih a floor rae of K f < K c and he same paymen daes and underlying floaing rae). The payoff of a collar a

2.8 Caps, floors, and collars 34 ime T i, i = 1, 2,..., n, is hus [ ) )] L i T i = Hδ max l Ti T K i δ c, 0 max K f l Ti T, 0 i δ [ ] Hδ K f l Ti T i δ, if l Ti T K i δ f, = 0, if K f l Ti T K i δ c, [ ] Hδ l Ti T K i δ c, if K c l Ti T. i δ The value of a collar wih cap rae K c and floor rae K f is of course given by L K c, K f ) = C K c ) F K f ), where he expressions for he values of caps and floors derived earlier can be subsiued in. An invesor who has borrowed funds on a floaing rae basis will by buying a collar ensure ha he paid ineres rae always lies in he inerval beween K f and K c. Clearly, a collar gives cheaper proecion agains high ineres raes han a cap wih he same cap rae K c ), bu on he oher hand he full benefis of very low ineres raes are sacrificed. In pracice, K f and K c are ofen se such ha he value of he collar is zero a he incepion of he conrac. 2.8.4 Exoic caps and floors Above we considered sandard, plain vanilla caps, floors, and collars. In addiion o hese insrumens, several conracs rade on he inernaional OTC markes wih cash flows ha are similar o plain vanilla conracs, bu deviae in one or more aspecs. The deviaions complicae he pricing mehods considerably. Le us briefly look a a few of hese exoic securiies. The examples are aken from Musiela and Rukowski 1997, Ch. 16). A bounded cap is like an ordinary cap excep ha he cap owner will only receive he scheduled payoff if he sum of he paymens received so far due o he conrac does no exceed a cerain pre-specified level. Consequenly, he ordinary cap paymens in 2.11) are o be muliplied wih an indicaor funcion. The payoff a he end of a given period will depend no only on he ineres rae in he beginning of he period, bu also on previous ineres raes. As many oher exoic insrumens, a bounded cap is herefore a pah-dependen asse. A dual srike cap is similar o a cap wih a cap rae of K 1 in periods when he underlying floaing rae l +δ says below a pre-specified level ˆl, and similar o a cap wih a cap rae of K 2, where K 2 > K 1, in periods when he floaing rae is above ˆl. A cumulaive cap ensures ha he accumulaed ineres rae paymens do no exceed a given level. A knock-ou cap will a any ime T i give he sandard payoff in 2.11) unless he floaing rae l +δ during he period [T i δ, T i ] has exceeded a cerain level. In ha case he payoff is zero. Opions on caps and floors are also raded. Since caps and floors hemselves are porfolios of) opions, he opions on caps and floors are so-called compound opions. An opion on a cap is called a capion and provides he holder wih he righ a a fuure poin in ime, T 0, o ener ino a cap saring a ime T 0 wih paymen daes T 1,..., T n ) agains paying a given exercise price.

2.9 Swaps and swapions 35 2.9 Swaps and swapions 2.9.1 Swaps Many differen ypes of swaps are raded on he OTC markes, e.g. currency swaps, credi swaps, asse swaps, bu in line wih he heme of his chaper we will focus on ineres rae swaps. An ineres rae) swap is an exchange of wo cash flow sreams ha are deermined by cerain ineres raes. In he simples and mos common ineres rae swap, a plain vanilla swap, wo paries exchange a sream of fixed ineres rae paymens and a sream of floaing ineres rae paymens. The paymens are in he same currency and are compued from he same hypoheical) face value or noional principal. The floaing rae is usually a money marke rae, e.g. a LIBOR rae, possibly augmened or reduced by a fixed margin. The fixed ineres rae is usually se so ha he swap has zero ne presen value when he paries agree on he conrac. While he wo paries can agree upon any mauriy, mos ineres rae swaps have a mauriy beween 2 and 10 years. The firs swap was conraced in 1981, and nowadays he inernaional swap markes are enormous, boh in erms of ransacions and ousanding conracs. The organizaion ISDA Inernaional Swaps and Derivaives Associaion) publishes key figures showing he size and developmen of he markes for swaps and ineres rae opions raded OTC. Afer 1997 he published figures are for all producs ogeher and are no informaive for he markes of he differen securiies. A he end of 1997, he oal noional principal face value) of all repored, acive ineres rae swaps amouned o 22.3 rillion U.S. dollars i.e. 22.300.000 million), and he swap marke was indispuably he larges OTC derivaive marke. 7 The oal OTC derivaive marke more han doubled in size from 1997 o 2000, and here is no reason o believe ha he growh rae of he swap marke iself is significanly differen. Le us briefly look a he uses of ineres rae swaps. An invesor can ransform a floaing rae loan ino a fixed rae loan by enering ino an appropriae swap, where he invesor receives floaing rae paymens neing ou he paymens on he original loan) and pays fixed rae paymens. This is called a liabiliy ransformaion. Conversely, an invesor who has len money a a floaing rae, i.e. owns a floaing rae bond, can ransform his o a fixed rae bond by enering ino a swap, where he pays floaing rae paymens and receives fixed rae paymens. This is an asse ransformaion. Hence, ineres rae swaps can be used for hedging ineres rae risk on boh cerain) asses and liabiliies. On he oher hand, ineres rae swaps can also be used for aking advanage of specific expecaions of fuure ineres raes, i.e. for speculaion. Swaps are ofen said o allow he wo paries o exploi heir comparaive advanages in differen markes. Concerning ineres rae swaps, his argumen presumes ha one pary has a comparaive advanage relaive o he oher pary) in he marke for fixed rae loans, while he oher pary has a comparaive advanage relaive o he firs pary) in he marke for floaing rae loans. However, hese markes are inegraed, and he exisence of comparaive advanages conflics wih modern financial heory and he efficiency of he money markes. Apparen comparaive advanages can be due o differences in defaul risk premia. For deails we refer he reader o he discussion in Hull 2003, Ch. 6). 7 For ineres rae swaps denoed in he Danish currency he corresponding number is 133 billion U.S. dollars i.e. 133.000 million), an increase of 16% relaive o he year before.

2.9 Swaps and swapions 36 Nex, we will discuss he valuaion of swaps. As for caps and floors, we assume ha boh paries in he swap have a defaul risk corresponding o he average defaul risk of major financial insiuions refleced by he money marke ineres raes. For a descripion of he impac on he paymens and he valuaion of swaps beween paries wih differen defaul risk, see Duffie and Huang 1996) and Huge and Lando 1999). Furhermore, we assume ha he fixed rae paymens and he floaing rae paymens occur a exacly he same daes hroughou he life of he swap. This is rue for mos, bu no all, raded swaps. For some swaps, he fixed rae paymens only occur once a year, whereas he floaing rae paymens are quarerly or semi-annual. The analysis below can easily be adaped o such swaps. In a plain vanilla ineres rae swap, one pary pays a sream of fixed rae paymens and receives a sream of floaing rae paymens. This pary is said o have a pay fixed, receive floaing swap or a fixed-for-floaing swap or simply a payer swap. The counerpar receives a sream of fixed rae paymens and pays a sream of floaing rae paymens. This pary is said o have a pay floaing, receive fixed swap or a floaing-for-fixed swap or simply a receiver swap. Noe ha he names payer swap and receiver swap refer o he fixed rae paymens. We consider a swap wih paymen daes T 1,..., T n, where T i+1 T i = δ. The floaing ineres rae deermining he paymen a ime T i is he money marke LIBOR) rae l Ti T i δ. In he following we assume ha here is no fixed exra margin on his floaing rae. If here were such an exra charge, he value of he par of he flexible paymens ha is due o he exra margin could be compued in he same manner as he value of he fixed rae paymens of he swap, see below. We refer o T 0 = T 1 δ as he saring dae of he swap. As for caps and floors, we call T 0, T 1,..., T n 1 he rese daes, and δ he frequency or he enor. Typical swaps have δ equal o 0.25, 0.5, or 1 corresponding o quarerly, semi-annual, or annual paymens and ineres raes. We will find he value of an ineres rae swap by separaely compuing he value of he fixed rae paymens V fix ) and he value of he floaing rae paymens V fl ). The fixed rae is denoed by K. This is a nominal, annual ineres rae, so ha he fixed rae paymens equal HKδ, where H is he noional principal or face value which is no swapped). The value of he remaining fixed paymens is simply 2.19) V fix = n i=i) HKδB Ti = HKδ n i=i) B Ti. The floaing rae paymens are exacly he same as he coupon paymens on a floaing rae bond, which was discussed in Secion 2.2, i.e. a ime T i i = 1, 2,..., n) he paymen is Hδl Ti T i δ. Noe ha his paymen is already known a ime T i δ. According o 2.1), he value of such a floaing bond a any ime [T 0, T n ) is given by H1 + δl T i) T i) δ )BT i). Since his is he value of boh he coupon paymens and he final repaymen of face value, he value of he coupon paymens only mus be V fl [ = Hδl T i) T i) δ BT i) + H B T i) ] B Tn, T 0 < T n. A and before ime T 0, he firs erm is no presen, so he value of he floaing rae paymens is simply 2.20) V fl [ = H B T0 ] B Tn, T 0.

2.9 Swaps and swapions 37 We will also develop an alernaive expression for he value of he floaing rae paymens of he swap. The ime T i δ value of he coupon paymen a ime T i is Hδl Ti T i δ BTi T i δ = Hδ l Ti T i δ 1 + δl Ti T i δ where we have applied 1.8) on page 8. Consider a sraegy of buying a zero-coupon bond wih face value H mauring a T i δ and selling a zero-coupon bond wih he same face value H bu mauring a T i. The ime T i δ value of his posiion is HB Ti δ T i δ HBTi T i δ = H H 1 + δl Ti T i δ, = Hδ l Ti T i δ 1 + δl Ti T i δ which is idenical o he value of he floaing rae paymen of he swap. Therefore, he value of his floaing rae paymen a any ime T i δ mus be, 2.21) H B Ti δ ) B Ti = HδB Ti B T i δ B T i δ 1 = HδB Ti L Ti δ,ti, where we have applied 1.9) on page 8. Thus, he value a ime T i δ of geing Hδl Ti ime T i is equal o HδB Ti L Ti δ,ti, i.e. he unknown fuure spo rae l Ti T i δ T i δ a in he payoff is replaced by he curren forward rae for L Ti δ,ti and hen discouned by he curren riskfree discoun facor. The value a ime > T 0 of all he remaining floaing coupon paymens can herefore be B Ti wrien as V fl = HδB T i) l T i) T i) δ + Hδ n i=i)+1 A or before ime T 0, he firs erm is no presen, so we ge 2.22) V fl = Hδ n i=1 B Ti L Ti δ,ti, > T 0. B Ti L Ti δ,ti, T 0. The value of a payer swap is while he value of a receiver swap is P = V fl R = V fix V fix, V fl. In paricular, he value of a payer swap a or before is saring dae T 0 can be wrien as 2.23) P = Hδ using 2.19) and 2.22), or as n i=1 2.24) P = H [ B T0 B Ti ) L Ti δ,ti K, T 0, ] B Tn n i=1 KδB Ti ), T 0, using 2.19) and 2.20). If we le Y i = Kδ for i = 1,..., n 1 and Y n = 1+Kδ, we can rewrie 2.24) as 2.25) P = H B T0 n i=1 Y i B Ti ), T 0.

2.9 Swaps and swapions 38 Also noe he following relaion beween a cap, a floor, and a payer swap having he same paymen daes and where he cap rae, he floor rae, and he fixed rae in he swap are all idenical: 2.26) C = F + P. This follows from he fac ha he paymens from a porfolio of a floor and a payer swap exacly mach he paymens of a cap. The equilibrium) swap rae l T δ 0 prevailing a ime T 0 for a swap wih frequency δ and paymens daes T i = T 0 + iδ, i = 1, 2,..., n, is defined as he unique value of he fixed rae ha makes he presen value of a swap saring a T 0 equal o zero, i.e. P T0 = R T0 = 0. Applying 2.23), we can wrie he swap rae as lδ T0 = n i=1 LTi δ,ti T 0 B Ti T 0 n, i=1 BTi T 0 which can also be wrien as a weighed average of he relevan forward raes: 2.27) lδ T0 = n i=1 w i L Ti δ,ti T 0, where w i = B Ti T 0 / n i=1 BTi T 0. Alernaively, we can le = T 0 in 2.24) yielding ) P T0 = H so ha he swap rae can be expressed as 1 B Tn T 0 Kδ n i=1 B Ti T 0, 2.28) lδ T0 = 1 BTn T 0 δ n i=1 BTi Subsiuing 2.28) ino he expression jus above i, he ime T 0 value of an agreemen o pay a fixed rae K and receive he prevailing marke rae a each of he daes T 1,..., T n, can be wrien in erms of he curren swap rae as 2.29) P T0 = H = n i=1 lδ T0 δ B Ti T 0 n ) i=1 B Ti T 0 ) T 0. Kδ ) Hδ lδ K T0. n A forward swap or deferred swap) is an agreemen o ener ino a swap wih a fuure saring dae T 0 and a fixed rae which is already se. Of course, he conrac also fixes he frequency, he mauriy, and he noional principal of he swap. The value a ime T 0 of a forward payer swap wih fixed rae K is given by he equivalen expressions 2.23) 2.25). The forward swap rae L δ,t0 is defined as he value of he fixed rae ha makes he forward swap have zero value a ime. The forward swap rae can be wrien as 2.30) Lδ,T 0 = BT0 B Tn δ n i=1 BTi = i=1 B Ti T 0 n i=1 LTi δ,ti B Ti n i=1 BTi Noe ha boh he swap rae and he forward swap rae depend on he frequency and he mauriy of he underlying swap. To indicae his dependence, le l δ n) denoe he ime swap )).

2.9 Swaps and swapions 39 rae for a swap wih paymen daes T i = + iδ, i = 1, 2,..., n. If we depic he swap rae as a funcion of he mauriy, i.e. he funcion n l δ n) only defined for n = 1, 2,... ), we ge a erm srucure of swap raes for he given frequency. Many financial insiuions paricipaing in he swap marke will offer swaps of varying mauriies under condiions refleced by heir posed erm srucure of swap raes. In Exercise 2.3, he reader is asked o show how he discoun facors B Ti T 0 can be derived from a erm srucure of swap raes. 2.9.2 Swapions A European swapion gives is holder he righ, bu no he obligaion, a he expiry dae T 0 o ener ino a specific ineres rae swap ha sars a T 0 and has a given fixed rae K. No exercise price is o be paid if he righ is uilized. The rae K is someimes referred o as he exercise rae of he swapion. We disinguish beween a payer swapion, which gives he righ o ener ino a payer swap, and a receiver swapion, which gives he righ o ener ino a receiver swap. Le us firs focus on a European receiver swapion. A ime T 0, he value of a receiver swap wih paymen daes T i = T 0 + iδ, i = 1, 2,..., n, and a fixed rae K is given by n ) R T0 = H Y i B Ti T 0 1, i=1 where Y i = Kδ for i = 1,..., n 1 and Y n = 1 + Kδ; cf. 2.25). Hence, he ime T 0 payoff of a receiver swapion is 2.31) R T0 = max R T0 0, 0) = H max n i=1 Y i B Ti T 0 1, 0 which is equivalen o he payoff of H European call opions on a bulle bond wih face value 1, n paymen daes, a period of δ beween successive paymens, and an annualized coupon rae K. The exercise price of hese opions equals he face value 1. ), The price of a European receiver swapion mus herefore be equal o he price of hese call opions. In many of he pricing models we develop in laer chapers, we can compue such prices quie easily. Similarly, a European payer swapion yields a payoff of 2.32) P T0 = max P T0 0, 0) = max R T0, 0) = H max 1 n i=1 Y i B Ti T 0, 0 This is idenical o he payoff from H European pu opions expiring a T 0 and having an exercise price of 1 wih a bond paying Y i a ime T i, i = 1, 2,..., n, as is underlying asse. Alernaively, we can apply 2.29) o express he payoff of a European payer swapion as n ) ) 2.33) P T0 = Hδ max lδ K, 0 T0, and similarly R T0 = for a European receiver swapion. i=1 n i=1 B Ti T 0 B Ti T 0 ) Hδ max K l ) T δ 0, 0 ).

2.9 Swaps and swapions 40 Also noe ha he following payer-receiver pariy holds for European swapions having he same underlying swap and he same exercise rae: 2.34) P R = P, T 0, cf. Exercise 2.4. In words, a payer swapion minus a receiver swapion is indisinguishable form a forward payer swap. While a large majoriy of raded swapions are European, so-called Bermuda swapions are also raded. A Bermuda swapion can be exercised a a number of pre-specified daes and, herefore, resembles an American opion. When he Bermuda swapion is exercised, he holder receives a posiion in a swap wih cerain paymen daes. Mos Bermuda swapions are consruced such ha he underlying swap has some fixed, poenial paymen daes T 1,..., T n. If he Bermuda swapion is exercised a, say, ime, only he remaining swap paymens will be effecive, i.e. he paymens a dae T i ),..., T n. Laer exercise resuls in a shorer swap. The possible exercise daes will usually coincide wih he poenial swap paymen daes. Exercise of a Bermuda payer receiver) swapion a dae T l resuls in a payoff a ha dae equal o he payoff of a European payer receiver) swapion expiring a ha dae wih a swap wih paymen daes T l+1,..., T n. Bermuda swapions are ofen issued ogeher wih a given swap. Such a package is called a cancellable swap or a puable swap. Typically, he Bermuda swapion canno be exercised over a cerain period in he beginning of he swap. When praciioners alk of, say, a 10 year non call 2 year Bermuda swapion, hey mean an opion on a 10 year swap, where he opion a he earlies can be exercised 2 years ino he swap and hen on all subsequen paymen daes of he swap. A less raded varian is a consan mauriy Bermuda swapion, where he opion holder upon exercise receives a swap wih he same ime o mauriy no maer when he opion is exercised. 2.9.3 Exoic swap insrumens The following examples of exoic swap marke producs are adaped from Musiela and Rukowski 1997) and Hull 2003): Floa-for-floaing swap: Two floaing ineres raes are swapped, e.g. he hree-monh LIBOR rae and he yield on a given governmen bond. Amorizing swap: The noional principal is reduced from period o period following a pre-specified scheme, e.g. so ha he noional principle a any ime reflecs he ousanding deb on a loan wih periodic insalmens as for an annuiy or a serial bond). Sep-up swap: The noional principal increases over ime in a pre-deermined way. Accrual swap: The scheduled paymens of one pary are only o be paid as long as he floaing rae lies in some inerval I. Assume for concreeness ha i is he fixed rae paymens ha have his feaure. A he swap paymen dae T i he effecive fixed rae paymen is hen HδKN 1 /N 2, where N 1 is he number of days in he period beween T i 1 and T i, where he floaing rae l +δ was in he inerval I, and N 2 is he oal number of days in he period. The inerval I may even differ from period o period eiher in a deerminisic way or depending on he evoluion of he floaing ineres rae so far.

2.10 Exercises 41 Consan mauriy swap: A he paymen daes a fixed rae is exchanged for he equilibrium) swap rae on a swap of a given, consan mauriy, i.e. he floaing rae is iself a swap rae. Exendable swap: One pary has he righ o exend he life of he swap under cerain condiions. Forward swapion: A forward swapion gives he righ o ener ino a forward swap, i.e. he swapion expires a ime before he saring dae of he swap T 0. The payoff is Hδ n i=1 Lδ, ) max T 0 K, 0 B Ti = n i=1 ) Lδ, ) B Ti Hδ max T 0 K, 0. Swap rae spread opion: The payoff is deermined by he difference beween equilibrium) swap raes for wo differen mauriies. Recall ha l δ T 0 m) denoes he swap rae for a swap wih paymen daes T 1,..., T m, where T i = T 0 + iδ. An m, n)-period European swap rae spread call opion wih an exercise rae K yields a payoff a ime T 0 of max lδ m) l ) δ n) K, 0 T0 T0. The corresponding pu has a payoff of max K [ lδ m) l ] ) δ n) T0 T0, 0. Yield curve swap: In a one-period yield curve swap one pary receives a a given dae T a swap rae l T δ m) and pays a rae K + l T δ n), boh compued on he basis of a given noional principal H. A muli-period yield curve swap has, say, L paymen daes T 1,..., T L. A ime T l one pary receives an ineres rae of l δ T l m) and pays an ineres rae of K + l δ T l n). In addiion, several insrumens combine elemens of ineres rae swaps and currency swaps. For example, in a differenial swap a domesic floaing rae is swapped for a foreign floaing rae. 2.10 Exercises EXERCISE 2.1 Show ha he no-arbirage price of a European call on a zero-coupon bond will saisfy ) max 0, B S KB T C K,T,S B S 1 K) provided ha all ineres raes are non-negaive. Here, T is he mauriy dae of he opion, K is he exercise price, and S is he mauriy dae of he underlying zero-coupon bond. Compare wih he corresponding bounds for a European call on a sock, cf. Hull 2003, Ch. 8). Derive similar bounds for a European call on a coupon bond. EXERCISE 2.2 Give a proof of he pu-call pariy for opions on coupon bonds in Theorem 2.4. EXERCISE 2.3 Le l δ T 0 k) be he equilibrium swap rae for a swap wih paymen daes T 1, T 2,..., T k, where T i = T 0 + iδ as usual. Suppose ha l δ T 0 1),..., l δ T 0 n) are known. Find a recursive procedure for deriving he associaed discoun facors B T 1 T 0, B T 2 T 0,..., B Tn T 0. EXERCISE 2.4 Show he pariy 2.34). Show ha a payer swapion and a receiver swapion wih idenical erms) will have idenical prices, if he exercise rae of he conracs is equal o he forward swap

2.10 Exercises 42 rae L δ,t 0. EXERCISE 2.5 Consider a swap wih saring dae T 0 and a fixed rae K. For T 0, show ha V fl /V fix = L δ,t 0 /K, where L δ,t 0 is he forward swap rae.

Chaper 3 Sochasic processes and sochasic calculus In he previous chaper we saw ha many ineres rae dependen securiies canno be priced uniquely jus by appealing o no-arbirage argumens. To derive prices and hedging sraegies we have o model he uncerainy abou he erm srucure of ineres raes a relevan fuure daes. In order o analyze he relaion beween ineres raes and oher macroeconomic variables such as aggregae consumpion or producion, we also have o ake he uncerainy abou he fuure values of hese variables ino accoun. For example, he uncerainy abou fuure consumpion will affec individuals supply and demand for bonds and, hence, affec he ineres raes se oday. In modern finance, sochasic processes are used o model he evoluion of uncerain variables over ime. Therefore, a basic knowledge of sochasic processes and how o do compuaions involving sochasic processes is needed in order o undersand, evaluae, and develop models of he erm srucure of ineres raes. This chaper is devoed o a relaively brief inroducion o sochasic processes and he mahemaical ools needed o do calculaions wih sochasic processes, he so-called sochasic calculus. We will omi many echnical deails ha are no imporan for a reasonable level of undersanding and focus on processes and resuls ha will become imporan in laer chapers. For more deails and proofs, he reader is referred o he exbooks of Øksendal 1998) and Karazas and Shreve 1988). 3.1 Probabiliy spaces The basic objec for sudies of uncerain evens is a probabiliy space, which is a riple Ω, F, P). Here, Ω is he sae space, which is he se of possible saes or oucomes of he uncerain objec. For example, if one sudies he oucome of a hrow of a dice, he sae space is Ω = {1, 2, 3, 4, 5, 6}. An even is a se of possible oucomes, i.e. a subse of he sae space. In he example wih he dice, some evens are {1, 2, 3}, {4, 5}, {1, 3, 5}, {6}, and {1, 2, 3, 4, 5, 6}. The second componen of a probabiliy space, F, is he se of evens o which a probabiliy can be assigned, i.e. he se of probabilizable evens. Hence, F is a se of subses of he sae space! I is required ha i) he enire sae space can be assigned a probabiliy, i.e. Ω F; ii) if some even F Ω can be assigned a probabiliy, so can is complemen F c Ω \ F, i.e. 43

3.2 Sochasic processes 44 F F F c F; and iii) given a sequence of probabilizable evens, he union is also probabilizable, i.e. F 1, F 2, F i=1 F i F. Ofen F is referred o as a sigma-field. The final componen of a probabiliy space is a probabiliy measure P, which formally is a funcion from he sigma-field F ino he inerval [0, 1]. To each even F F, he probabiliy measure assigns a number PF ) in he inerval [0, 1]. This number is called he P-probabiliy or simply he probabiliy) of F. A probabiliy measure mus saisfy he following condiions: i) PΩ) = 1 and P ) = 0, where denoes he empy se; ii) he probabiliy of he sae being in he union of disjoin ses is equal o he sum of he probabiliies for each of he ses, i.e. given F 1, F 2, F wih F i F j = for all i j, we have P i=1 F i) = i=1 PF i). Many differen probabiliy measures can be defined on he same sigma-field, F, of evens. In he example of he dice, a probabiliy measure P corresponding o he idea ha he dice is fair is defined by P{1}) = P{2}) = = P{6}) = 1/6. Anoher probabiliy measure, Q, can be defined by Q{1}) = 1/12, Q{2}) = = Q{5}) = 1/6, and Q{6}) = 3/12, which may be appropriae if he dice is believed o be unfair. Two probabiliy measures P and Q defined on he same sae space and sigma-field Ω, F) are called equivalen if he wo measures assign probabiliy zero o exacly he same evens, i.e. if PA) = 0 QA) = 0. The wo probabiliy measures in he dice example are equivalen. In he sochasic models of financial markes swiching beween equivalen probabiliy measures urns ou o be imporan. 3.2 Sochasic processes The sae of many sysems or objecs changes over ime in a manner ha canno be prediced wih cerainy. This is also rue for many economic objecs such as sock prices, ineres raes, and exchange raes. Such an objec can be described by a sochasic process, which is a family of random variables wih one random variable for each ime we observe he sae of he objec. We will denoe a generic sochasic process by he symbol x, which is hen given as a collecion x ) T of random variables defined on a common probabiliy space Ω, F, P). We will only consider realvalued sochasic process, i.e. all he random variables ake values in a subse of) R K for some ineger K 1. The se T consiss of all he poins in ime a which we care abou he sae of he objec, which is represened by he value of he process. Associaed wih a sochasic process is informaion abou exacly how he sae can change over ime. 3.2.1 Differen ypes of sochasic processes A sochasic process for he sae of an objec a every poin in ime in a given inerval is called a coninuous-ime sochasic process. This corresponds o he case where he se T akes he form of an inerval [0, T ] or [0, ). In conras a sochasic process for he sae of an objec a counably many separaed poins in ime is called a discree-ime sochasic process. This is

3.2 Sochasic processes 45 for example he case when T = {0, 1, 2,..., T } or {0, 1, 2,... }. If he sae can ake on all values in a given inerval e.g. all real numbers), he process is called a coninuous-variable sochasic process. On he oher hand, if he sae can ake on counably many separaed values, he process is called a discree-variable sochasic process. The invesors in he financial markes can rade a more or less any poin in ime. Due o pracical consideraions and ransacion coss, no invesor will rade coninuously. However, wih many invesors here will be some rades a almos any poin in ime, so ha prices and ineres raes ec. will also change almos coninuously. Therefore, i seems o be a beer approximaion of real life o describe such economic variables by coninuous-ime sochasic processes han by discree-ime sochasic processes. Coninuous-ime sochasic processes are in many aspecs also easier o handle han discree-ime sochasic processes. In pracice, hese economic variables can only ake on counably many values, e.g. sock prices are muliples of he smalles possible uni 0.01 currency unis in many counries), and ineres raes are only saed wih a given number of decimals. Bu since he possible values are very close ogeher, i seems reasonable o use coninuous-variable processes in he modeling of hese objecs. In addiion, he mahemaics involved in he analysis of coninuous-variable processes is simpler and more elegan han he mahemaics for discree-variable processes. In sum, we will use coninuous-ime, coninuousvariable sochasic processes hroughou o describe he evoluion in prices and raes. Therefore he remaining secion of his chaper will be devoed o ha ype of sochasic processes. 3.2.2 Basic conceps Le us consider a coninuous-ime, coninuous-variable sochasic process x = x ) R+, where he random variable x represens he sae or value of he objec a ime. Here we assume ha we are ineresed in he sae a every poin in ime in R + = [0, ), where ime is measured relaive o some given saring ime, ime 0. In his case an oucome is an enire se of values {x 0}, which we will call a sample) pah. A pah of a sochasic process is a possible realizaion of he evoluion of he process over ime. The sae space Ω is he se of all pahs. Evens are subses of Ω, i.e. ses of pahs. The following are examples of some evens: {x 10 for all 1}, {x 5 0}, and {x 1 a 1, x 3/2 a 2 }. Aached o all possible evens is a probabiliy given by a probabiliy measure P. We assume, furhermore, ha all he random variables x ake on values in he same se S, which we call he value space of he process. More precisely his means ha S is he smalles se wih he propery ha P{x S}) = 1. If S R, we call he process a one-dimensional, real-valued process. If S is a subse of R K bu no a subse of R K 1 ), he process is called a K- dimensional, real-valued process, which can also be hough of as a collecion of K one-dimensional, real-valued processes. Noe ha as long as we resric ourselves o equivalen probabiliy measures, he value space will no be affeced by changes in he probabiliy measure. As ime goes by, we can observe he evoluion in he objec which he sochasic process describes. A any given ime, he previous values x ) [0, ), where x S, will be known a leas in he models we consider). These values consiue he hisory of he process up o ime. The fuure values are sill sochasic.

3.2 Sochasic processes 46 3.2.3 Markov processes and maringales As ime passes we will ypically revise our expecaions of he fuure values of he process or, more precisely, revise he probabiliy disribuion we aribue o he value of he process a any fuure poin in ime. Suppose we sand a ime and consider he value of a process x a a fuure ime >. The disribuion of he value of x is characerized by probabiliies Px A) for subses A of he value space S. If for all, R + wih < and all A S, we have ha P ) x A x s ) s [0,] = P x A x ), hen x is called a Markov process. Broadly speaking, his condiion says ha, given he presence, he fuure is independen of he pas. The hisory conains no informaion abou he fuure value ha canno be exraced from he curren value. Markov processes are ofen used in financial models o describe he evoluion in prices of financial asses, since he Markov propery is consisen wih he so-called weak form of marke efficiency, which says ha exraordinary reurns canno be achieved by use of he precise hisorical evoluion in he price of an asse. 1 If exraordinary reurns could be obained in his manner, all invesors would ry o profi from i, so ha prices would change immediaely o a level where he exraordinary reurn is non-exisen. Therefore, i is reasonable o model prices by Markov processes. In addiion, models based on Markov processes are ofen more racable han models wih non-markov processes. A sochasic process is said o be a maringale if, a all poins in ime, he expeced change in he value of he process over any given fuure period is equal o zero. In oher words, he expeced fuure value of he process is equal o he curren value of he process. Because expecaions depend on he probabiliy measure, he concep of a maringale should be seen in connecion wih he applied probabiliy measure. More rigorously, a sochasic process x = x ) 0 is a P-maringale if for all 0 we have ha E P [x s ] = x, for all s >. Here, E P denoes he expeced value compued under he P-probabiliies given he informaion available a ime, ha is, given he hisory of he process up o and including ime. Someimes he probabiliy measure will be clear from he conex and can be noaionally suppressed. 3.2.4 Coninuous or disconinuous pahs We will only consider sochasic processes having pahs ha are coninuous funcions of ime, so ha one can depic he evoluion of he process by a coninuous curve. The mos fundamenal process wih his propery is he so-called sandard Brownian moion or Wiener process, which we will describe in deail in he nex secion. From he sandard Brownian moion many oher ineresing coninuous-pah processes can be consruced as we will see in laer secions. Sochasic processes which have pahs wih disconinuiies jumps) also exis. The jumps of such processes are ofen modeled by Poisson processes or relaed processes. I is well-known ha large, sudden movemens in financial variables occur from ime o ime, for example in connecion 1 This does no conflic wih he fac ha he hisorical evoluion is ofen used o idenify some characerisic properies of he process, e.g. for esimaion of means and variances.

3.3 Brownian moions 47 wih sock marke crashes. There may be many explanaions of such large movemens, for example a large unexpeced change in he produciviy in a paricular indusry or he economy in general, perhaps due o a echnological break-hrough. Anoher source of sudden, large movemens is changes in he poliical or economic environmen such as inervenions by he governmen or cenral bank. Sock marke crashes are someimes explained by he bursing of a bubble which does no necessarily conflic wih he usual assumpion of raional invesors). Wheher such sudden, large movemens can be explained by a sequence of small coninuous movemens in he same direcion or jumps have o be included in he models is an empirical quesion, which is sill open. There are numerous financial models of sock markes ha allow for jumps in sock prices, e.g. Meron 1976) discusses he pricing of sock opions in such a framework. On he oher hand, here are only very few models allowing for jumps in ineres raes. 2 This can be jusified empirically by he observaion ha sudden, large movemens are no nearly as frequen in he bond markes as in he sock markes. There are also heoreical argumens supporing hese findings. In a general equilibrium model of he economy, Wu 1999) shows among oher hings ha jumps in he overall produciviy of he economy will cause jumps in sock prices, bu no in bond prices or ineres raes. Of course, models for corporae bonds mus be able o handle he possible defaul of he issuing company, which in some cases comes as a surprise o he financial marke. Therefore, such models will ypically involve jump processes; see e.g. Lando 1998). In he main par of he ex we will focus on defaul-free conracs and use coninuous-pah processes. 3.3 Brownian moions All he sochasic processes we shall apply in he financial models in he following chapers build upon a paricular class of processes, he so-called Brownian moions. A one-dimensional) sochasic process z = z ) 0 is called a sandard Brownian moion, if i saisfies he following condiions: i) z 0 = 0, ii) for all, 0 wih < : z z N0, ) [normally disribued incremens], iii) for all 0 0 < 1 < < n, he random variables z 1 z 0,..., z n z n 1 independen [independen incremens], are muually iv) z has coninuous pahs. Here Na, b) denoes he normal disribuion wih mean a and variance b. A sandard Brownian moion is defined relaive o a probabiliy measure P, under which he incremens have he properies above. For example, for all < and all h R we have ha ) z z h P < h 1 = Nh) e a2 /2 da, 2π where N ) denoes he cumulaive disribuion funcion for an N0, 1)-disribued random sochasic variable. To be precise, we should use he erm P-sandard Brownian moion, bu he probabiliy 2 For an example see Babbs and Webber 1994).

3.3 Brownian moions 48 measure is ofen clear from he conex. Noe ha a sandard Brownian moion is a Markov process, since he incremen from oday o any fuure poin in ime is independen of he hisory of he process. A sandard Brownian moion is also a maringale, since he expeced change in he value of he process is zero. The name Brownian moion is in honor of he Scoish boanis Rober Brown, who in 1828 observed he apparenly random movemens of pollen submerged in waer. The ofen used name Wiener process is due o Norber Wiener, who in he 1920s was he firs o show he exisence of a sochasic process wih hese properies and who iniiaed a mahemaically rigorous analysis of he process. As early as in he year 1900, he sandard Brownian moion was used in a model for sock price movemens by he French researcher Louis Bachelier, who derived he firs opion pricing formula. The defining characerisics of a sandard Brownian moion look very nice, bu hey have some drasic consequences. I can be shown ha he pahs of a sandard Brownian moion are nowhere differeniable, which broadly speaking means ha he pahs bend a all poins in ime and are herefore sricly speaking impossible o illusrae. However, one can ge an idea of he pahs by simulaing he values of he process a differen imes. If ε 1,..., ε n are independen draws from a sandard N0,1) disribuion, we can simulae he value of he sandard Brownian moion a ime 0 0 < 1 < 2 < < n as follows: z i = z i 1 + ε i i i 1, i = 1,..., n. Wih more ime poins and hence shorer inervals we ge a more realisic impression of he pahs of he process. Figure 3.1 shows a simulaed pah for a sandard Brownian moion over he inerval [0, 1] based on a pariion of he inerval ino 200 subinervals of equal lengh. 3 Noe ha since a normally disribued random variable can ake on infiniely many values, a sandard Brownian moion has infiniely many pahs ha each has a zero probabiliy of occurring. The figure shows jus one possible pah. Anoher propery of a sandard Brownian moion is ha he expeced lengh of he pah over any fuure ime inerval no maer how shor) is infinie. In addiion, he expeced number of imes a sandard Brownian moion akes on any given value in any given ime inerval is also infinie. Inuiively, hese properies are due o he fac ha he size of he incremen of a sandard Brownian moion over an inerval of lengh is proporional o. When is close o zero, is significanly larger han, so he changes are large relaive o he lengh of he ime inerval over which he changes are measured. The expeced change in an objec described by a sandard Brownian moion equals zero and he variance of he change over a given ime inerval equals he lengh of he inerval. This can easily be generalized. As before le z = z ) 0 be a one-dimensional sandard Brownian moion 3 Mos spreadshees and programming ools have a buil-in procedure ha generaes uniformly disribued numbers over he inerval [0, 1]. Such uniformly disribued random numbers can be ransformed ino sandard normally disribued numbers in several ways. One example: Given uniformly disribued numbers U 1 and U 2, he numbers ε 1 and ε 2 defined by ε 1 = 2 ln U 1 sin2πu 2 ), ε 2 = 2 ln U 1 cos2πu 2 ) will be independen sandard normally disribued random numbers. This is he so-called Box-Muller ransformaion. See e.g. Press, Teukolsky, Veerling, and Flannery 1992, Sec. 7.2).

3.3 Brownian moions 49 0.8 0.6 0.4 0.2 0-0.2-0.4-0.6-0.8 0 0.2 0.4 0.6 0.8 1 Figure 3.1: A simulaed pah of a sandard Brownian moion based on 200 subinervals. and define a new sochasic process x = x ) 0 by 3.1) x = x 0 + µ + σz, 0, where x 0, µ, and σ are consans. The consan x 0 is he iniial value for he process x. I follows from he properies of he sandard Brownian moion ha, seen from ime 0, he value x is normally disribued wih mean µ and variance σ 2, i.e. x Nµ, σ 2 ). The change in he value of he process beween wo arbirary poins in ime and, where <, is given by x x = µ ) + σz z ). The change over an infiniesimally shor inerval [, + ] wih 0 is ofen wrien as 3.2) dx = µ d + σ dz, where dz can loosely be inerpreed as a N0, d)-disribued random variable. precise mahemaical meaning, i mus be inerpreed as a limi of he expression To give his a x + x = µ + σz + z ) for 0. The process x is called a generalized Brownian moion or generalized Wiener process. The parameer µ reflecs he expeced change in he process per uni of ime and is called he drif rae or simply he drif of he process. The parameer σ reflecs he uncerainy abou he fuure values of he process. More precisely, σ 2 reflecs he variance of he change in he process per uni of ime and is ofen called he variance rae of he process. σ is a measure for he sandard deviaion of he change per uni of ime and is referred o as he volailiy of he process. A generalized Brownian moion inheris many of he characerisic properies of a sandard Brownian moion. For example, also a generalized Brownian moion is a Markov process, and he

3.3 Brownian moions 50 1,4 1,2 1 0,8 0,6 0,4 0,2 0-0,2-0,4-0,6 0 0,2 0,4 0,6 0,8 1 sigma = 0.5 sigma = 1.0 Figure 3.2: Simulaion of a generalized Brownian moion wih µ = 0.2 and σ = 0.5 or σ = 1.0. The sraigh line shows he rend corresponding o σ = 0. The simulaions are based on 200 subinervals. pahs of a generalized Brownian moion are also coninuous and nowhere differeniable. However, a generalized Brownian moion is no a maringale unless µ = 0. The pahs can be simulaed by choosing ime poins 0 0 < 1 < < n and ieraively compue x i = x i 1 + µ i i 1 ) + ε i σ i i 1, i = 1,..., n, where ε 1,..., ε n are independen draws from a sandard normal disribuion. Figures 3.2 and 3.3 show simulaed pahs for differen values of he parameers µ and σ. The sraigh lines represen he deerminisic rend of he process, which corresponds o imposing he condiion σ = 0 and hence ignoring he uncerainy. Boh figures are drawn using he same sequence of random numbers ε i, so ha hey are direcly comparable. The parameer µ deermines he rend, and he parameer σ deermines he size of he flucuaions around he rend. If he parameers µ and σ are allowed o be ime-varying in a deerminisic way, he process x is said o be a ime-inhomogeneous generalized Brownian moion. In differenial erms such a process can be wrien as defined by 3.3) dx = µ) d + σ) dz. Over a very shor inerval [, + ] he expeced change is approximaely µ), and he variance of he change is approximaely σ) 2. More precisely, he incremen over any inerval [, ] is given by 3.4) x x = µu) du + σu) dz u.

3.4 Diffusion processes 51 1.4 1.2 1 0.8 0.6 0.4 0.2 0-0.2-0.4-0.6 0 0.2 0.4 0.6 0.8 1 sigma = 0.5 sigma = 1.0 Figure 3.3: Simulaion of a generalized Brownian moion wih µ = 0.6 and σ = 0.5 or σ = 1.0. The sraigh line shows he rend corresponding o σ = 0. The simulaions are based on 200 subinervals. The las inegral is a so-called sochasic inegral, which we will define and describe in a laer secion. There we will also sae a heorem, which implies ha, seen from ime, he inegral σu) dz u is a normally disribued random variable wih mean zero and variance σu) 2 du. 3.4 Diffusion processes For boh sandard Brownian moions and generalized Brownian moions, he fuure value is normally disribued and can herefore ake on any real value, i.e. he value space is equal o R. Many economic variables can only have values in a cerain subse of R. For example, prices of financial asses wih limied liabiliy are non-negaive. The evoluion in such variables canno be well represened by he sochasic processes sudied so far. In many siuaions we will insead use so-called diffusion processes. A one-dimensional) diffusion process is a sochasic process x = x ) 0 for which he change over an infiniesimally shor ime inerval [, + d] can be wrien as 3.5) dx = µx, ) d + σx, ) dz, where z is a sandard Brownian moion, bu where he drif µ and he volailiy σ are now funcions of ime and he curren value of he process. 4 This expression generalizes 3.2), where µ and σ were assumed o be consans, and 3.3), where µ and σ were funcions of ime only. An equaion 4 For he process x o be mahemaically meaningful, he funcions µx, ) and σx, ) mus saisfy cerain condiions. See e.g. Øksendal 1998, Ch. 7) and Duffie 2001, App. E).

3.5 Iô processes 52 like 3.5), where he sochasic process eners boh sides of he equaliy, is called a sochasic differenial equaion. Hence, a diffusion process is a soluion o a sochasic differenial equaion. If boh funcions µ and σ are independen of ime, he diffusion is said o be ime-homogeneous, oherwise i is said o be ime-inhomogeneous. For a ime-homogeneous diffusion process, he disribuion of he fuure value will only depend on he curren value of he process and how far ino he fuure we are looking no on he paricular poin in ime we are sanding a. For example, he disribuion of x +δ given x = x will only depend on x and δ, bu no on. This is no he case for a ime-inhomogeneous diffusion, where he disribuion will also depend on. In he expression 3.5) one may hink of dz as being N0, d)-disribued, so ha he mean and variance of he change over an infiniesimally shor inerval [, + d] are given by E [dx ] = µx, ) d, Var [dx ] = σx, ) 2 d, where E and Var denoe he mean and variance, respecively, condiionally on he available informaion a ime he hisory up o and including ime ). To be more precise, he change in a diffusion process over any inerval [, ] is 3.6) x x = µx u, u) du + σx u, u) dz u, where σx u, u) dz u is a sochasic inegral, which we will discuss in Secion 3.6. However, we will coninue o use he simple and inuiive differenial noaion 3.5). The drif rae µx, ) and he variance rae σx, ) 2 are really he limis E [x + x ] µx, ) = lim, 0 σx, ) 2 Var [x + x ] = lim. 0 A diffusion process is a Markov process as can be seen from 3.5), since boh he drif and he volailiy only depend on he curren value of he process and no on previous values. A diffusion process is no a maringale, unless he drif µx, ) is zero for all x and. A diffusion process will have coninuous, bu nowhere differeniable pahs. The value space for a diffusion process and he disribuion of fuure values will depend on he funcions µ and σ. In Secion 3.8 we will give some imporan examples of diffusion processes, which we shall use in laer chapers o model he evoluion of some economic variables. 3.5 Iô processes I is possible o define even more general processes han hose in he class of diffusion processes. A one-dimensional) sochasic process x is said o be an Iô process, if he local incremens are on he form 3.7) dx = µ d + σ dz, where he drif µ and he volailiy σ hemselves are sochasic processes. A diffusion process is he special case where he values of he drif µ and he volailiy σ are given by and x. For a general

3.6 Sochasic inegrals 53 Iô process, he drif and volailiy may also depend on pas values of he x process. I follows ha Iô processes are generally no Markov processes. They are generally no maringales eiher, unless µ is idenically equal o zero and σ saisfies some echnical condiions). The processes µ and σ mus saisfy cerain regulariy condiions for he x process o be well-defined. We will refer he reader o Øksendal 1998, Ch. 4). The expression 3.7) gives an inuiive undersanding of he evoluion of an Iô process, bu i is more precise o sae he evoluion in he inegral form 3.8) x x = µ u du + σ u dz u, where he las erm again is a sochasic inegral. Iô processes ha are no diffusion processes are used in some models of he erm srucure of ineres raes, see e.g. Chapers 10 and 11. 3.6 Sochasic inegrals 3.6.1 Definiion and properies of sochasic inegrals In 3.6) and 3.8) and similar expressions a erm of he form σ u dz u appears. An inegral of his ype is called a sochasic inegral or an Iô inegral. We will only consider sochasic inegrals where he inegraor z is a Brownian moion, alhough sochasic inegrals involving more general processes can also be defined. For given <, he sochasic inegral σ u dz u is a random variable. Assuming ha σ u is known a ime u, he value of he inegral becomes known a ime. The process σ is called he inegrand. The sochasic inegral can be defined for very general inegrands. The simples inegrands are hose ha are piecewise consan. Assume ha here are poins in ime 0 < 1 < < n, so ha σ u is consan on each subinerval [ i, i+1 ). The sochasic inegral is hen defined by 3.9) n 1 ) σ u dz u = σ i zi+1 z i. i=0 If he inegrand process σ is no piecewise consan, a sequence of piecewise consan processes σ 1), σ 2),... exiss, which converges o σ. For each of he processes σ m), he inegral σ m) u dz u is defined as above. The inegral σ u dz u is hen defined as a limi of he inegrals of he approximaing processes: 3.10) σ u dz u = lim σ u m) dz u. m We will no discuss exacly how his limi is o be undersood and which inegrand processes we can allow. Again he ineresed reader is referred o Øksendal 1998). The disribuion of he inegral σ u dz u will, of course, depend on he inegrand process and can generally no be compleely characerized, bu he following heorem gives he mean and he variance of he inegral: Theorem 3.1 The sochasic inegral σ u dz u has he following properies: [ Var [ E σ u dz u ] = 0, σ u dz u ] = E [σ 2 u] du.

3.6 Sochasic inegrals 54 If he inegrand is a deerminisic funcion of ime, σu), he inegral will be normally disribued, so ha he following resul holds: Theorem 3.2 If z is a Brownian moion, and σu) is a deerminisic funcion of ime, he random variable σu) dz u is normally disribued wih mean zero and variance σu) 2 du. Proof: We presen a skech of he proof. Dividing he inerval [, ] ino subinervals defined by he ime poins 0 < 1 < < n, we can approximae he inegral wih he sum n 1 σu) dz u σ i ) ) z i+1 z i. i=0 The incremen of he Brownian moion over any subinerval is normally disribued wih mean zero and a variance equal o he lengh of he subinerval. Furhermore, he differen erms in he sum are muually independen. I is well-known ha a sum of normally disribued random variables is iself normally disribued, and ha he mean of he sum is equal o he sum of he means, which in he presen case yields zero. Due o he independence of he erms in he sum, he variance of he sum is also equal o he sum of he variances, i.e. Var n 1 σ i ) ) ) n 1 z i+1 z i = σ i ) 2 i+1 i ), i=0 which is an approximaion of he inegral σu) 2 du. The resul now follows from an appropriae limi where he subinervals shrink o zero lengh. i=0 Noe ha he process y = y ) 0 defined by y = 0 σ u dz u is a maringale, since [ ] E [y ] = E σ u dz u 0 = E [ 0 = E [ = 0 = y, 0 σ u dz u ] σ u dz u + σ u dz u ] [ ] σ u dz u + E σ u dz u so ha he expeced fuure value is equal o he curren value. 3.6.2 Leibniz rule for sochasic inegrals Leibniz differeniaion rule for ordinary inegrals is as follows: If f, s) is a deerminisic funcion, and we define Y ) = T f, s) ds, hen Y ) = f, ) + T f, s) ds.

3.6 Sochasic inegrals 55 If we use he noaion Y ) = dy d, we can rewrie his resul as ) T df dy = f, ) d +, s) ds d, d and f and formally cancelling he d-erms, we ge = df d dy = f, ) d + T df, s) ds. We will now consider a similar resul in he case where f, s) and, hence, Y ) are sochasic processes. We will make use of his resul in Chaper 10 and only in ha chaper). Theorem 3.3 For any s [ 0, T ], le f s = f s ) [0,s] be he Iô process defined by he dynamics df s = α s d + β s dz, where α and β are sufficienly well-behaved sochasic processes. Then he dynamics of he sochasic process Y = T f s ds is given by [ ) ] T ) T dy = α s ds f d + β s ds dz. Since he resul is usually no included in sandard exbooks on sochasic calculus, a skech of he proof is included. The proof applies he generalized Fubini-rule for sochasic processes, which was saed and demonsraed in he appendix of Heah, Jarrow, and Moron 1992). The Fubini-rule says ha he order of inegraion in double inegrals can be reversed, if he inegrand is a sufficienly well-behaved funcion we will assume ha his is indeed he case. Proof: Given any arbirary 1 [ 0, T ]. Since we ge Y 1 = = T 1 f s 0 ds + T 1 f s 0 ds + = Y 0 + 1 0 1 0 f s 1 = f s 0 + T [ 1 1 1 0 0 1 0 α s d + 1 ] T α s d ds + [ ] T α s ds 1 [ ] T α s ds f s 0 ds 1 [ 1 0 1 d + 0 β s dz, 1 1 d + 0 0 ] β s dz ds [ 0 ] T β s ds dz [ 1 1 [ ] T β s ds dz ] α s ds d 1 [ 1 0 ] β s ds dz [ 1 ] T [ 1 ] T = Y 0 + α s ds d + β s ds dz 0 0 1 1 [ s ] 1 [ s ] f s 0 ds α s d ds β s dz ds 0 0 0 0 0 [ 1 ] T [ 1 ] T 1 = Y 0 + α s ds d + β s ds dz fs s ds 0 0 0 1 = Y 0 + 0 [ ) ] T α s ds f 1 d + 0 [ ] T β s ds dz,

3.7 Iô s Lemma 56 where he Fubini-rule was employed in he second and fourh equaliy. The resul now follows from he final expression. 3.7 Iô s Lemma In our dynamic models of he erm srucure of ineres raes, we will ake as given a sochasic process for he dynamics of some basic quaniy such as he shor-erm ineres rae. Many oher quaniies of ineres will be funcions of ha basic variable. To deermine he dynamics of hese oher variables, we shall apply Iô s Lemma, which is basically he chain rule for sochasic processes. We will sae he resul for a funcion of a general Iô process, alhough we will mos frequenly apply he resul for he special case of a funcion of a diffusion process. Theorem 3.4 Le x = x ) 0 be a real-valued Iô process wih dynamics dx = µ d + σ dz, where µ and σ are real-valued processes, and z is a one-dimensional sandard Brownian moion. Le gx, ) be a real-valued funcion which is wo imes coninuously differeniable in x and coninuously differeniable in. Then he process y = y ) 0 defined by y = gx, ) is an Iô-process wih dynamics g 3.11) dy = x, ) + g x x, )µ + 1 2 ) g 2 x 2 x, )σ 2 d + g x x, )σ dz. The proof is based on a Taylor expansion of gx, ) combined wih appropriae limis, bu a formal proof is beyond he scope of his book. Once again, we refer o Øksendal 1998, Ch. 4) and similar exbooks. The resul can also be wrien in he following way, which may be easier o remember: 3.12) dy = g x, ) d + g x x, ) dx + 1 2 2 g x 2 x, )dx ) 2. Here, in he compuaion of dx ) 2, one mus apply he rules d) 2 = d dz = 0 and dz ) 2 = d, so ha dx ) 2 = µ d + σ dz ) 2 = µ 2 d) 2 + 2µ σ d dz + σ 2 dz ) 2 = σ 2 d. The inuiion behind hese rules is as follows: When d is close o zero, d) 2 is far less han d and can herefore be ignored. Since dz N0, d), we ge E[d dz ] = d E[dz ] = 0 and Var[d dz ] = d) 2 Var[dz ] = d) 3, which is also very small compared o d and is herefore ignorable. Finally, we have E[dz ) 2 ] = Var[dz ] E[dz ]) 2 = d, and i can be shown ha 5 Var[dz ) 2 ] = 2d) 2. For d close o zero, he variance is herefore much less han he mean, so dz ) 2 can be approximaed by is mean d. In Secion 3.8, we give examples of he applicaion of Iô s Lemma. We will use Iô s Lemma exensively hroughou he res of he book. I is herefore imporan o be familiar wih he way i works. I is a good idea o rain yourself by doing he exercises a he end of his chaper. 5 This is based on he compuaion Var[z + z ) 2 ] = E[z + z ) 4 ] E[z + z ) 2 ] ) 2 = 3 ) 2 ) 2 = 2 ) 2 and a passage o he limi.

3.8 Imporan diffusion processes 57 150 140 130 120 110 100 90 80 70 0 0.2 0.4 0.6 0.8 1 sigma = 0.2 sigma = 0.5 Figure 3.4: Simulaion of a geomeric Brownian moion wih iniial value x 0 = 100, relaive drif rae µ = 0.1, and a relaive volailiy of σ = 0.2 and σ = 0.5, respecively. The smooh curve shows he rend corresponding o σ = 0. The simulaions are based on 200 subinervals of equal lengh, and he same sequence of random numbers has been used for he wo σ-values. 3.8 Imporan diffusion processes In his secion we will discuss paricular examples of diffusion processes ha are frequenly applied in modern financial models, as hose we consider in he following chapers. 3.8.1 Geomeric Brownian moions A sochasic process x = x ) 0 is said o be a geomeric Brownian moion if i is a soluion o he sochasic differenial equaion 3.13) dx = µx d + σx dz, where µ and σ are consans. The iniial value for he process is assumed o be posiive, x 0 > 0. A geomeric Brownian moion is he paricular diffusion process ha is obained from 3.5) by insering µx, ) = µx and σx, ) = σx. Pahs can be simulaed by compuing x i = x i 1 + µx i 1 i i 1 ) + σx i 1 ε i i i 1. Figure 3.4 shows a single simulaed pah for σ = 0.2 and a pah for σ = 0.5. For boh pahs we have used µ = 0.1 and x 0 = 100, and he same sequence of random numbers. The expression 3.13) can be rewrien as dx x = µ d + σ dz,

3.8 Imporan diffusion processes 58 which is he relaive percenage) change in he value of he process over he nex infiniesimally shor ime inerval [, + d]. If x is he price of a raded asse, hen dx /x is he rae of reurn on he asse over he nex insan. The consan µ is he expeced rae of reurn per period, while σ is he sandard deviaion of he rae of reurn per period. In his conex i is ofen µ which is called he drif raher han µx ) and σ which is called he volailiy raher han σx ). Sricly speaking, one mus disinguish beween he relaive drif and volailiy µ and σ, respecively) and he absolue drif and volailiy µx and σx, respecively). An asse wih a consan expeced rae of reurn and a consan relaive volailiy has a price ha follows a geomeric Brownian moion. For example, such an assumpion is used for he sock price in he famous Black-Scholes-Meron model for sock opion pricing, cf. Secion 5.7, and a geomeric Brownian moion is also used o describe he evoluion in he shor-erm ineres rae in some models of he erm srucure of ineres rae, cf. Secion 7.6. Nex, we will find an explici expression for x, i.e. we will find a soluion o he sochasic differenial equaion 3.13). We can hen also deermine he disribuion of he fuure value of he process. We apply Iô s Lemma wih he funcion gx, ) = ln x and define he process y = gx, ) = ln x. Since g x, ) = 0, g x x, ) = 1 x, 2 g x 2 x, ) = 1 x 2, we ge from Theorem 3.4 ha dy = 0 + 1 µx 1 ) 1 σ 2 x 2 d + 1 σx dz = µ 12 ) x 2 x σ2 d + σ dz. x 2 Hence, he process y = ln x is a generalized Brownian moion. In paricular, we have y y = µ 12 ) σ2 ) + σz z ), which implies ha ln x = ln x + µ 12 ) σ2 ) + σz z ). Taking exponenials on boh sides, we ge 3.14) x = x exp {µ 12 ) } σ2 ) + σz z ). This is rue for all > 0. In paricular, x = x 0 exp {µ 12 ) } σ2 + σz. Since exponenials are always posiive, we see ha x can only have posiive values, so ha he value space of a geomeric Brownian moion is S = 0, ). Suppose now ha we sand a ime and have observed he curren value x of a geomeric Brownian moion. Which probabiliy disribuion is hen appropriae for he uncerain fuure value, say a ime? Since z z N0, ), we see from 3.14) ha he fuure value x given x ) will be lognormally disribued. The probabiliy densiy funcion for x given by { 1 fx) = x 2πσ 2 ) exp 1 2σ 2 ) given x ) is ) x ln µ 12 ) ) } 2 x σ2 ), x > 0,

3.8 Imporan diffusion processes 59 and he mean and variance are cf. Appendix A. E [x ] = x e µ ), [ ] Var [x ] = x 2 e 2µ ) e σ2 ) 1, The geomeric Brownian moion in 3.13) is ime-homogeneous, since neiher he drif nor he volailiy are ime-dependen. We will also make use of he ime-inhomogeneous varian, which is characerized by he dynamics 3.15) dx = µ)x d + σ)x dz, where µ and σ are deerminisic funcions of ime. Following he same procedure as for he imehomogeneous geomeric Brownian moion, one can show ha he inhomogeneous varian saisfies { 3.16) x = x exp µu) 12 ) } σu)2 du + σu) dz u. According o Theorem 3.2, σu) dz u is normally disribued wih mean zero and variance σu) 2 du. Therefore, he fuure value of he ime-inhomogeneous geomeric Brownian moion is also lognormally disribued. In addiion, we have E [x ] = x e µu) du, Var [x ] = x 2 e 2 µu) du e σu) 2 du. 1 ). 3.8.2 Ornsein-Uhlenbeck processes Anoher sochasic process we shall apply in models of he erm srucure of ineres rae is he so-called Ornsein-Uhlenbeck process. A sochasic process x = x ) 0 is said o be an Ornsein-Uhlenbeck process, if is dynamics is of he form 3.17) dx = [ϕ κx ] d + β dz, where ϕ, β, and κ are consans wih κ > 0. Alernaively, his can be wrien as 3.18) dx = κ [θ x ] d + β dz, where θ = ϕ/κ. An Ornsein-Uhlenbeck process exhibis mean reversion in he sense ha he drif is posiive when x < θ and negaive when x > θ. The process is herefore always pulled owards a long-erm level of θ. However, he random shock o he process hrough he erm β dz may cause he process o move furher away from θ. The parameer κ conrols he size of he expeced adjusmen owards he long-erm level and is ofen referred o as he mean reversion parameer or he speed of adjusmen. To deermine he disribuion of he fuure value of an Ornsein-Uhlenbeck process we proceed as for he geomeric Brownian moion. We will define a new process y as some funcion of x such ha y = y ) 0 is a generalized Brownian moion. I urns ou ha his is saisfied for

3.8 Imporan diffusion processes 60 y = gx, ), where gx, ) = e κ x. From Iô s Lemma we ge [ g dy = x, ) + g x x, ) ϕ κx ) + 1 2 ] g 2 x 2 x, )β 2 This implies ha = [ κe κ x + e κ ϕ κx ) ] d + e κ β dz = ϕe κ d + βe κ dz. Afer subsiuion of he definiion of y and y expression y = y + ϕe κu du + βe κu dz u. d + g x x, )β dz and a muliplicaion by e κ, we arrive a he 3.19) x = e κ ) x + ϕe κ u) du + βe κ u) dz u ) = e κ ) x + θ 1 e κ ) + βe κ u) dz u. This holds for all > 0. In paricular, we ge ha he soluion o he sochasic differenial equaion 3.17) can be wrien as 3.20) x = e κ x 0 + θ 1 e κ) + 0 βe κ u) dz u. According o Theorem 3.2, he inegral βe κ u) dz u is normally disribued wih mean zero and variance ) β 2 e 2κ u) du = β2 2κ 1 e 2κ ). We can hus conclude ha x given x ) is normally disribued, wih mean and variance given by ) E [x ] = e κ ) x + θ 1 e κ ) 3.21), ) Var [x ] = β2 1 e 2κ ) 3.22). 2κ The value space of an Ornsein-Uhlenbeck process is R. For, he mean approaches θ, and he variance approaches β 2 /2κ). For κ, he mean approaches θ, and he variance approaches 0. For κ 0, he mean approaches he curren value x, and he variance approaches β 2 ). The disance beween he level of he process and he long-erm level is expeced o be halved over a period of = ln 2)/κ, since E [x ] θ = 1 2 x θ) implies ha e κ ) = 1 2 and, hence, = ln 2)/κ. The effec of he differen parameers can also be evaluaed by looking a he pahs of he process, which can be simulaed by x i = x i 1 + κ[θ x i 1 ] i i 1 ) + βε i i i 1. Figure 3.5 shows a single pah for differen combinaions of x 0, κ, θ, and β. In each sub-figure one of he parameers is varied and he ohers fixed. The base values of he parameers are x 0 = 0.08, θ = 0.08, κ = ln 2 0.69, and β = 0.03. All pahs are compued using he same sequence of random numbers ε 1,..., ε n and are herefore direcly comparable. None of he pahs shown involve negaive values of he process, bu oher pahs will, see e.g. Figure 3.6. As a maer of

3.8 Imporan diffusion processes 61 0.12 0.1 0.08 0.12 0.1 0.08 0.06 0.04 0.06 0.04 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0.02 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 x0 = 0.06 x0 = 0.08 x0 = 0.12 ka = 0.17 ka = 0.69 ka = 2.77 a) Differen iniial values x 0 b) Differen κ-values; x 0 = 0.04 0.14 0.12 0.1 0.08 0.06 0.04 0.02 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0.14 0.12 0.1 0.08 0.06 0.04 0.02 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 h = 0.04 h = 0.08 h = 0.12 be = 0.01 be = 0.03 be = 0.05 c) Differen θ-values d) Differen β-values Figure 3.5: Simulaed pahs for an Ornsein-Uhlenbeck process. The basic parameer values are x 0 = θ = 0.08, κ = ln 2 0.69, and β = 0.03. fac, i can be shown ha an Ornsein-Uhlenbeck process wih probabiliy one will sooner or laer become negaive. We will also apply he ime-inhomogeneous Ornsein-Uhlenbeck process, where he consans ϕ and β are replaced by deerminisic funcions: 3.23) dx = [ϕ) κx ] d + β) dz = κ [θ) x ] d + β) dz. Following he same line of analysis as above, i can be shown ha he fuure value x given x is normally disribued wih mean and variance given by 3.24) E [x ] = e κ ) x + ϕu)e κ u) du, 3.25) Var [x ] = βu) 2 e 2κ u) du. One can also allow κ o depend on ime, bu we will no make use of ha exension. One of he earlies bu sill frequenly applied) dynamic models of he erm srucure of ineres raes is based on he assumpion ha he shor-erm ineres rae follows an Ornsein-Uhlenbeck

3.8 Imporan diffusion processes 62 process, cf. Secion 7.4. In an exension of ha model, he shor-erm ineres rae is assumed o follow a ime-inhomogeneous Ornsein-Uhlenbeck process, cf. Secion 9.4. 3.8.3 Square roo processes Anoher sochasic process frequenly applied in erm srucure models is he so-called square roo process. A one-dimensional sochasic process x = x ) 0 is said o be a square roo process, if is dynamics is of he form 3.26) dx = [ϕ κx ] d + β x dz = κ [θ x ] d + β x dz, where ϕ = κθ. Here, ϕ, θ, β, and κ are posiive consans. We assume ha he iniial value of he process x 0 is posiive, so ha he square roo funcion can be applied. The only difference o he dynamics of an Ornsein-Uhlenbeck process is he erm x in he volailiy. The variance rae is now β 2 x which is proporional o he level of he process. A square roo process also exhibis mean reversion. A square roo process can only ake on non-negaive values. To see his, noe ha if he value should become zero, hen he drif is posiive and he volailiy zero, and herefore he value of he process will wih cerainy become posiive immediaely afer zero is a so-called reflecing barrier). I can be shown ha if 2ϕ β 2, he posiive drif a low values of he process is so big relaive o he volailiy ha he process canno even reach zero, bu says sricly posiive. 6 Hence, he value space for a square roo process is eiher S = [0, ) or S = 0, ). Pahs for he square roo process can be simulaed by successively calculaing x i = x i 1 + κ[θ x i 1 ] i i 1 ) + β x i 1 ε i i i 1. Variaions in he differen parameers will have similar effecs as for he Ornsein-Uhlenbeck process, which is illusraed in Figure 3.5. Insead, le us compare he pahs for a square roo process and an Ornsein-Uhlenbeck process using he same drif parameers κ and θ, bu where he β- parameer for he Ornsein-Uhlenbeck process is se equal o he β-parameer for he square roo process muliplied by he square roo of θ, which ensures ha he processes will have he same variance rae a he long-erm level. Figure 3.6 compares wo pairs of pahs of he processes. In par a), he iniial value is se equal o he long-erm level, and he wo pahs coninue o be very close o each oher. In par b), he iniial value is lower han he long-erm level, so ha he variance raes of he wo processes differ from he beginning. For he given sequence of random numbers, he Ornsein-Uhlenbeck process becomes negaive, while he square roo process of course says posiive. In his case here is a clear difference beween he pahs of he wo processes. Since a square roo process canno become negaive, he fuure values of he process canno be normally disribued. In order o find he acual disribuion, le us ry he same rick as for he Ornsein-Uhlenbeck process, ha is we look a y = e κ x. By Iô s Lemma, dy = κe κ x d + e κ ϕ κx ) d + e κ β x dz = ϕe κ d + βe κ x dz, 6 To show his, he resuls of Karlin and Taylor 1981, p. 226ff) can be applied.

3.8 Imporan diffusion processes 63 0.11 0.08 0.1 0.09 0.08 0.07 0.06 0.05 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0.06 0.04 0.02 0-0.02 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 OU sq roo OU sq roo a) Iniial value x 0 = 0.08, same random numbers as in Figure 3.5 b) Iniial value x 0 = 0.06, differen random numbers Figure 3.6: A comparison of simulaed pahs for an Ornsein-Uhlenbeck process and a square roo process. For boh processes, he parameers θ = 0.08 and κ = ln 2 0.69 are used, while β is se o 0.03 for he Ornsein-Uhlenbeck process and o 0.03/ 0.08 0.1061 for he square roo process. so ha y = y + ϕe κu du + βe κu x u dz u. Compuing he ordinary inegral and subsiuing he definiion of y, we ge 3.27) x = x e κ ) + ϕ κ ) 1 e κ ) + β e κ u) x u dz u. Since x eners he sochasic inegral we canno immediaely deermine he disribuion of x given x from his equaion. We can, however, use i o obain he mean and variance of x. Due o he fac ha he sochasic inegral has mean zero, cf. Theorem 3.1, we easily ge ) 3.28) E [x ] = e κ ) x + θ 1 e κ ) = ϕ κ + x ϕ) e κ ). To compue he variance we apply he second equaion of Theorem 3.1: ] Var [x ] = Var [β e κ u) x u dz u 3.29) = β 2 = β 2 = β2 ϕ κ e 2κ u) E [x u ] du e 2κ u) ϕ κ + x ϕ) e κu )) du e 2κ u) du + β2 κ = β2 ϕ ) 2κ 2 1 e 2κ ) + β2 κ = β2 x κ x ϕ κ x ϕ κ e κ ) e 2κ ) ) + β2 θ 2κ ) e 2κ +κ e κu du ) e κ ) e 2κ ) ) 1 e κ ) ) 2.

3.9 Muli-dimensional processes 64 Noe ha he mean is idenical o he mean for an Ornsein-Uhlenbeck process, whereas he variance is more complicaed for he square roo process. For, he mean approaches θ, and he variance approaches θβ 2 /2κ). For κ, he mean approaches θ, and he variance approaches 0. For κ 0, he mean approaches he curren value x, and he variance approaches β 2 x ). I can be shown ha, given he value x, he value x wih > is non-cenrally χ 2 -disribued. More precisely, he probabiliy densiy funcion for x is f x x x) = f χ 2 a,b 2cx), where c = 2κ β 2 1 e κ ) ), b = cx e κ ), a = 4ϕ β 2, and where f χ 2 a,b ) denoes he probabiliy densiy funcion for a non-cenrally χ 2 -disribued random variable wih a degrees of freedom and non-cenraliy parameer b. A frequenly applied dynamic model of he erm srucure of ineres raes is based on he assumpion ha he shor-erm ineres rae follows a square roo process, cf. Secion 7.5. Since ineres raes are posiive and empirically seem o have a variance rae which is posiively correlaed o he ineres rae level, he square roo process gives a more realisic descripion of ineres raes han he Ornsein-Uhlenbeck process. On he oher hand, models based on square roo processes are more complicaed o analyze han models based on Ornsein-Uhlenbeck processes. 3.9 Muli-dimensional processes So far we have only considered one-dimensional processes, i.e. processes wih a value space which is R or a subse of R. Some models in he following chapers will involve muli-dimensional processes, which have values in a subse of) R K for some ineger K > 1. A muli-dimensional process can also be considered as a vecor of one-dimensional processes. In his secion we will briefly inroduce some muli-dimensional processes and a muli-dimensional version of Iô s Lemma. A noe on he noaion: vecors are prined in boldface. Marices are indicaed by a double line under he symbol. We rea all vecors as column vecors. ) The symbol denoes ransposiion, a so ha e.g. a, b) represens he column) vecor. If a is a column) vecor, hen a is he b corresponding row vecor. A K-dimensional sandard) Brownian moion z = z 1,..., z K ) is a sochasic process where he individual componens z i are muually independen one-dimensional sandard) Brownian moions. If we le 0 = 0,..., 0) denoe he zero vecor in R K and le I denoe he ideniy marix of dimension K K he marix wih ones in he diagonal and zeros in all oher enries), hen we can wrie he defining properies of a K-dimensional Brownian moion z as follows: i) z 0 = 0, ii) for all, 0 wih < : z z N0, )I) [normally disribued incremens],

3.9 Muli-dimensional processes 65 iii) for all 0 0 < 1 < < n, he random variables z 1 z 0,..., z n z n 1 independen [independen incremens], are muually iv) z has coninuous pahs in R K. Here, Na, b) denoes a K-dimensional normal disribuion wih mean vecor a and variancecovariance marix b. As for sandard Brownian moions, we can also define muli-dimensional generalized Brownian moions, which simply are vecors of independen one-dimensional generalized Brownian moions. A K-dimensional diffusion process x = x 1,..., x K ) is a process wih incremens of he form 3.30) dx = µx, ) d + σx, ) dz, where µ is a funcion from R K R + ino R K, and σ is a funcion from R K R + ino he space of K K-marices. As before, z is a K-dimensional sandard Brownian moion. The evoluion of he muli-dimensional diffusion can also be wrien componenwise as 3.31) dx i = µ i x, ) d + σ i x, ) dz K = µ i x, ) d + σ ik x, ) dz k, i = 1,..., K, k=1 where σ i x, ) is he i h row of he marix σx, ), and σ ik x, ) is he i, k) h enry i.e. he enry in row i, column k). Since dz 1,..., dz K are muually independen and all N0, d) disribued, he expeced change in he i h componen process over an infiniesimal period is E [dx i ] = µ i x, ) d, i = 1,..., K, so ha µ i can be inerpreed as he drif of he i h componen. Furhermore, he covariance beween changes in he i h and he j h componen processes over an infiniesimal period becomes K ) K Cov dx i, dx j ) = Cov σ ik x, ) dz k, σ jl x, ) dz l k=1 l=1 k=1 l=1 K K = σ ik x, )σ jl x, ) Cov dz k, dz l ) = K σ ik x, )σ jk x, ) d k=1 = σ i x, ) σ j x, ) d, i, j = 1,..., K, where we have applied he usual rules for covariances and he independence of he componens of z. In paricular, he variance of he change in he i h componen process of an infiniesimal period is given by K Var [dx i ] = Cov dx i, dx i ) = σ ik x, ) 2 d = σ i x, ) 2 d, i = 1,..., K. k=1 The volailiy of he i h componen is given by σ i x, ). I is clear from hese compuaions ha he elemens of he marix σx, ) deermine all variances and covariances over infiniesimal periods. Noe ha he individual componen processes are generally no muually independen

3.9 Muli-dimensional processes 66 since he drif and volailiy of one componen will generally depend on he values of he oher componens. Also noe ha means, variances, and covariances over non-infiniesimal inervals such as [, ] will generally depend on he enire pah of he process over his inerval. Finally, we sae a muli-dimensional version of Iô s Lemma, where a one-dimensional process is defined as a funcion of ime and a muli-dimensional process. Theorem 3.5 Le x = x ) 0 be an Iô process in R K wih dynamics which componenwise is equivalen o dx = µ d + σ dz, dx i = µ i d + σ i dz = µ i d + K σ ik dz k, i = 1,..., K, where z 1,..., z K are independen sandard Brownian moions, and µ i and σ ik are well-behaved sochasic processes. Le gx, ) be a real-valued funcion for which all he derivaives g, g x, and i are coninuous. Then he process y = y ) 0 defined by is also an Iô process wih dynamics dy = g K x, ) + i=1 3.32) K + i=1 where we have inroduced he noaion k=1 y = gx, ) g x i x, )µ i + 1 2 g x i x, )σ i1 dz 1 + + K i=1 K i=1 j=1 γ ij = σ i1 σ j1 + + σ ik σ jk, which is exacly he covariance beween he processes x i and x j. The resul can also be wrien as 3.33) dy = g x, ) d + K i=1 g x i x, ) dx i + 1 2 K 2 g x, )γ ij d x i x j g x i x, )σ ik dz K, K K i=1 j=1 2 g x i x j 2 g x i x j x, )dx i )dx j ), exis and where in he compuaion of dx i )dx j ) one mus use he rules d) 2 = d dz i = 0 for all i, dz i dz j = 0 for i j, and dz i ) 2 = d for all i. 3.34) Furhermore, he resul can be expressed using vecor and marix noaion: dy = where ) g g x, ) + x x, ) µ + 1 [ 2 ] )) ) 2 r g g σ x 2 x, ) σ d + x x, ) σ dz, g x x, ) g x x 1, ) =..., g x x, ) K 2 g x x 2, ) 1 2 g 2 g x 2 x x 2 x 1 x, ), ) =. 2 g x K x 1 x, ) 2 g x 1 x 2 x, )... 2 g x 1 x K x, ) 2 g x x 2, )... 2 g 2 x 2 x K x, ).....,.. 2 g x K x 2 x, )... 2 g x x 2, ) K

3.10 Change of probabiliy measure 67 and r denoes he race of a quadraic marix, i.e. he sum of he diagonal elemens. For example, ra) = K i=1 A ii. 3.10 Change of probabiliy measure When we represen he evoluion of a given economic variable by a sochasic process and discuss he disribuional properies of his process, we have implicily fixed a probabiliy measure P. For example, when we use he square-roo process x = x ) in 3.26) for he dynamics of a paricular ineres rae, we have aken as given a probabiliy measure P under which he sochasic process z = z ) is a sandard Brownian moion. Since he process x is presumably mean o represen he uncerain dynamics of he ineres rae in he world we live in, we refer o he measure P as he real-world probabiliy measure. Of course, i is he real-world dynamics and disribuional properies of economic variables ha we are ulimaely ineresed in. Neverheless, i urns ou ha in order o compue and undersand prices and raes i is ofen convenien o look a he dynamics and disribuional properies of hese variables assuming ha he world was differen from he world we live in, e.g. if invesors were risk-neural insead of risk-averse. A differen world is represened mahemaically by a differen probabiliy measure. Hence, we need o be able o analyze sochasic variables and processes under differen probabiliy measures. In his secion we will briefly discuss how we can change he probabiliy measure. If he sae space Ω has only finiely many elemens, we can wrie i as Ω = {ω 1,..., ω n }. As before, he se of evens, i.e. subses of Ω, ha can be assigned a probabiliy is denoed by F. Le us assume ha he single-elemen ses {ω i }, i = 1,..., n, belong o F. In his case we can represen a probabiliy measure P by a vecor p 1,..., p n ) of probabiliies assigned o each of he individual elemens: p i = P {ω i }), i = 1,..., n. Of course, we mus have ha p i [0, 1] and ha n i=1 p i = 1. The probabiliy assigned o any oher even can be compued from hese basic probabiliies. For example, he probabiliy of he even {ω 2, ω 4 } is given by P {ω 2, ω 4 }) = P {ω 2 } {ω 4 }) = P {ω 2 }) + P {ω 4 }) = p 2 + p 4. Anoher probabiliy measure Q on F is similarly given by a vecor q 1,..., q n ) wih q i [0, 1] and n i=1 q i = 1. We are only ineresed in equivalen probabiliy measures. In his seing, he wo measures P and Q will be equivalen whenever p i > 0 q i > 0 for all i = 1,..., n. Wih a finie sae space here is no poin in including saes ha occur wih zero probabiliy so we can assume ha all p i, and herefore all q i, are sricly posiive. where We can represen he change of probabiliy measure from P o Q by he vecor ξ = ξ 1,..., ξ n ), ξ i = q i p i, i = 1,..., n. We can hink of ξ as a random variable ha will ake on he value ξ i if he sae ω i is realized. Someimes ξ is called he Radon-Nikodym derivaive of Q wih respec o P and is denoed by dq/dp. Noe ha ξ i > 0 for all i and ha he P-expecaion of ξ is [ ] dq n n E P = E P q i n [ξ] = p i ξ i = p i = q i = 1. dp p i i=1 i=1 i=1

3.10 Change of probabiliy measure 68 Consider a random variable x ha akes on he value x i if sae i is realized. The expeced value of x under he measure Q is given by E Q [x] = n q i x i = i=1 n i=1 p i q i p i x i = n p i ξ i x i = E P [ξx]. Now le us consider he case where he sae space Ω is infinie. Also in his case he change from a probabiliy measure P o an equivalen probabiliy measure Q is represened by a sricly posiive random variable ξ = dq/dp wih E P [ξ] = 1. Again he expeced value under he measure Q of a random variable x is given by E Q [x] = E P [ξx]. In our economic models we will model he dynamics of uncerain objecs over some ime span [0, T ]. For example, we migh be ineresed in deermining bond prices wih mauriies up o T years. Then we are ineresed in he sochasic process on his ime inerval, i.e. x = x ) [0,T ]. The sae space Ω is he se of possible pahs of he relevan processes over he period [0, T ] so ha all he relevan uncerainy has been resolved a ime T and he values of all relevan random variables will be known a ime T. The Radon-Nikodym derivaive ξ = dq/dp is also a random variable and is herefore known a ime T and usually no before ime T. To indicae his he Radon-Nikodym derivaive is ofen denoed by ξ T = dq dp. We can define a sochasic process ξ = ξ ) [0,T ] by seing ξ = E P [dq/dp]. This definiion is consisen wih ξ T being idenical o dq/dp, since all uncerainy is resolved a ime T so ha he ime T expecaion of any variable is jus equal o he variable. Noe ha he process ξ is a i=1 P-maringale, since for any < T we have [ [ ]] dq E P [ξ ] = E P E P = E P dp [ ] dq = ξ. dp Here he firs and he hird equaliies follow from he definiion of ξ. The second equaliy follows from he law of ieraed expecaions, which broadly speaking says ha he expecaion oday of wha we expec omorrow for a given random variable realized laer is equal o oday s expecaion of ha random variable. This is a very inuiive resul. For a more formal saemen and proof, see Øksendal 1998). The following resul urns ou o be very useful in our dynamic models of he economy. Le x = x ) [0,T ] be any sochasic process. Then we have 3.35) E Q [x ] = E P [ ξ ξ x Suppose ha he underlying uncerainy is represened by a sandard Brownian moion z = z ) under he real-world probabiliy measure P), as will be he case in all he models we will consider. Le λ = λ ) [0,T ] be any sufficienly well-behaved sochasic process. 7. Here, z and λ mus have he same dimension. ]. For noaional simpliciy, we assume in he following ha hey are onedimensional, bu he resuls generalize naurally o he muli-dimensional case. We can generae an equivalen probabiliy measure Q λ in he following way. Define he process ξ λ = ξ λ ) [0,T ] by { 3.36) ξ λ = exp 0 λ s dz s 1 2 0 } λ 2 s ds. 7 Basically, λ mus be square-inegrable in he sense ha [ T { 0 λ2 d is finie wih probabiliy 1 and ha λ saisfies Novikov s condiion, i.e. he expecaion E P exp 1 }] T 2 0 λ2 d is finie.

3.10 Change of probabiliy measure 69 Then ξ0 λ = 1, ξ λ is sricly posiive, and i can be shown ha ξ λ is a P-maringale see Exercise 3.5)so ha E P [ξt λ] = ξλ 0 = 1. Consequenly, an equivalen probabiliy measure Q λ can be defined by he Radon-Nikodym derivaive { dq λ T dp = ξλ T = exp λ s dz s 1 } T λ 2 s ds. 2 From 3.35), we ge ha 3.37) E Qλ [x ] = E P [ ξ λ ξ λ x ] [ { = E P x exp 0 0 λ s dz s 1 2 λ 2 s ds for any sochasic process x = x ) [0,T ]. A cenral resul is Girsanov s Theorem: Theorem 3.1 Girsanov) The process z λ = z λ ) [0,T ] defined by 3.38) z λ = z + 0 λ s ds, 0 T, is a sandard Brownian moion under he probabiliy measure Q λ. In differenial noaion, }] dz λ = dz + λ d. If x = x ) is an Iô-process wih dynamics dx = µ d + σ dz, hen dx = µ d + σ dz λ λ d ) = µ σ λ ) d + σ dz λ. Hence, µ σλ is he drif under he probabiliy measure Q λ, which is differen from he drif under he original measure P unless σ or λ are idenically equal o zero. In conras, he volailiy remains he same as under he original measure. In many financial models, he relevan change of measure is such ha he disribuion under Q λ of he fuure value of he cenral processes is of he same class as under he original P measure, bu wih differen momens. For example, consider he Ornsein-Uhlenbeck process dx = ϕ κx ) d + σ dz and perform he change of measure given by a consan λ = λ. Then he dynamics of x under he measure Q λ is given by dx = ˆϕ κx ) d + σ dz λ, where ˆϕ = ϕ σλ. Consequenly, he fuure values of x are normally disribued boh under P and Q λ. From 3.21) and 3.22), we see ha he variance of x and P, bu he expeced values will differ recall ha θ = ϕ/κ): E P [x ] = e κ ) x + ϕ ) 1 e κ ), κ ) 1 e κ ). E Qλ [x ] = e κ ) x + ˆϕ κ given x ) is he same under Q λ However, in general, a shif of probabiliy measure may change no only some or all momens of fuure values, bu also he disribuional class.

3.11 Exercises 70 3.11 Exercises EXERCISE 3.1 Suppose x = x ) is a geomeric Brownian moion, dx = µx d + σx dz. Wha is he dynamics of he process y = y ) defined by y = x ) n? Wha can you say abou he disribuion of fuure values of he y process? EXERCISE 3.2 Adaped from Björk 1998). Define he process y = y ) by y = z 4, where z = z ) is a sandard Brownian moion. Find he dynamics of y. Show ha y = 6 z 2 s ds + 4 0 0 z 3 s dz s. Show ha E[y ] E[z 4 ] = 3 2, where E[ ] denoes he expecaion given he informaion a ime 0. EXERCISE 3.3 Adaped from Björk 1998). Define he process y = y ) by y = e az, where a is a consan and z = z ) is a sandard Brownian moion. Find he dynamics of y. Show ha y = 1 + 1 2 a2 y s ds + a 0 0 y s dz s. Define m) = E[y ]. Show ha m saisfies he ordinary differenial equaion m ) = 1 2 a2 m), m0) = 1. Show ha m) = e a2 /2 and conclude ha E [e az ] = e a2 /2. EXERCISE 3.4 Consider he wo general sochasic processes x 1 = x 1) and x 2 = x 2) defined by he dynamics dx 1 = µ 1 d + σ 1 dz 1, dx 2 = µ 2 d + ρ σ 2 dz 1 + 1 ρ 2 σ 2 dz 2, where z 1 and z 2 are independen one-dimensional sandard Brownian moions. Inerpre µ i, σ i, and ρ. Define he processes y = y ) and w = w ) by y = x 1x 2 and w = x 1/x 2. Wha is he dynamics of y and w? Concreize your answer for he special case where x 1 and x 2 are geomeric Brownian moions wih consan correlaion, i.e. µ i = µ ix i, σ i = σ ix i, and ρ = ρ wih µ i, σ i, and ρ being consans. EXERCISE 3.5 Find he dynamics of he process ξ λ defined in 3.36).

Chaper 4 Asse pricing and erm srucures: discree-ime models 4.1 Inroducion Bonds and oher fixed income securiies have some special characerisics ha make hem disincively differen from oher financial asses such as socks and sock marke derivaives. However, in he end, all financial asses serve he same purpose: shifing consumpion opporuniies hrough ime and saes. Hence, he pricing of fixed income securiies follow he same general principles as he pricing of all oher financial asses. In his chaper and he nex, we will discuss some imporan general conceps and resuls in asse pricing heory and ouline he consequences of hese resuls for he erm srucure of ineres rae and he pricing of fixed income securiies. The fundamenal conceps are arbirage, sae prices, risk-neural probabiliy measures, and marke compleeness. We will also address he imporan links o individuals uiliy maximizaion problems and marke equilibrium. The vas majoriy of modern erm srucure models are formulaed in coninuous ime. This is primarily due o he compuaional advanages offered by he well-developed ools of sochasic calculus for coninuous-ime processes ha ofen leads o relaively simple and explici expressions for he quaniies we are ineresed in. However, in order o develop an undersanding of he key conceps and resuls, i is ofen preferable firs o look a discree-ime models. Hence, in his chaper we will discuss general asse pricing heory and he erm srucure of ineres raes in discree-ime models. The nex chaper gives a similar analysis for he coninuous-ime seing. The simples discree-ime model wih uncerainy is a one-period, finie-sae model. In Secion 4.2 below we will give an inroducion o asse pricing in such a framework. Secion 4.3 exends he analysis o a muli-period, bu sill discree-ime, framework. The presenaion given in hese secions focuses on he conceps and resuls of mos imporance for he erm srucure of ineres raes. Hence, some issues ha are also imporan for undersanding financial economics are ignored. For example, we do no discuss how asse pricing resuls can be formulaed in erms of relaions beween expeced reurns and beas, as is well-known from he Capial Asse Pricing Model. The ideas and resuls addressed were originally inroduced in papers such as Arrow 1951, 1953, 1970), Debreu 1954), Negishi 1960), and Ross 1978). Some exbook presenaions are given by Huang and Lizenberger 1988), Cochrane 2001), and Duffie 2001). Secion 4.4 oulines 71

4.2 A one-period model 72 some discree-ime models of he erm srucure of ineres raes. Finally, Secion 4.5 offers some concluding remarks. 4.2 A one-period model Consider a one-period economy wih uncerainy abou he sae of he economy a he end of he period. We assume ha he sae space is Ω = {1, 2,..., S} so ha here are S possible saes of which one will be realized a he end of he period. In his seing an even is simply a subse of Ω and F is he collecion of all subses of Ω. Le p j > 0 be he real-world probabiliy ha sae j is realized a he end of he period. Of course, we require ha p 1 +... p S = 1. The vecor p = p 1,..., p S ) represens a probabiliy measure P on Ω, F). [Alhough vecors are someimes wrien as row vecors, hink of hem as column vecors.] Any S-dimensional vecor x = x 1,..., x S ) can be hough of as a one-dimensional, real-valued random variable x. The expeced value under he probabiliy measure P) of a random variable x is E[x] = p x = S j=1 p jx j, where is he do produc beween wo vecors. 4.2.1 Asses, porfolios, and arbirage We assume ha N asses are raded. A he end of he period each asse pays a dividend ha may depend on he realized sae. Denoe he dividend of asse i in sae j by D ij. We can represen all he dividends of he risky asses by he N S marix D = [D ij ]. The ranspose is denoed by D. Alernaively, we can hink of he dividend of asse i as a random variable D i and gaher he uncerain dividends of all N asses in he N-dimensional random variable D = D 1,..., D N ). The curren price of asse i is denoed P i and we can gaher all he prices in he vecor P = P 1,..., P N ). The agens can rade he asses a he beginning of he period. A he end of he period hey receive he dividends from heir posiions aken a he beginning of he period. A porfolio is a combinaion of posiions in differen asses. We represen a porfolio by an N-dimensional vecor θ = θ 1,..., θ N ) wih θ i being he number of unis held of asse i. The oal dividend in sae j from a porfolio θ is D θ j = N D ij θ i. i=1 The S-dimensional vecor of porfolio dividends is D θ = D1 θ,..., DS θ ), which can be wrien as D θ = D θ. We can also hink of he porfolio dividend as a random variable D θ = θ D. Le P θ denoe he price of he porfolio θ. We assume ha here are no resricions on he porfolios ha invesors may form and ha here are no rading coss. An arbirage is a porfolio θ saisfying one of he following wo condiions: i) P θ < 0 and D θ 0, ii) P θ 0 and D θ 0 wih a leas one sricly posiive elemen, i.e. D θ j > 0 for some j {1,..., S}. An arbirage offers somehing for nohing. This is clearly aracive o any greedy invesor. Therefore, a marke wih arbirage canno be a marke in equilibrium. We wan o characerize prices

4.2 A one-period model 73 ha do no admi arbirage. An consequence of absence of arbirage is ha prices are linear, i.e. he price of a porfolio is given by N P θ = θ i P i = θ P. i=1 This resul is someimes referred o as he law of one price. To see ha his relaion is implied by absence of arbirage, suppose ha P θ < θ P. Then an arbirage can be formed by purchasing he porfolio θ and, for each j = 1,..., S selling θ j unis of asse j a a uni price of P j. The end-of-period ne dividend from his posiion will be zero no maer which sae is realized. The oal iniial price of he posiion is P θ θ P, which is negaive. Hence, in he absence of arbirage, we canno have ha P θ < θ P. The inequaliy P θ > θ P can be ruled ou by a similar argumen. 4.2.2 Invesors We consider an economy wih a single perishable consumpion good. All asses pay ou in unis of he consumpion good, i.e. we look a a real economy. Consider an individual who has some iniial endowmen of e 0 consumpion goods and is endowed wih some sae-dependen endof-period income of e j consumpion goods, j = 1,..., S. A he beginning of he period he invess in some porfolio θ such ha his end-of-period consumpion in sae j will be c j = e j + D θ j = e j + N D ij θ i. Afer paying for he porfolio, he agen is lef wih a consumpion a he beginning of he period of c 0 = e 0 P θ = e 0 i=1 N θ i P i. Le us wrie c 0, c) for he consumpion plan, where c denoes he sae-dependen fuure consumpion. We assume ha he preferences of he individual is represened by a uiliy funcion U : R + R S + R wih Uc 0, c) = Uc 0, c 1,..., c S ) denoing he uiliy of he consumpion plan c 0, c). We assume ha he invesor is greedy, i.e. U is sricly increasing in all argumens. The uiliy maximizaion problem can now be saed as max θ 1,...,θ N Uc 0, c 1,..., c S ) s.. c j = e j + D θ j = e j + c 0 = e 0 N θ i P i, i=1 c 0, c 1,..., c S 0. i=1 N D ij θ i, j = 1,..., S, If an arbirage porfolio θ exised, he invesor could increase eiher he curren consumpion or he fuure consumpion in some saes)) wihou violaing he budge consrains by acquiring kθ. There will be no finie soluion o he uiliy maximizaion problem. Hence, we assume absence of arbirage. Furhermore, assume ha he non-negaiviy consrain on consumpion is no binding i=1

4.2 A one-period model 74 and ha U is concave such ha we can focus on firs-order condiions. The firs order condiion for θ i is 4.1) P i U c 0 + S j=1 U c j D ij = 0 P i = S j=1 U c j U c 0 D ij, i = 1,..., N. Here he marginal uiliies U c j, j = 0, 1,..., S, are o be evaluaed a he opimal consumpion plan of he agen. Le us look a he special case of ime-addiive expeced uiliy, where here is a von Neumann- Morgensern uiliy funcion u : R + R such ha Uc 0, c 1,..., c S ) = uc 0 ) + E [ e δ uc) ] = uc 0 ) + e δ S j=1 p j uc j ), where δ is a subjecive ime preference rae. The firs-order condiion 4.1) hen implies ha 4.2) P i = S j=1 e δ p j u [ c j ) e δ u ] c) u D ij = E c 0 ) u c 0 ) D i, i = 1,..., N. Again i is imporan o noe ha he marginal uiliies mus be evaluaed a he opimal consumpion plan. Le us look a he firs-order condiion wih ime-addiive expeced uiliy. Assume ha c 0, c 1,..., c S ) denoes he opimal consumpion plan for an agen. Then any deviaion from his plan will give he agen a lower uiliy. One deviaion is obained by invesing in ε > 0 addiional unis of asse i a he beginning of he period. This leaves an iniial consumpion of c 0 εp i. On he oher hand, he end-of-period consumpion in sae j becomes c j + εd ij. We know ha uc 0 εp i ) + e δ S j=1 p j uc j + εd ij ) uc 0 ) + e δ S j=1 uc j ). Subracing he righ-hand side from he lef-hand side and dividing by ε yields uc 0 εp i ) uc 0 ) ε Leing ε go o zero, we obain which implies ha + e δ S P i u c 0 ) + e δ P i e δ j=1 S j=1 S j=1 p j uc j + εd ij ) uc j ) ε p j D ij u c j ) 0, p j D ij u c j ) u c 0 ). On he oher hand, if we consider selling ε > 0 unis of asse i a he beginning of he period, he same reasoning can be used o show ha P i e δ S j=1 p j D ij u c j ) u c 0 ). Hence, he relaion mus hold as an equaliy, jus as in 4.2). 0.

4.2 A one-period model 75 4.2.3 Sae-price vecors and deflaors A sae-price vecor is an S-dimensional vecor ψ = ψ 1,..., ψ S ) which has only sricly posiive elemens and has he propery ha he price of any asse or porfolio wih dividends D 1,..., D S in he S saes is given by S j=1 D jψ j. In paricular, for he N basic asses we have P = Dψ. For any porfolio θ, we have 4.3) P θ = ψ D θ. If we can consruc a porfolio paying 1 in sae j and nohing in all oher saes, he price of his porfolio will be equal o ψ j, hence he name sae price. If we can consruc a riskless dividend of 1 he price mus be equal o S j=1 ψ j so if we le r denoe he coninuously compounded riskless ineres rae over he period, we have ha e r = 1 S j=1 ψ j = 1 ψ 1. A sae-price deflaor is an S-dimensional vecor ζ = ζ 1,..., ζ S ) which has only sricly posiive elemens and has he propery ha for any porfolio θ he price can be wrien as 4.4) P θ = S p j ζ j Dj θ. j=1 We can also hink of a sae-price deflaor ζ as a random variable ζ ha, for each j = 1,..., S, akes on he value ζ j if sae j is realized. Then we can wrie he above equaion as 4.5) P θ = E [ ζd θ]. In paricular for each of he basic N asses, we have P i = E[ζD i ]. In vecor noaion, we have 4.6) P = E [Dζ]. For a riskless porfolio wih a dividend of 1, we have e r = E [ζ]. Some auhors use he name sochasic discoun facor or pricing kernel insead of sae-price deflaor. Given a sae-price vecor ψ, we can consruc a sae-price deflaor ζ = ζ 1,..., ζ S ) by ζ j = ψ j /p j, i.e. he sae price divided by he sae probabiliy. Conversely, we can define a sae-price vecor from any sae-price deflaor, so here is a one-o-one correspondence beween sae-price vecors and sae-price deflaors. Wih infiniely many saes, as considered in he coninuous-ime case, we canno define sae-price vecors, bu we can sill define sae-price deflaors in erms of random variables. Assuming no arbirage, i follows from he firs order condiion of an arbirary greedy and risk-averse invesor, Eq. 4.1), ha a sae-price vecor is defined by ψ = U c c 0, c 1,..., c S ) 1 U c c 0, c 1,..., c S ),..., 0 U c S c 0, c 1,..., c S ) U c 0 c 0, c 1,..., c S ) ).

4.2 A one-period model 76 Here U is he uiliy funcion of an invesor and c 0, c 1,..., c S ) is he opimal consumpion plan for ha invesor. For he special case of ime-addiive von Neumann-Morgensern preferences we ge ha a sae-price vecor is given by e δ p 1 u c 1 ) ψ = u,..., e δ p S u ) c S ) c 0 ) u c 0 ) wih he associaed sae-price deflaor given by he random variable ζ = e δ u c) u c 0 ), which will ake he value ζ j = e δ u c j )/u c 0 ) if sae j is realized. We emphasize ha such a sae-price deflaor can be defined for any individual s uiliy funcion u and opimal consumpion allocaion c 0, c). In general, he erms e δ u c)/u c 0 ) may no be idenical for differen invesors so ha muliple sae-price vecors and deflaors can be consruced in his way. More on his issue laer. We have shown ha when prices admi no arbirage, a leas one sae-price vecor and one sae-price deflaor exis. The converse is also rue as shown by he following argumen. Suppose ha a sae-price vecor ψ exiss. Le θ be a porfolio wih non-negaive dividends, i.e. D θ j 0 for all j = 1,..., S. Since he elemens of he sae-price vecor are sricly posiive, he price of he porfolio, P θ = ψ D θ, will also be non-negaive. If, furhermore, one of he elemens of D θ is sricly posiive, he price will also be sricly posiive. Therefore arbirage is ruled ou. We summarize our findings as follows resul: Theorem 4.1 Prices admi no arbirage if and only if a sae-price vecor or deflaor) exiss. Example 4.1 Consider a one-period economy where wo basic financial asses are raded wihou porfolio consrains or ransacion coss. There are hree equally likely end-of-period saes of he economy and he prices and sae-coningen dividends of he wo asses are given in Table 4.1. By forming porfolios we can generae all dividend vecors of he form θ 1 + 2θ 2, θ 1 + 2θ 2, 2θ 2 ), i.e. of he form a, a, b). A sae price vecor ψ = ψ 1, ψ 2, ψ 3 ) has ψ i > 0 and solves he equaion sysem ) ψ 1 1 0 1 ) 2 2 2 ψ 2 0.5 =. 1.8 ψ 3 The firs of he wo equaions in his sysem requires ha ψ 1 + ψ 2 = 0.5. Subsiuing ha ino he second equaion, we ge ψ 3 = 0.4. The sae price for sae 3 is herefore uniquely deermined, whereas here are infiniely many valid sae prices for he firs wo saes. We can wrie he se of admissible sae-price vecors as {ψ 1, ψ 2, ψ 3 ) ψ 1 = 0.5 ψ 2, 0 < ψ 1 < 0.5, ψ 3 = 0.4}. For each of he admissible sae-price vecors we can deermine a sae-price deflaor ζ by leing ζ j = ψ j /p j, where in his example we have sae probabiliies of p j = 1/3. Consequenly, ζ 3 = 1.2 is uniquely deermined, and ζ 1 = ψ 1 /1/3) = 3ψ 1 = 1.5 3ψ 2 = 1.5 ζ 2. Hence, he se of admissible sae-price deflaors are given by {ζ 1, ζ 2, ζ 3 ) ζ 1 = 1.5 ζ 2, 0 < ζ 1 < 1.5, ζ 3 = 1.2}.

4.2 A one-period model 77 sae-coningen dividend price sae 1 sae 2 sae 3 Asse 1 1 1 0 0.5 Asse 2 2 2 2 1.8 Table 4.1: The prices and sae-coningen dividends of he asses considered in Example 4.1. Only he dividend vecors ha can be generaed by forming porfolios of he wo asses can be uniquely priced wih he available informaion. For example, he dividend vecor 1, 1, 5) has he unique no-arbirage price of 1, 1, 5) ψ 1, ψ 2, ψ 3 ) = ψ 1 + ψ 2 + 5ψ 3, which equals 2.5 for any sae-price vecor. On he oher hand, he dividend vecor 1, 2, 5) is no aainable from he raded asses and canno be uniquely priced. Since 1, 2, 5) ψ 1, ψ 2, ψ 3 ) = [ψ 1 + ψ 2 + 5ψ 3 ] + ψ 2 = 2.5 + ψ 2 and ψ 2 is beween 0 and 0.5, any price beween 2.5 and 3 is consisen wih absence of arbirage. Wha is he inuiion here? Well, he dividend 1, 2, 5) dominaes he dividend 1, 1, 5), which was shown above o have he price 2.5. Consequenly, he price of 1, 2, 5) mus be sricly higher han 2.5. On he oher hand he dividend 1, 2, 5) is dominaed by he dividend 2, 2, 5), which has a unique price of 2ψ 1 + ψ 2 ) + 5ψ 3 = 3. Hence, he price of 1, 2, 5) mus sricly smaller han 3. 4.2.4 Risk-neural probabiliies As discussed earlier a probabiliy measure Q on he sae space Ω = {1, 2,..., S} can be represened by a vecor of sae probabiliies q 1,..., q S ) wih q j 0 for all j = 1,..., S and, of course, S j=1 q j = 1. The expeced value of a random variable x = x 1,..., x S ) under he probabiliy measure Q is given E Q [x] = S j=1 q jx j. We will now inroduce a specific probabiliy measure Q, which is differen from he real-world probabiliy measure P. For a general inroducion o he change of probabiliy measure, see Secion 3.10. Suppose ha a riskless porfolio can be consruced and ha i provides a coninuously compounded rae of reurn over he period given by r. A probabiliy measure Q is called a risk-neural probabiliy measure if he following wo condiions are saisfied: i) P and Q are equivalen, i.e. aach zero probabiliy o he same saes; since we have assumed p j > 0 for all j = 1,..., S, his is he same as requiring ha q j > 0 for all j = 1,..., S, ii) he price of any porfolio θ is given by P θ = E Q [ e r D θ] = e r S j=1 q j D θ j, i.e. he price of any asse or porfolio equals he expeced discouned dividend using he risk-free ineres rae as he discoun rae. If all invesors were risk-neural, hey would exacly value an uncerain dividend by he discouned expeced value. If Q is a risk-neural probabiliy measure, hen he asse prices in he real-world

4.2 A one-period model 78 are jus as hey would have been in an economy in which all agens are risk-neural and he sae probabiliies are given by he q j s. Given a sae-price vecor ψ and he associaed sae-price deflaor ζ, we can consruc a risk-neural probabiliy measure by defining q j = ψ j S k=1 ψ k = e r ψ j = e r p j ζ j for each j = 1,..., S. Then all he q j s are sricly posiive and sum o one. Furhermore, 4.3) implies ha P θ = ψ D θ = S ψ j Dj θ = j=1 so Q is indeed a risk-neural probabiliy measure. S e r q j Dj θ = E Q [ e r D θ], j=1 The change of measure from he real-world probabiliy measure P o he risk-neural probabiliy measure Q is given by he raios q j /p j = e r ζ j. The Radon-Nikodym derivaive dq/dp for he change of measure is hence given by he random variable ξ = dq dp = er ζ. Noe ha q j > p j if and only if ζ j > e r = E[ζ], i.e. if he sae price for sae j is higher han average. Conversely, suppose a risk-neural probabiliy measure Q wih associaed probabiliies q 1,..., q S exiss. Le ξ = dq/dp denoe he Radon-Nikodym derivaive, i.e. ξ j = q j /p j. Then we can consruc a sae-price vecor ψ and a sae-price deflaor ζ as ψ j = e r q j, ζ j = e r ξ j, j = 1,..., S. We summarize hese observaions in he following heorem: Theorem 4.2 Assume ha a riskless asse exiss. Then here is a one-o-one correspondence beween sae-price vecors or deflaors) and risk-neural probabiliy measures. Combining he wo previous heorems, we reach a he nex conclusion. Theorem 4.3 Assume ha a riskless asse exiss. risk-neural probabiliy measure exiss. Prices admi no arbirage if and only if a Example 4.2 In Example 4.1 we derived he admissible sae-price vecors and sae-price deflaors in he marke given in Table 4.1. Clearly, asse 2 is a riskless asse and we see ha e r = 1 0.9 = 10 9 1.1111. To any sae-price deflaor ζ we can associae a risk-neural probabiliy measure Q defined by he Radon-Nikodym derivaive ξ = dq/dp = e r ζ, i.e. ξ j = e r ζ j. Recall ha he real-world sae probabiliies p j are all equal o 1/3. We ge and q 3 = ξ 3 p 3 = e r ζ 3 p 3 = 10 9 1.2 1 3 = 4 9 0.4444 q 1 = ξ 1 p 1 = e r ζ 1 p 1 = 10 9 1.5 ζ 2) 1 3 = 5 9 q 2.

4.2 A one-period model 79 Consequenly, he se of risk-neural probabiliy measures consisen wih he dividends and prices of he wo asses is given by { q 1, q 2, q 3 ) q 1 = 5 9 q 2, 0 < q 1 < 5 9, q 3 = 4 }. 9 4.2.5 Redundan asses Le M denoe he se of dividend vecors ha can be generaed by forming porfolios of he N basic asses, i.e. M = { D θ θ R N }. Noe ha M is a subse of R S. Before we coninue, we need o recall some conceps of linear algebra. Two vecors a and b are called linearly independen if k 1 a + k 2 b = 0 implies k 1 = k 2 = 0, i.e. a and b canno be linearly combined ino a zero vecor. The rows and columns of a marix can be considered as vecors. I can be shown ha for any marix, he maximum number of linearly independen rows and he maximum number of linearly independen columns are he same. We define he rank of a marix as his number. The rank of a k l marix has o be less han or equal o he minimum of k and l. If he rank is equal o he minimum of k and l, he marix is said o be of full rank. An asse is said o be redundan if is dividend vecor can be wrien as a linear combinaion of he dividend vecors of oher asses. Removing he rows corresponding o hose asses from D we obain a marix ˆD of dimension k S, where k S since we canno have more han S linearly independen S-dimensional vecors. Le he k-dimensional vecor ˆP denoe he price vecor of he k remaining asses. Since only redundan asses have been removed we have { M = ˆD ˆθ ˆθ } R k, i.e. we can aain he same dividend vecors by forming porfolios of only he non-redundan asses as by forming porfolios of all he asses. Le us illusrae by a small example. Example 4.3 Consider a one-period economy wih wo possible end-of-period saes and hree raded asses. The dividends are given in Table 4.2. Wih hree asses and wo saes a leas one asse is redundan. The dividend vecor of asse 1 can be wrien as a non-rivial linear combinaion of he dividend vecors of asse 2 and 3 since ) ) ) 3 1 1 = 6 3. 0 1 2 Asse 1 is herefore redundan. Exacly he same se of dividend vecors can be generaed by forming porfolios of asses 2 and 3 as by forming porfolios of all hree asses. On he oher hand, asse 2 is redundan since ) ) ) 1 = 1 3 + 1 1. 1 3 0 2 2 And asse 3 is redundan since ) ) ) 1 = 1 3 1 + 2. 2 3 0 1

4.2 A one-period model 80 sae-coningen dividend sae 1 sae 2 Asse 1 3 0 Asse 2 1 1 Asse 3 1 2 Table 4.2: The sae-coningen dividends of he asses considered in Example 4.3. Hence, eiher of he hree asses can be removed wihou affecing he se of dividend vecors ha can be generaed by forming porfolios. Noe ha once one of he asses has been removed, neiher of he wo remaining asses will be redundan anymore. Wheher an asse is redundan or no depends on he se of oher asses available for rade. If an asse is redundan, is dividend vecor can be replicaed by a cerain porfolio of he oher raded asses. In he absence of arbirage, he price of he redundan asse mus be equal o he price of ha replicaing porfolio. This is he basic idea behind all opion pricing models. Le us illusrae i in a simple one-period binomial model. Example 4.4 Suppose here are wo possible end-of-period saes. Three asses are raded in an arbirage-free marke. Asse 1 is a riskless asse wih a price of 1 and an end-of-period dividend of R = e r. Asse 2 has a price of S and offers a dividend of us in sae 1 and ds in sae 2. To avoid arbirage we mus have ha d < R < u. Asse 3 is a call-opion on asse 2 wih an exercise price of K. The dividend of asse 3 is herefore C u maxus K, 0) in sae 1 and C d maxds K, 0) in sae 2. We can replicae he opion by a porfolio of θ 1 unis of asse 1 and θ 2 unis of asse 2, where θ 1 and θ 2 are chosen so ha Solving hese wo equaions we ge θ 1 R + θ 2 us = C u, θ 1 R + θ 2 ds = C d. θ 1 = R 1 uc d dc u, θ 2 = C u C d u d u d)s. The no-arbirage price of he opion is herefore given by C θ 1 + θ 2 S = 1 R d R u d C u + u R ) u d C d = e r qc u + 1 q)c d ) = e r E Q [C 1 ], where q = R d)/u d) defines he risk-neural probabiliy measure Q, and C 1 denoes he random end-of-period dividend of he opion. 4.2.6 Complee markes The financial marke is said o be complee if M = R S, i.e. any sae-coningen payoff can be generaed by forming porfolios of he raded asses. Oherwise, he marke is called incomplee.

4.2 A one-period model 81 The marke is complee, if and only if for any x R S, we can find θ R N such ha D θ = x. Hence, we have he following resul: Theorem 4.4 The marke is complee if and only if he rank of he N S dividend marix D is equal o S. Clearly, a necessary bu no sufficien) condiion for a complee marke is ha N S, i.e. ha here are a leas as many asses as saes. The pruned dividend marix ˆD will in his case be a non-singular S S marix. We can always define he S-dimensional vecor 4.7) ψ = ˆD ˆD ) 1 ˆD ˆP, which will saisfy ˆDψ = ˆD ˆD ˆD ˆD ) 1 ˆP = ˆP. The redundan asses are uniquely priced by no-arbirage so ha we have P = Dψ. I i clear ha ψ is in fac exacly equal o he dividend generaed by a porfolio ˆθ ) 1 = ˆD ˆD ˆP, i.e. i is a sae-price vecor in he se of aainable dividend vecors, a leas if i has only sricly posiive elemens. In he special case of a complee and arbirage-free marke he elemens of ψ will be sricly posiive. Why? Form a porfolio ha pays off 1 in sae j and 0 in all oher saes. This is possible since he marke is complee. The price of his porfolio is ψj. To avoid arbirage, ψ j mus be sricly posiive. This argumen works for all j = 1,..., S. Hence, ψ is a sae-price vecor. In fac Exercise 4.2 shows ha ψ is he only sae-price vecor when he marke is complee. In he case of a complee marke he marix ˆD is non-singular, as noed above. Using marix algebra we may hen rewrie ψ as 4.8) ψ = ˆD 1 ˆP. Nex, define he S-dimensional vecor ζ by 4.9) ζ = ˆD [ ]) 1 E ˆD ˆD ˆP. To see he meaning of his, le us for simpliciy assume ha none of he basic asses are redundan so ha ζ = D E [DD ]) 1 P. Recall ha D is he N-dimensional random variable for which he j h componen is given by he random dividend of asse j. Hence, DD is an N N marix of random variables wih he i, j) h enry given by D i D j, i.e. he produc of he random dividend of asse i and he random dividend of asse j. The expecaion of a marix of random variables is equal o he marix of expecaions of he individual random variables. So E [DD ] is also an N N marix. For he general case we see from he definiion ha ζ is in fac he dividend vecor generaed by he porfolio ˆθ [ ]) 1 = E ˆD ˆD ˆP of he non-redundan asses. We can hink of ζ as a random variable ζ given by 4.10) ζ = ˆD [ ]) 1 E ˆD ˆD ˆP.

4.2 A one-period model 82 We can see ha E [ ˆDζ ] = E [ ˆD ˆD [ ]) 1 [ [ ]) 1 E ˆD ˆD ˆP] = E ˆD ˆD ] E ˆD ˆD ˆP = ˆP. I follows ha ζ is in fac a sae-price deflaor if i akes sricly posiive values. I can be shown ha no oher sae-price deflaor can be wrien as he dividend of a porfolio of raded asses. In a complee marke, ζ will be a sae-price deflaor and i will be unique. Recall ha here is a one-o-one relaion beween sae-price vecors and sae-price deflaors. In general ζ is no he sae-price deflaor associaed wih ψ. However, his will be so if he marke is complee. In a complee marke i is possible o generae a riskless dividend as a porfolio of he raded asses. Therefore, we can alk of risk-neural probabiliy measures. Due o he one-o-one correspondence beween sae-price vecors/deflaors and risk-neural probabiliy measures, we can conclude ha a complee marke will have a unique risk-neural probabiliy measure. This is he probabiliy measure Q given by he sae probabiliies qj = er ψj, where ψ j is he j h elemen of he unique sae-price vecor ψ. If he marke is incomplee, here will be more han one risk-neural measure. This is he case in Example 4.2. Theorem 4.5 Suppose ha prices admi no arbirage. Then he marke is complee if and only if here is a unique sae-price vecor or deflaor). Equivalenly, he marke is complee if and only if here is a unique risk-neural probabiliy measure. 4.2.7 Equilibrium and represenaive agens in complee markes Consider an economy wih L agens characerized by a uiliy funcion U l and he endowmen e l 0, e l ). The aggregae iniial and fuure endowmens are ē 0 = L e l 0, ē j = l=1 L e l j, j = 1,..., S. Take he raded asses as given. An equilibrium for he economy consiss of a price vecor P and l=1 porfolios θ l, l = 1,..., L, saisfying he wo condiions i) for each l = 1,..., L, θ l is opimal for agen l given P, ii) markes clear, i.e. L l=1 θl i = 0 for all i = 1,..., N. To an equilibrium corresponds an equilibrium consumpion allocaion c l 0, c l ), l = 1,..., L. Clearly, an equilibrium se of prices canno admi arbirage. Wih complee markes he sae-price deflaor is unique. In paricular, for any agens k and l and any sae j, we mus have ha ζ j = e δ k u k ck j ) u k ck 0 ) = e δ l u l cl j ) u l cl 0 ) under he assumpion of ime-addiive expeced uiliy. Noe ha in every sae he raios of marginal uiliies scaled by he ime preference facor) of all agens are idenical. For any wo saes j and j, we ge u k ck j ) u k ck j ) = u l cl j ) u l cl j ).

4.2 A one-period model 83 This equaliy shows ha he aggregae risk in he economy is shared in an efficien way in a complee marke. Suppose ha aggregae consumpion is higher in sae j han in sae j, i.e. C j > C j. Then here mus be some agen, say agen l, who consumes more in sae j han in sae j, c l j > cl j. Consequenly, u l cl j ) < u l cl j ). Bu hen u k ck j ) < u k ck j ) and hus ck j > ck j for all agens k. I follows ha in a complee marke, he opimal consumpion of any agen is an increasing funcion of he aggregae consumpion level, c l = fc) for some increasing funcion f. Individuals fuure consumpion levels move ogeher. Similarly for he curren consumpion level. A consumpion allocaion c 1 0, c 1 ),..., c L 0, c L ) is said o be feasible if L l=1 cl j ēl j for all j = 0, 1,..., S. A consumpion allocaion is called Pareo-opimal if he aggregae endowmen canno be allocaed o consumpion in anoher way ha leaves all agens a leas as good off and some agen sricly beer off. In mahemaical erms a consumpion allocaion c 1 0, c 1 ),..., c L 0, c L ) is Pareo-opimal if i is feasible and here is no oher feasible consumpion plan ĉ 1 0, ĉ 1 ),..., ĉ L 0, ĉ L ) such ha U l ĉ l 0, ĉ l ) U l c l 0, c l ) for all l = 1,..., L wih sric inequaliy for some l. An imporan resul is he Firs Welfare Theorem: Theorem 4.6 If he financial marke is complee, hen every equilibrium consumpion allocaion is Pareo-opimal. Boh for heoreical and pracical applicaions i is very cumbersome o deal wih he individual uiliy funcions and opimal consumpion plans of many differen agens. I would be much simpler if we could jus consider a single agen. As we shall see nex, he assumpion of a single represenaive agen can be jusified in a complee marke. Le η = η 1,..., η L ) be a vecor of sricly posiive numbers. Define he funcion U η : R + R S + R by { L } U η x 0, x) = sup η l U l y0, l y l ) yj 1 +... yj L x j, yj l 0 j = 0, 1,..., S; l = 1,..., L). l=1 We can hink of U η as he uiliy funcion of a single agen wih he η s being weighs on he individual uiliy funcions. The economy is said o have a represenaive agen, if for each equilibrium in he L agen economy here is a vecor η such ha he equilibrium ineres rae and asse prices are he same in he economy wih he single agen wih uiliy U η defined by { L } U η x 0, x) = sup η l U l y0, l y l ) yj 1 +... yj L x j, yj l 0 j = 0, 1,..., S; l = 1,..., L) l=1 and endowmen ē 0, ē) is he same as in he L agen equilibrium. Noe ha in he represenaive agen economy here can be no rade in he financial asses who should be he oher pary in he rade?) and he consumpion of he represenaive agen mus equal he oal endowmen. The nex heorem says ha a represenaive agen exiss if he financial marke is complee. Theorem 4.7 Suppose all individuals are greedy and risk-averse. If he marke is complee, he economy has a represenaive agen. Assuming ha all agens have ime-addiive expeced uiliies, hen he represenaive agen will also have ime-addiive expeced uiliy, i.e. U η x 0, x) = u η,0 x 0 ) + E [u η,1 x)] = u η,0 x 0 ) + S p j u η,1 x j ), j=1

4.3 A muli-period, discree-ime model 84 where { L } u η,0 x 0 ) = sup η l u l y l ) y 1,..., y L > 0 wih y 1 + + y L x, l=1 { L } u η,1 x) = sup e δl η l u l y l ) y 1,..., y L > 0 wih y 1 + + y L x. l=1 Ofen a funcional form for u η,0 and u η,1 is assumed direcly wih he propery ha u η,1 c) = e δ u η,0 c) = e δ uc), where δ can be inerpreed as he average ime preference rae in he economy. Then he unique sae-price deflaor follows by evaluaing he derivaive of u η a he aggregae endowmen, ζ j = u η,1ē j ) u η,0 ē 0) = e δ u ē j ) u ē 0 ). In Chaper 6 we shall apply his echnique o relae he erm srucure of ineres rae o aggregae endowmen/consumpion. 4.3 A muli-period, discree-ime model In a muli-period, discree-ime model we consider prices, dividends, porfolios ec. a counably many poins in ime. We denoe his se of ime poins by T = {0, 1, 2,..., T }, where we assume ha T is some finie ineger represening he las day of he world. 1 We assume a finie sae space Ω, where a sae in his case should be seen as a pah or possible evoluion of he economy hrough all ime poins in T. We can illusrae such a seing by a mulinomial ree. For concreeness look a Figure 4.1 which shows a wo-period ree. Here he dos represen he differen possible saes of he economy a he differen poins in ime. The lines represen hose ransiions of he sae from one poin in ime o he nex ha can occur wih a sricly posiive probabiliy. As always here is no uncerainy abou he sae of he economy a ime 0, so here is only one possible sae, s 0. A ime 1 here are hree possible saes of he world, s 11, s 12, s 13. Given he informaion available a ime 0, here are five possible saes of he world a ime 2, namely s 21, s 22, s 23, s 24, s 25. However, a ime 1 he agens of he economy have received addiional informaion ha may allow hem o rule ou some of he saes. If he agens observe ha he economy is in sae s 11 a ime 1, hen only he saes s 21 and s 22 can be realized wih sricly posiive probabiliy a ime 2. If he economy is in sae s 12 a ime 1, hen s 21, s 23, and s 24 are possible a ime 2. Finally, if he economy is in sae s 13 a ime 1, hen he economy will surely end up in sae s 25 a ime 2. There are six possible evoluions of he economy over he wo periods, i.e. he se Ω has six elemens, which we can wrie as hree-dimensional vecors wih he firs elemen represening he sae of he economy a ime 0, he second elemen he sae a ime 1, and he hird elemen he sae a ime 2. Of course, since we assume ha here is only one possible sae a ime 0, we could leave ou he firs elemen in hese vecors.) In he curren example, we have Ω = {s 0, s 11, s 21 ), s 0, s 11, s 22 ), s 0, s 12, s 21 ), s 0, s 12, s 23 ), s 0, s 12, s 24 ), s 0, s 13, s 25 )}. 1 All he resuls will also hold in he infinie horizon case corresponding o T =, bu only under some addiional echnical condiions ha are no paricularly enlighening and herefore omied from his shor presenaion.

4.3 A muli-period, discree-ime model 85.8 s 21 s 11.2 s 22.3.1 s 0.4 s 12.4 s 23.3.5 s 24 s 13 1.0 s 25 = 0 = 1 = 2 Figure 4.1: An example of a wo-period economy. A probabiliy measure is in his case compleely specified by he ransiion probabiliies beween saes a adjacen daes. Near each line in he figure we have wrien he ransiion probabiliy, i.e. he probabiliy ha he sae of he economy shifs as indicaed by ha line. For example, ps 11, s 22 ) = 0.2 denoes he probabiliy ha he economy shifs from sae s 11 a ime 1 o sae s 22 a ime 2. The lines indicae exacly hose ransiions ha can happen wih sricly posiive probabiliy. The probabiliy of any oucome, i.e. any elemen in Ω, and of any even, i.e. any se of elemens in Ω, can be compued from hese ransiion probabiliies. For example, he probabiliy of he oucome s 0, s 12, s 24 ) is equal o P {s 0, s 12, s 24 )}) = ps 0, s 12 )ps 12, s 24 ) = 0.4 0.5 = 0.2. We will reurn o his example below. I is inuiively clear from Figure 4.1 ha a muli-period model can be seen as a sequence of one-period economies. Hence, mos of he basic resuls from he one-period seing carries over o he muli-period seing. 4.3.1 Asses, rading sraegies, and arbirage Prices of financial asses change over ime in a non-deerminisic way, so we will model prices by sochasic processes. The price process of asse i is denoed by P i = P i ) T. Financial asses are characerized by he paymens or dividends hey enile heir owners o. We model he dividends of an asse by a sochasic process D i = D i ) T. D i represens he dividend paymen of asse i a ime. We use he convenion ha P i is he ex-dividend price a ime, i.e. he price of all dividends sricly laer han. The cum-dividend ime price is hen P i + D i. Given ha here are no dividends afer ime T, we mus have P it = 0. Since we wan o derive prices of fuure dividends we can safely assume ha here are no dividends a ime 0. We assume ha here are N asses available for rading. We collec he prices and dividend processes for he N asses in

4.3 A muli-period, discree-ime model 86 N-dimensional processes P = P ) T and D = D ) T wih P = P 1,..., P N ) and similarly for D. A porfolio is a posiion in he N asses, which we can represen mahemaically by an N-vecor wih elemen i denoing he number of unis of asse i. A rading sraegy is a porfolio for each poin in ime, i.e. an N-dimensional sochasic process θ = θ ) T where θ = θ 1,..., θ N ) denoes he porfolio held a ime or raher immediaely afer rading a ime. An invesor following a rading sraegy θ will have a porfolio given by θ 1 immediaely before ime, receive dividends θ 1 D a ime, and hen rebalance he porfolio o θ immediaely afer ime. The ne gain a ime is herefore equal o 4.11) D θ = θ 1 D θ θ 1 ) P = θ 1 P + D ) θ P, = 1, 2,.... The iniial price of he rading sraegy is P θ = θ 0 P 0. We can le D0 θ = P θ so ha he dividend process is defined a all T. An arbirage is a rading sraegy θ ha generaes a dividend process ha is non-negaive and wih a sricly posiive probabiliy of having a posiive value a a leas) one poin in ime. An arbirage can have a zero iniial dividend, i.e. a zero price, and always non-negaive fuure payoffs wih a possibiliy of a sricly posiive payoff or a sricly posiive iniial dividend negaive iniial price) and zero fuure payoffs or boh a posiive iniial dividend negaive price) and non-negaive, bu possibly posiive fuure payoffs. In any case an arbirage gives somehing for nohing. Prices ha allow arbirages o be consruced canno represen an equilibrium. If prices do no admi arbirage, hen we have linear pricing, i.e. he law of one price is valid. The price of a package of asses is equal o he sum of he prices of he individual asses in he package. 4.3.2 Invesors Invesors use he financial marke o shif consumpion across ime and saes. Le us focus on invesors wih ime-addiive expeced uiliy. Suppose an agen has an iniial endowmen of e 0 consumpion unis and a fuure ime- and sae-dependen endowmen or income wih e denoing he random endowmen received a ime. Sanding a ime 0 he problem of an individual invesor can herefore be wrien as max uc 0 ) + θ T e δ E[uc )] =1 s.. c 0 = e 0 θ 0 P 0, c = e + D θ, = 1,..., T, c 0, c 1,..., c T 0. Here c denoes he random, i.e. sae-dependen, consumpion a ime. Applying 4.11), we can also wrie he ime consumpion as c = e + θ 1 P + D ) θ P.

4.3 A muli-period, discree-ime model 87 I can be shown ha he firs-order condiion wih respec o θ 0 implies ha he ime 0 prices are given by he vecor P 0 = e δ E [u c 1 )P 1 + D 1 )] u c 0 ) [ e δ u ] c 1 ) = E u P 1 + D 1 ). c 0 ) Similarly, he uiliy maximizaion problem ha he agen faces a ime implies ha [ e δ u ] c +1 ) 4.12) P = E u P +1 + D +1 ), = 0, 1,..., T 1. c ) This is he muli-period equivalen of 4.2). For an individual asse, say asse j, his reads as [ e δ u ] c +1 ) 4.13) P j = E u P j,+1 + D j,+1 ). c ) Noe again ha c and c +1 in hese expressions are he opimal consumpion raes of he agen. We can jusify 4.13) by considering a perurbaion of he opimal sraegy of he agen. By invesing in ε > 0 exra unis of asse j a ime and reselling he same number of unis of he asse a ime + 1, he agen can shif consumpion from o + 1. A all oher daes he consumpion rae will be unchanged. Since c and c +1 are opimal, he perurbed sraegy will generae a lower uiliy so ha which implies ha u c εp j ) + e δ E [u c +1 + ε[p j,+1 + D j,+1 ])] uc ) + e δ E [uc +1 )], u c εp j ) uc ) ε Leing ε 0, we ge ha and hence [ ] u + e δ c+1 + ε[p j,+1 + D j,+1 ]) uc +1 ) E 0. ε P j u c ) + e δ E [u c +1 ) P j,+1 + D j,+1 )] 0, P j e δ E [ u c +1 ) u c ) ] P j,+1 + D j,+1 ). The reverse inequaliy can be shown by he perurbaion where consumpion is shifed from + 1 o by invesing less in asse j a ime. 4.3.3 Sae-price vecors and deflaors We can define a sae-price vecor for each of he one-period models ha make up a muliperiod model. In his way we can define wha we could call a sae-price vecor process for he muli-period model. For he example in Figure 4.1, we can define a sae-price vecor ψ 11 for he one-period submodel ha has s 11 as he sae of he economy a he beginning of he period and s 21 and s 22 as he possible end-of-period saes. Noe ha ψ 11 is a wo-dimensional vecor. Similarly, we can define a hree-dimensional sae-price vecor ψ 12 for he one-period submodel wih s 12 as he iniial sae and a one-dimensional sae-price vecor ψ 13 for he one-period submodel wih s 13 as he iniial sae. Finally, we can define ψ 0 as he sae-price vecor for he firs of he wo subperiods. In his way we have consruced a sae-price vecor sochasic process ψ = ψ ) =0,1 or in general ψ = ψ ) =0,1,...,T 1. Seen from ime 0 he sae-price vecor for he firs period

4.3 A muli-period, discree-ime model 88 is known, bu he sae-price vecors for he fuure periods are random, i.e. dependen on which sae is realized. We will no presen a formal definiion of a sae-price vecor process and, in fac, we will no discuss use his concep in he following. This is mainly because he relaed concep of a sae-price deflaor is more racable in muli-period models and in he infinie sae, coninuous-ime model we are heading o, i is no even possible o define a sae-price vecor process. A sae-price deflaor is a sricly posiive sochasic process ζ = ζ ) T wih ζ 0 = 1 and he propery ha for any rading sraegy θ, he sae-price deflaed gain process G θ,ζ = G θ,ζ defined by G θ,ζ which implies ha = P θ ζ + s=1 Dθ s ζ s is a maringale. In paricular, for <, G θ,ζ P θ ζ + P θ ζ + s=1 4.14) P θ i = E D θ s ζ s = E P θ ζ ζ + s=1 D θ ζ s s ζ s=+1 Ds θ ζ s,. = E[G θ,ζ Noe ha for any sae-price deflaor hese relaions hold for all asses and porfolios, i.e. all price and dividend pairs. A paricularly simple version of 4.14) occurs for = + 1: [ 4.15) P θ ζ+1 = E P θ ζ +1 + D θ ) ] +1. Anoher special case is obained for = T, using PT θ = 0: [ T ] 4.16) P θ = E Ds θ ζ s. ζ s=+1 Comparing his o 4.5), we can see ha our muli-period definiion of a sae-price deflaor generalizes our one-period definiion. ) ], i.e. If we can consruc a zero-coupon bond ha pays off only a one poin in ime, say, and he payoff is D θ = 1, he price a ime is given by 4.17) B = E [ ζ ζ In muli-period seings a sae-price deflaor is someimes referred o as a sae-price densiy, a sochasic discoun facor, a maringale pricing process, a pricing kernel, or relaed names. These names are someimes also used for he raios ζ s /ζ. Similarly o Theorem 4.1 we have he following resul: Theorem 4.8 Prices admi no arbirage if and only if a sae-price deflaor exiss. Le us ouline he proof. If here is no arbirage, we can consruc a sae-price deflaor from he firs-order condiion of any agen s uiliy maximizaion problem. Comparing 4.15) wih 4.12), we see ha we can define a sae-price deflaor by ]. ζ 0 = 1, ζ +1 = e δ u c +1 ) ζ u, c )

4.3 A muli-period, discree-ime model 89 where c ) =0,1,...,T is he opimal consumpion plan of he agen. Conversely, given a sae-price deflaor ζ, he price of a rading sraegy θ equals [ T ] P θ = E ζ D θ. =1 Recall ha ζ is sricly posiive for all. If D θ is non-negaive, he price P θ mus be non-negaive. If here is a sricly posiive probabiliy of obaining a sricly posiive dividend a some poin in ime, hen he expecaion on he righ-hand side and herefore he price mus be sricly posiive. Hence, no arbirages can be consruced. 4.3.4 Risk-neural probabiliy measures Now assume ha for any sae of he economy a any poin in ime i is possible o consruc a riskless one-period invesmen. We denoe by r he coninuously compounded rae of reurn per period on a riskless invesmen beween ime and + 1. In oher words, a riskless invesmen of 1 a ime grows o e r a ime + 1. Here r may be sochasic, i.e. i may depend on he sae of he world a ime. Generally he value of r is no known unil ime. If we inves 1 a ime 0 in he firs-period riskless asse, we have e r0 a ime 1. If we inves ha in he second-period riskless asse, we will have e r0 e r1 = e r0+r1 a ime 2. If we coninue his roll-over sraegy, we will have e r0+r1+ +r 1 a ime. We can hink of his invesmen opporuniy as an asse wih a dividend process D = D ) and a price process A = A ) given by D 1 = = D T 1 = 0, D T = e r0+r1+ +rt 1, A 0 = 1, A = e r0+r1+ +r 1, = 1,..., T 1, A T = 0. We refer o his asse as he bank accoun. Alernaively, we could hink of his invesmen opporuniy as an asse wih periodic dividends D = e r 1 1 and a consan price A = 1, = 0, 1,..., T 1.) Noe ha for a sae-price deflaor ζ we have for any < T ha [ ] ζ e r0+r1+...r 1 = E e r0+r1+ +r ζ 1, and, consequenly, 4.18) 1 = E [ ζ ζ e r+r+1+ +r 1 A probabiliy measure Q is said o be a risk-neural probabiliy measure for he given marke) if i) Q is equivalen o P, and ii) for any asse i, he discouned gain process Ḡi = Ḡi) defined by ]. is a Q-maringale. Ḡ i = P i e 1 s=0 rs + s=1 D is e s 1 v=0 rv

4.3 A muli-period, discree-ime model 90 In paricular, if Q is a risk-neural probabiliy measure, hen for <, we have Ḡi = E Q [Ḡi ], i.e. P i e 1 s=0 rs + which implies ha s=1 P i e 1 s=0 rs = E Q D is e s 1 v=0 rv = E Q P i e 1 P i e 1 s=0 rs + s=0 rs + s=+1 s=1 D is e s 1 v=0 rv, D is e s 1 v=0 rv, or, equivalenly, 4.19) P i = E Q P i e 1 s= rs + s=+1 D is e s 1 v= rv. Wih = + 1, his reduces o 4.20) P i = e r E Q [P i,+1 + D i,+1 ]. Wih = T and P it = 0, we ge 4.21) P i = E Q [ T s=+1 ] D is e s 1 v= rv. If we can idenify a risk-neural probabiliy measure and compue he righ-hand side expecaion in our hese equaions, we have he price of he asse. For a zero-coupon bond wih a uni paymen a ime, we can wrie he price a ime < as 4.22) B = E Q [e 1 s= rs ]. Theorem 4.9 Assume ha a riskless invesmen exiss. A risk-neural probabiliy measure can be consruced from any sae-price deflaor. Conversely, a sae-price deflaor can be consruced from any risk-neural probabiliy measure. Proof: Suppose ζ is a sae-price deflaor. Define he random variable ξ T = ζ T e r0+r1+ +rt 1. Noe ha ξ T is non-negaive and, from 4.18) wih = 0 and = T, we ge E[ξ T ] = 1. Therefore we can define an equivalen probabiliy measure Q by he Radon-Nikodym derivaive dq/dp = ξ T. Define ξ = E [ξ T ]. From 4.18) wih = T we have ha [ ] ζt r+ +rt 1 1 = E e ζ and hence [ ] [ E [ξ T ] = E ζt e r0+ +rt 1] ζt = ζ e r0+ +r 1 r+...rt 1 E e ζ so ha 4.23) ξ = ζ e r0+ +r 1, = 0, 1,..., T. = ζ e r0+ +r 1,

4.3 A muli-period, discree-ime model 91 We now have ha E Q P i e 1 s= rs + s=+1 D is e s 1 v= rv = E ξ T P i e = E ξ 1 s= rs + s=+1 1 P i e 1 s= rs E [ξ T ] + 1 ξ ξ = E ξ P i e ξ = E = P i, ζ ζ P i + 1 s= rs + s=+1 ζ s s=+1 D is ζ D is e s 1 v= rv s=+1 D is e s 1 v= rv E s [ξ T ] ξ s D is e s 1 v= rv ξ as was o be shown. Here he firs equaliy is due o he shif of measure which implies ha [ ] E Q ξt [X] = E X ξ for any random variable X, cf. Secion 3.10; he second equaliy is due o he law of ieraed expecaions; he hird equaliy is due o he definiion of ξ for any ; he fourh equaliy comes from insering 4.23); and he final equaliy holds since ζ is a sae-price deflaor. Conversely, suppose Q is a risk-neural probabiliy measure. Then some sricly posiive random variable ξ T exiss such ha dq/dp = ξ T. The process ζ = ζ ) defined by ζ = E [ξ T ]e r0+ +r 1) is hen a sae-price deflaor. The following resul now follows immediaely from Theorems 4.8 and 4.9. Theorem 4.10 Assume ha a riskless invesmen exiss. Prices admi no arbirage if and only if a risk-neural probabiliy measure exiss. 4.3.5 Redundan asses In he muli-period model an asse is said o be redundan if is dividend process can be generaed by a rading sraegy in he oher asses. Jus as in he one-period seing, redundan asses are uniquely priced by no-arbirage. The simples example is a wo-period binomial model. Example 4.5 Consider a wo-period binomial arbirage-free model in which he price processes of wo raded asses are shown in Figure 4.2. The firs asse is riskless and yields a gross reurn of R = e r in each period. The second asse has an iniial price of S and in each period he prices is eiher muliplied by a facor u or a facor d. We can eiher assume ha he values shown a he end of he second period are dividends or prices ha have been deermined from a larger model

4.3 A muli-period, discree-ime model 92 wih addiional periods following he wo shown here. In addiion o hese wo asses, a hird asse is raded. This is a European call-opion on he second asse wih expiraion a = 2 and an exercise price of K. We can replicae and hence price his asse by repeaedly using he idea from he one-period binomial model in Example 4.4. The hree possible dividends of he opion are C uu = maxu 2 S K, 0), C ud = C du = maxuds K, 0), and C dd = maxd 2 S K, 0). If, a ime 1, he price of asse 2 is us hen we can replicae he subsequen dividend of he opion by a porfolio of θ 1u unis of asse and θ 2u unis of asse 2, where which implies ha θ 1u R 2 + θ 2u u 2 S = C uu, θ 1u R 2 + θ 2u uds = C ud, θ 1u = R 2 uc ud dc uu, θ 2u = C uu C ud u d u d)us. The unique no-arbirage price of he opion a ime 1 in he up-sae is equal o he cos of seing up his porfolio, i.e. C u = θ 1u R + θ 2u us = 1 R R d u d C uu + u R ) u d C ud = e r qc uu + 1 q)c ud ), where q = R d)/u d) as in he one-period binomial model. Similarly, i can be shown ha he unique no-arbirage price of he opion a ime 1 in he down-sae is C d = e r qc ud + 1 q)c dd ). Seing up a replicaing porfolio over he firs period is exacly as in he one-period example. Consequenly, he unique no-arbirage price of he opion a ime 0 is C = e r qc u + 1 q)c d ) = e 2r q 2 C uu + 2q1 q)c ud + 1 q) 2 C dd ) = e 2r E Q [C 2 ], where Q is he risk-neural probabiliy measure defined hrough q and C 2 denoes he random erminal dividend of he opion. 4.3.6 Complee markes In his muli-period seing he marke is called complee if for any process x = x ) here is a rading sraegy θ such ha D θ = x for all. The idea is he same as in he single-period model. In a complee marke any dividend process can be generaed by some rading sraegy in he given asses. In he one-period model Theorem 4.4 ells us ha wih S possible saes we need a leas S sufficienly differen asses for he marke o be complee. To generalize his resul o he muli-period seing we mus be careful. Consider once again he wo-period model in Figure 4.1. Here here are six possible oucomes, i.e. Ω has six elemens. Hence, one migh hink ha we need access o rade in a leas six sufficienly differen asses in order for he marke o be complee. This is no correc. We can do wih fewer asses. This is based on wo observaions: i) he uncerainy is no revealed compleely a once, bu lile by lile, and ii) we can rade dynamically in he asses. In he example here are hree possible ransiions of he economy from ime 0 o ime 1. From our one-period analysis we know ha hree sufficienly differen asses are enough o span his uncerainy. From ime 1 o ime 2 here are eiher wo, hree, or one

4.3 A muli-period, discree-ime model 93 R, us) R 2, u 2 S) 1, S) R 2, uds) R, ds) R 2, d 2 S) = 0 = 1 = 2 Figure 4.2: An wo-period binomial model for derivaive pricing. possible ransiion of he economy, depending on which sae he economy is in a ime 1. A mos, we need hree sufficienly differen asses o span he uncerainy over his period. In oal, we can generae any dividend process if we jus have access o hree sufficienly differen asses in boh periods. As in he one-period model, sufficienly differen means ha he marix of he dividends of he asses in a given period is of full rank. Analogous o he one-period model i is possible o define a sochasic process ζ = ζ ) =0,1,...,T wih ζ 0 = 1 such ha i) a rading sraegy exiss ha generaes a dividend process equal o ζ, and ii) he deflaed gain process G θ,ζ is a maringale for any rading sraegy θ. Consequenly, if ζ is sricly posiive, i will be a valid sae-price deflaor. If he marke is complee, his will be he case, and ζ will be he unique sae-price deflaor. Furhermore, he probabiliy measure Q associaed wih ζ will be he unique risk-neural probabiliy measure. Theorem 4.11 Suppose prices admi no arbirage. Then markes are complee if and only if here is a unique risk-neural probabiliy measure and, hence, a unique sae-price deflaor. 4.3.7 Equilibrium and represenaive agens in complee markes The definiions of equilibrium, Pareo-opimaliy, and represenaive agens in he muli-period model are sraighforward generalizaions of he one-period definiions. We have he same basic resuls: Theorem 4.12 If he financial marke is complee, hen every equilibrium consumpion allocaion is Pareo-opimal. Theorem 4.13 Suppose all individuals are greedy and risk-averse. If he marke is complee, he economy has a represenaive agen. If he marke is complee, he unique sae-price deflaor is given by he marginal uiliy of he represenaive agen evaluaed a he aggregae endowmen.

4.4 Discree-ime, finie-sae models of he erm srucure 94 4.4 Discree-ime, finie-sae models of he erm srucure The vas majoriy of dynamic erm srucure models ha have been suggesed and analyzed in he lieraure is formulaed in coninuous ime. As we shall see in he analysis of he coninuousime models in laer chapers no all bu many) relevan quaniies can be derived in he form of a nice closed-form expression. For example, in none of hese models is i possible o wrie up a simple closed-form equaion for he value of American-syle derivaives. Ofen such values can be compued by a simple, alhough cumbersome, procedure in a discree-ime model such as a binomial ree. A frequenly applied way o price such asses is o consruc a binomial or rinomial ree ha in some sense provides a close approximaion o he coninuous-ime model and hen price he asses in he ree using a backward recursive echnique. Hence, i is indeed imporan o know how o price asses in discree-ime, finie-sae models. Some erm srucure models have been formulaed originally in discree ime, e.g. Ho and Lee 1986), Pedersen, Shiu, and Thorlacius 1989), and Black, Derman, and Toy 1990) have developed binomial erm srucure models. These models can also be seen as discree-ime approximaions o coninuous-ime models. Le us focus on a simple binomial erm srucure model. I follows from 4.22) ha in order o compue he prices of zero-coupon bond and hence any defaul-free bond, we jus need o know he dynamics of he one-period or shor-erm ) ineres rae r under a risk-neural probabiliy measure Q. Binomial erm srucure models herefore sar by assuming a binomial model for he evoluion of he one-period ineres rae. Figure 4.3 gives an example of a hree-period binomial ree for he ineres rae. Near each line beween wo nodes he risk-neural ransiion probabiliy is shown. Here we wan o focus on how o use a ree for pricing so we ake hese risk-neural probabiliies as given. 7%.4 6%.5.6 5%.6.5 5%.6.4.7.4 4%.3.8 3%.2 = 0 = 1 = 2 = 3 Figure 4.3: An hree-period binomial erm srucure model. Le us sar by pricing a one-period zero-coupon bond which gives a dividend of 1 a he end

4.5 Concluding remarks 95 of he firs-period no maer which sae he economy is in. We denoe he price of his bond by B 1 0. From 4.22) we have ha B 1 0 = E Q [ e r0] = e r0 = e 0.05 0.9512. Now le us look a a zero-coupon bond mauring a ime 2. We can find he curren price B 2 0 by working backwards. Firs we find he wo possible prices a ime 1. If he shor rae is 6% a ime 1, he price will be B 2 1,u = e 0.06 0.9418. If he shor rae a ime 1 is 4%, he price will be B 2 1,d = e 0.04 0.9608. Using 4.20) we can compue he curren 2-period bond price as B 2 0 = e r0 E Q [ B 2 1] = e 0.05 0.6 0.9418 + 0.4 0.9608) 0.9307, which corresponds o a coninuously compounded yield of ln B 2 0)/2 5.098%. In a similar recursive manner we can compue he price of a hree-period zero-coupon bond. We ge a price of B 3 0 0.8561 corresponding o a yield of 5.180%. So from he shor rae ree we can generae a whole yield curve a ime 0. Similarly, we can generae a yield curve for each of he fuure nodes of he ree. We can also use he ree o price oher fixed-income securiies. 4.5 Concluding remarks This chaper has inroduced general asse pricing conceps and relaion in a discree-ime seing and discussed he consequences for he modeling of he erm srucure of ineres raes. We have focused on discree-ime models wih finiely many saes, bu he resuls generalize o discree-ime models wih infiniely many saes. Some auhors prefer o work wih such models, bu hey can again be seen as approximaions of coninuous-ime models. For an overview of discree-ime erm srucure models, see Backus, Foresi, and Telmer 1998). If you suppor he view ha coninuous-ime models are easier o inerpre, undersand, and apply for pricing and hedging, his chaper is mainly giving some imporan inuiion for he coninuous-ime models ha we sar working wih in he nex chaper. However, as menioned earlier, even if you prefer coninuous-ime models, you may end up using an approximaing discree-ime model o compue some ineresing quaniies. 4.6 Exercises EXERCISE 4.1 Consider a one-period, hree-sae economy wih wo asses raded. Asse 1 has a price of 0.9 and pays a dividend of 1 no maer wha sae is realized. Asse 2 has a price of 2 and pays 1, 2, and 4 in sae 1, 2, and 3, respecively. The real-world probabiliies of he saes are 0.2, 0.6, and 0.2, respecively. Assume absence of arbirage. a) Find he sae-price vecor ψ defined in 4.7). Wha porfolio generaes a dividend of ψ? Wha is he sae-price deflaor corresponding o ψ? b) Find he sae-price deflaor ζ defined in 4.9). Wha porfolio generaes a dividend of ζ? c) Find he risk-neural probabiliy measure associaed wih ψ. Wha is he sae-price vecor corresponding o ζ? d) Find a porfolio wih dividends 4, 3, and 1 in saes 1, 2, and 3, respecively. Wha is he price of he porfolio?

4.6 Exercises 96 EXERCISE 4.2 Suppose he marke is complee and arbirage-free. Show ha ψ defined in Secion 4.2.6 is he only sae-price vecor. Hin: Suppose ha ψ and ψ are wo sae-price vecors. Since he marke is complee we mus have ha why?) ψ ψ = ψ ψ and ψ ψ = ψ ψ. Use hese equaliies o show ha ψ = ψ. EXERCISE 4.3 Suppose η 1 and η 2 are sricly posiive numbers and ha u 1c) = u 2c) = c 1 γ /1 γ) for any non-negaive real number c. Define he funcion u η : R + R by u ηx) = sup {η 1u 1y 1) + η 2u 2y 2) y 1 + y 2 x; y 1, y 2 0}. Show ha u ηx) = kx 1 γ /1 γ) for some consan k. Wha is he implicaion for he uiliy of represenaive agens?

Chaper 5 Asse pricing and erm srucures: an inroducion o coninuous-ime models 5.1 Inroducion In he previous chaper we inroduced he conceps of no arbirage, sae-price deflaors, and risk-neural probabiliies in a discree-ime seing. Among he key findings were a one-o-one relaionship beween sae-price deflaors and risk-neural probabiliy measures, absence of arbirage is equivalen o he exisence of a sae-price deflaor and a risk-neural probabiliy measure, in a complee arbirage-free marke, here is a unique sae-price deflaor and a unique risk-neural probabiliy measure. The sae-price deflaor can be represened in erms of a represenaive agen s marginal uiliy evaluaed a he aggregae consumpion level. In Secion 5.2 we will generalize hese conceps and relaions o a coninuous-ime seing. Up o some echnical condiions, he same resuls hold as in he discree-ime seing. In he discree-ime modeling of he previous chaper we showed how risk-neural probabiliy measures can be used o capure he pricing mechanism of he marke. We saw ha an asse can be priced by compuing a cerain risk-neural expecaion involving he fuure shor-erm ineres raes and he dividends of he asse. Secion 5.3 shows ha even oher probabiliy measures will be helpful in he compuaion of he prices of derivaive asses. In Secion 5.4 we discuss he implicaions of he general asse pricing heory for he pricing of forwards and fuures. We give a general characerizaion of boh he forward price and he fuures price and discuss he difference beween hese prices. In Secion 5.5 we will briefly describe and discuss he special feaures of American-syle derivaives, i.e. derivaives ha enile he owner o exercise he conrac a all poins in ime before he final expiraion dae, or a leas a several pre-specified poins in ime. Closed-form expressions for he prices of American-syle derivaives are only presen in he very rivial cases where premaure exercise can be ruled ou. Hence, in general, one mus resor o numerical mehods o compue boh prices and hedge raions for American-syle derivaives. The asse pricing relaions in Secion 5.2 were developed under quie general assumpions on he ineres raes and he expeced reurns and volailiy erms of he raded asses. In Secion 5.6 97

5.2 Asse pricing in coninuous-ime models 98 we will specialize his framework by assuming ha he values of ineres raes, expeced reurns, and volailiies are given as funcions of some common sae variable and ha his sae variable follows a diffusion process. The sae variable may be one- or muli-dimensional. We will refer o such models as diffusion models. The main advanage of his added srucure is ha he price of any given asse in a broad class of asses can now be wrien as some funcion of he sae variable and ime, and ha his funcion can be compued by solving a parial differenial equaion. Someimes i is easier o solve parial differenial equaions han o compue he necessary riskneural expecaions involving he fuure asse dividends and shor-erm ineres raes. All he classic and also many modern dynamic erm srucure models are diffusion models so his secion also serves as an inroducion o many concree erm srucure models. The bes known model for he pricing of derivaive securiies is he Black-Scholes-Meron model developed by Black and Scholes 1973) and Meron 1973). The model was originally creaed for he pricing of sock opions, bu a varian of he model he so-called Black model inroduced by Black 1976)) has laer been applied o he pricing of many oher derivaive securiies, including some of he ineres rae dependen securiies discussed in Chaper 2. In Secion 5.7 we will briefly go hrough he Black-Scholes-Meron model and Black s varian and sae Black s formula for he pricing of seleced fixed income securiies. However, we will also poin ou ha he assumpions underlying Black s model are compleely unaccepable when i comes o ineres rae dependen securiies. Secion 5.8 gives an overview of he concree coninuous-ime erm srucure models we will describe in deail in Chapers 7 11. The disincion beween absolue pricing models and relaive pricing models is discussed. We will also ouline some crieria for he choice of which of he numerous erm srucure models o use. 5.2 Asse pricing in coninuous-ime models The cenral conceps of no arbirage, sae-price deflaors, and risk-neural probabiliy measures can easily be generalized o coninuous-ime models. In discree-ime models, absence of arbirage, exisence of a sae-price deflaor, and exisence of a risk-neural probabiliy measure were shown o be equivalen. This is almos rue in coninuous-ime, bu here are some suble deails. I is easy o show ha he exisence of eiher a sae-price deflaor or a risk-neural probabiliy measure will imply absence of arbirage. The one-o-one correspondence beween sae-price deflaors and risk-neural probabiliy measures coninues o hold, a leas under echnical condiions. However, absence of arbirage does no imply he exisence of a sae-price deflaor or a risk-neural probabiliy measure. Loosely speaking, we need o rule ou boh arbirage and near arbirage rading sraegies o ensure ha a risk-neural measure exiss. However, if he asse prices are well-behaved in a cerain sense, we can indeed consruc a risk-neural probabiliy measure. So he basic message is ha for well-behaved markes, no arbirage is sill equivalen o he exisence of a risk-neural probabiliy measure and o he exisence of a sae-price deflaor. Furhermore, in an arbirage-free complee marke he risk-neural probabiliy measure is unique and so is he sae-price deflaor. The presenaion below is srucured very similarly o ha of Duffie 2001, Ch. 6), bu many echnical deails are lef ou here. These deails are no imporan for a basic undersanding of

5.2 Asse pricing in coninuous-ime models 99 coninuous-ime asse pricing models. Many of he definiions and resuls in he coninuous-ime framework are originally due o Harrison and Kreps 1979) and Harrison and Pliska 1981, 1983). We assume ha he basic uncerainy in he economy is represened by he evoluion of a d- dimensional sandard Brownian moion, z = z ) [0,T ]. Think of dz as d exogenous shocks o he economy a ime. All he uncerainy ha affecs he invesors sems from hese exogenous shocks. This includes financial uncerainy, i.e. uncerainy abou he evoluion of prices and ineres raes, fuure expeced reurns, volailiies, and correlaions, bu also non-financial uncerainy, e.g. uncerainy abou prices of consumpion goods and uncerainy abou fuure labor income of he agens. The sae space Ω is in his case he se of all pahs of he Brownian moion z. Noe ha since a Brownian moion has infiniely many possible pahs, we have an infinie sae space. 5.2.1 Asses, rading sraegies, and arbirage For noaional simpliciy we shall firs develop he main resuls for he case where he available asses pay no dividends before ime T, leing P T denoe he ime T dividend and, consequenly, cum-dividend price. Laer we will discuss he necessary modificaions in he presence of inermediae dividends. We model a financial marke wih one insananeously riskless and N risky asses. The insananeously riskless asse is simply he coninuous-ime version of he bank accoun process. Le r denoe he coninuously compounded, insananeously riskless ineres rae a ime, i.e. he rae of reurn over an infiniesimal inerval [, + d] is r d. The bank accoun coninuously rolls over such insananeously riskless invesmens. Le A denoe he price process of he bank accoun, hen da = A r d, or, equivalenly, A = A 0 e 0 ru du. We need o assume ha he process r = r ) is such ha T 0 r d is finie wih probabiliy one. Noe ha he bank accoun is only insananeously riskless since fuure ineres raes are generally no known. We refer o r as he shor-erm ineres rae or simply he shor rae. Some auhors use he phrase spo rae o disinguish his rae from forward raes. If he zero-coupon yield curve a ime is given by τ y +τ for τ > 0, we can hink of r as he limiing value lim τ 0 y +τ, which corresponds o he inercep of he yield curve and he verical axis in a τ, y)-diagram. The shor rae is sricly speaking a zero mauriy ineres rae. The mauriy of he shores governmen bond raded in he marke may be several monhs, so ha i is impossible o observe he shor rae direcly from marke prices. The shor rae in he bond markes can be esimaed by he inercep of a yield curve esimae, which can be obained by he mehods discussed in Secion 1.6 on page 14. In he money markes, raes are se for deposis and loans of very shor mauriies, ypically as shor as one day. While his is surely a reasonable proxy for he zero-mauriy ineres rae in he money markes, i is no necessarily a good proxy for he riskless governmen bond) shor rae. The reason is ha money marke raes apply for unsecured loans beween financial insiuions and hence hey reflec he defaul risk of hose invesors. herefore expeced o be higher han similar bond marke raes. Money marke raes are

5.2 Asse pricing in coninuous-ime models 100 The prices of he N risky asses are modeled as general Iô processes, cf. Secion 3.5. The price process P i = P i ) of he i h risky asse is assumed o be of he form d dp i = P i µ i d + σ ij dz j. Here µ i = µ i ) denoes he relaive) drif, and σ ij = σ ij ) reflecs he sensiiviy of he relaive price o he j h exogenous shock. We can wrie he price dynamics of all he N risky asses compacly in vecor noaion as dp = diagp ) [ ] µ d + σ dz, j=1 where P = P 1 P 2. P N µ = P 1 0... 0 0 P 2... 0, diagp ) =......,.. 0 0... P N σ 11 σ 12... σ 1d σ 21 σ 22... σ 2d, σ =........ σ N1 σ N2... σ Nd µ 1 µ 2. µ N We assume ha he processes µ i and σ ij are well-behaved, e.g. generaing prices wih finie variances. The economic inerpreaion of µ i is he expeced rae of reurn per ime period year) over he nex insan. The marix σ deermines he insananeous variances and covariances and, hence, also he correlaions) of he risky asse prices. In paricular, σ σ d is he N N variance-covariance marix of he raes of reurns over he nex insan [, + d]. A rading sraegy is a pair α, θ), where α = α ) is a real-valued process represening he unis held of he insananeously riskless asse and θ is an N-dimensional process represening he unis held of he N risky asses. To be precise, θ = θ 1,..., θ N ), where θ i = θ i ) wih θ i represening he unis of asse i held a ime. The value of a rading sraegy a ime is given by V α,θ = α A + θ P. The gains from he rading sraegy over an infiniesimal inerval [, + d] is α da + θ dp = α r e 0 rs ds d + θ dp. A rading sraegy is called self-financing if he fuure value is equal o he sum of he iniial value and he accumulaed rading gains so ha no money has been added or wihdrawn. In erms of mahemaics, a rading sraegy α, θ) is self-financing if or, in differenial erms, V α,θ = V α,θ 0 + 5.1) dv α,θ 0 α s r s e s 0 ru du ds + θ s dp s ) = α r e 0 ru du d + θ dp.

5.2 Asse pricing in coninuous-ime models 101 An arbirage is a self-financing rading sraegy α, θ) saisfying one of he following wo condiions: i) V α,θ 0 < 0 and V α,θ T 0 wih probabiliy one, ii) V α,θ 0 0, V α,θ T 0 wih probabiliy one, and V α,θ T > 0 wih sricly posiive probabiliy. This definiion also covers shorer erm riskless gains. Suppose for example ha we can consruc a rading sraegy wih a non-posiive iniial value i.e. a non-posiive price), always non-negaive values, and a sricly posiive value a some ime < T. Then his sricly posiive value can be invesed in he bank accoun in he period [, T ] generaing a sricly posiive erminal value. In a coninuous-ime seing i is possible o consruc some sraegies ha generae somehing for nohing. These are he so-called doubling sraegies. Think of a series of coin osses n = 1, 2,.... The n h coin oss akes place a ime 1 1/n + 1). In he n h oss, you ge α2 n 1 if heads comes up, and looses α2 n 1 oherwise. You sop being he firs ime heads comes up. Suppose heads comes up he firs ime in oss number k + 1). Then in he firs k osses you have los a oal of α1 + 2 + + 2 k 1 ) = α2 k 1). Since you win α2 k in oss number k + 1, your oal profi will be α2 k α2 k 1) = α. Since he probabiliy ha heads comes up evenually is equal o one, you will gain α wih probabiliy one. Similar sraegies can be consruced in coninuous-ime models of financial markes, bu are clearly impossible o implemen in real life. These sraegies can be ruled ou by requiring ha rading sraegies have values ha are bounded from below, i.e. ha some consan K exiss such ha V α,θ K for all. This is a reasonable resricion since no one can borrow an infinie amoun of money. If you have a limied borrowing poenial, he doubling sraegy described above canno be implemened. 5.2.2 Invesors In a coninuous-ime seing i is naural o assume ha each agen consumes according o a nonnegaive coninuous-ime process c = c ) and ha he life-ime uiliy from a given consumpion process is of he ime-addiive form E[ T 0 e δ uc ) d]. In his case c is he consumpion rae a ime, i.e. i is measured as he number of consumpion goods consumed per ime period. The oal number of unis of he good consumed over an inerval [, + ] is + c s ds which for small is approximaely equal o c. As in he discree-ime models he agens can shif consumpion across ime and saes by applying appropriae rading sraegies. We will no specify in mahemaical deails he decision problem of an agen in coninuous ime. In he discree-ime seing we derived he imporan equaion 4.13) ha relaes curren prices o marginal uiliies of consumpion and fuure prices and dividends. Le us consruc a similar heurisic argumen in he coninuous-ime seing. Suppose c = c ) is he opimal consumpion process for some agen. Any deviaion from his sraegy will generae a lower uiliy. One deviaion occurs if he agen a ime 0 increases his invesmen in asse j by ε unis. The exra coss of εp j0 implies a reduced consumpion now. Le us suppose ha he agen finances his exra invesmen by cuing down his consumpion rae in he ime inerval [0, ] for some small posiive by εp j0 /. The exra ε unis of asse j is resold a ime < T, yielding a revenue of εp j. This finances an increase in he consumpion rae over [, + ] by εp j /. Since we have assumed so far ha he asses pay no dividends before

5.2 Asse pricing in coninuous-ime models 102 ime T, he consumpion raes ouside he inervals [0, ] and [, + ] will be unaffeced. Given he opimaliy of c = c ), we mus have ha [ E e u δs c s εp ) ) j0 uc s ) ds + 0 Dividing by ε and leing ε 0, we obain [ E P j0 Leing 0, we arrive a 0 + e δs u c s ) ds + P j e u δs c s + εp ) ) ] j uc s ) ds 0. + e δs u c s ) ds E [ P j0 u c 0 ) + P j e δ u c ) ] 0, ] 0. or, equivalenly, P j0 u c 0 ) E [ e δ P u c ) ]. The reverse inequaliy can be shown similarly so ha we have ha P j0 u c 0 )) E[e δ P u c )] or more generally [ ] 5.2) P j = E e δ ) u c ) u c ) P j, T. Wih inermediae dividends his relaion is slighly differen. 5.2.3 Sae-price deflaors A sae-price deflaor is a sricly posiive process ζ = ζ ) wih ζ 0 = 1 and he propery ha he produc of he sae-price deflaor and he price of an asse is a maringale, i.e. ζ P i ) is a maringale for any i = 1,..., N and ζ exp{ 0 r u du}) is a maringale. In paricular, for all < T, we have or P i ζ = E [P i ζ ], [ ] ζ P i = E P i. ζ Due o he lineariy of he value of a rading sraegy, he same relaion holds for any self-financing rading sraegy: [ ] V α,θ ζ = E V α,θ. ζ Le us wrie he dynamics of a sae-price deflaor as dζ = ζ [m d + v dz ] for some relaive drif m and some sensiiviy vecor v. Define ζ = ζ A = ζ exp{ 0 r u du}. By Iô s Lemma, dζ = ζ [m + r ) d + v dz ]. Since ζ = ζ ) is a maringale, we mus have m = r, i.e. he relaive drif of a sae-price deflaor is equal o he negaive of he shor-erm ineres rae. We will laer give a characerizaion of he sensiiviy vecor v. Also in coninuous-ime models, he exisence of a sae-price deflaor implies ha prices admi no arbirage, as saed in he following heorem.

5.2 Asse pricing in coninuous-ime models 103 Theorem 5.1 If a sae-price deflaor exiss, prices admi no arbirage. The proof is basically he same as in he discree-ime case. Proof: The following argumen ignores he lower bound on he value process for a rading sraegy. Consul Duffie 2001, p. 105) o see how o incorporae he lower bound. I is no erribly difficul, bu involves local maringales and super-maringales which we will no discuss here. Suppose α, θ) is a self-financing rading sraegy wih V α,θ T ] [ V α,θ 0 = E ζ T V α,θ T 0. Then mus be non-negaive. If, furhermore, here is a posiive probabiliy of V α,θ T being sricly posiive, hen V α,θ 0 mus be sricly posiive. Consequenly, arbirage is ruled ou. In he discree-ime models we showed ha absence of arbirage implies he exisence of a saeprice deflaor. We did ha by observing ha in he absence of arbirage he opimal consumpion sraegy of any agen is finie and well-defined and he marginal rae of ineremporal subsiuion of he agen can hen be used as a sae-price deflaor. In coninuous-ime models we can see from 5.2) ha ζ = e δ u c )/u c 0 ) is a good candidae for a sae-price deflaor whenever he opimal consumpion process c of he agen is well-behaved, as i presumably will be in he absence of arbirage. However, here are some echnical subleies one mus consider when going from no arbirage o he exisence of a sae-price deflaor. We will very briefly address his issue in a laer subsecion. 5.2.4 Risk-neural probabiliy measures For our marke wih no inermediae dividends, a probabiliy measure Q is said o be a riskneural probabiliy measure if he following hree condiions are saisfied: i) Q is equivalen o P, ii) for any asse, he discouned price process P i = P i exp{ 0 r s ds} is a Q-maringale, iii) he Radon-Nikodym derivaive dq/dp has finie variance. In paricular, if Q is a risk-neural probabiliy measure, hen P i = E Q [e ] r s ds P i for any <. Le us check wheher he same relaion holds for any self-financing rading sraegy α, θ), i.e. where V α,θ V α,θ = E Q [ ] V α,θ, = V α,θ exp{ 0 r s ds} for all. Firs, we noe he following lemma: Lemma 5.1 A rading sraegy α, θ) is self-financing if and only if V α,θ = V α,θ 0 + 0 θ s d p s.

5.2 Asse pricing in coninuous-ime models 104 α,θ Proof: Wha we aim o show is ha d V = θ dp if and only if 5.1) holds. Firs noe ha Iô s Lemma implies ha he dynamics of he discouned prices is given by Rearranging we ge d P = diag P ) [ µ r 1) d + σ dz ]. 5.3) d P = diag P ) [ µ d + σ dz ] r P d = e 0 ru du diagp ) [ µ d + σ dz ] r e 0 ru du P d = e 0 ru du dp r e 0 ru du P d. Similarly, he dynamics of he discouned value of a rading sraegy is α,θ d V = e 0 ru du dv α,θ r V α,θ e 0 ru du d = e 0 ru du dv α,θ r α + e ) 0 ru du θ P = e 0 ru du dv α,θ r α d θ Using 5.3), we can rewrie he las erm and we obain d r e ) 0 ru du P d α,θ d V = e 0 ru du dv α,θ r α d e 0 ru du θ dp + θ d P. α,θ Consequenly, d V = θ d P if and only if he firs hree erms on he righ-hand side yield zero, which is exacly when he rading sraegy is self-financing. The quesion is herefore wheher or no 5.4) E Q [ θ s d P s ] = 0 holds. This seems reasonable, since expeced price changes are zero under Q, bu cerain echnical resricions on θ need o be imposed, cf. Duffie 2001, p. 109). Analogous o he discree-ime models, we have he following resul: Theorem 5.2 If a risk-neural probabiliy measure exiss, prices admi no arbirage. Proof: Suppose α, θ) is a self-financing rading sraegy saisfying echnical condiions ensuring ha 5.4) holds. Then V α,θ 0 = E Q [ e T 0 r d V α,θ T Noe ha if V α,θ T is non-negaive wih probabiliy one under he real-world probabiliy measure P, hen i will also be non-negaive wih probabiliy one under a risk-neural probabiliy measure Q since Q and P are equivalen. We see from he equaion above ha if V α,θ T is non-negaive, so is V α,θ 0. If, in addiion, V α,θ T is sricly posiive wih a sricly posiive possibiliy, hen V α,θ 0 mus be sricly posiive again using he equivalence of P and Q). Arbirage is ruled ou. ]. In discree-ime models we have a one-o-one relaionship beween risk-neural probabiliy measures and sae-price deflaors: Given a risk-neural probabiliy measure, we can consruc a sae-price deflaor, and conversely. This is also rue in coninuous-ime under a echnical condiion as he following heorem shows.

5.2 Asse pricing in coninuous-ime models 105 Theorem 5.3 Given a risk-neural probabiliy measure Q. Le ξ = E [dq/dp] and define ζ = ξ exp{ 0 r s ds}. If ζ has finie variance for all T, hen ζ = ζ ) is a sae-price deflaor. Conversely, given a sae-price deflaor ζ, define ξ = exp{ 0 r s ds}ζ. If ξ T has finie variance, hen a risk-neural probabiliy measure Q is defined by dq/dp = ξ T. Proof: Suppose ha Q is a risk-neural probabiliy measure. The change of measure implies ha E [ζ s P is ] = e [ 0 ru du E ξ s P is e s ru du] = e 0 ru du ξ E Q [P is e s ru du] = e 0 ru du ξ P i = ζ P i, where he second equaliy follows from 3.35). Hence, ζ is a sae-price deflaor. The finie variance condiion on ζ and he finie variance of prices) ensure he exisence of he expecaions. Conversely, suppose ha ζ is a sae-price deflaor and define ξ as in he saemen of he heorem. Then E[ξ T ] = E [e ] T 0 rs ds ζ T = 1, where he las equaliy is due o he fac ha he produc of he sae-price deflaor and he bank accoun value is a maringale. Furhermore, ξ T is sricly posiive so dq/dp = ξ T defines an equivalen probabiliy measure Q. By assumpion ξ T has finie variance. I remains o check ha discouned prices are Q-maringales. Again using 3.35), we ge E Q [e ] [ ] [ ] r s ds ξ P i = E e r s ds ζ P i = E P i ξ ζ so his condiion is also me. Hence, Q is a risk-neural probabiliy measure. = P i, 5.2.5 From no arbirage o sae-price deflaors and risk-neural measures Theorem 4.8 esablished ha in discree-ime models no-arbirage is equivalen o he exisence of a sae-price deflaor and hence o he exisence of a risk-neural probabiliy measure. However, in coninuous-ime models absence of arbirage does no imply he exisence of a sae-price deflaor or a risk-neural measure. We mus require a lile more han absence of arbirage. As shown by Delbaen and Schachermayer 1994, 1999) he condiion ha prices admi no free lunch wih vanishing risk is equivalen o he exisence of a risk-neural probabiliy measure and hence, following Theorem 5.3, he exisence of a sae-price deflaor. We will no go ino he precise and very echnical definiion of a free lunch wih vanishing risk. Jus noe ha while an arbirage is a free lunch wih vanishing risk, here are rading sraegies which are no arbirages bu neverheless are free lunches wih vanishing risk. More imporanly, we will see in he nex secion ha in markes wih sufficienly nice price processes, we can indeed consruc a risk-neural probabiliy measure. 5.2.6 Marke prices of risk If Q is a risk-neural probabiliy measure, he discouned prices are Q-maringales. The discouned risky asse prices are given by P = P e 0 rs ds.

5.2 Asse pricing in coninuous-ime models 106 An applicaion of Iô s Lemma shows ha he dynamics of he discouned prices is 5.5) d P = diag P ) [ µ r 1) d + σ dz ]. Suppose ha Q is a risk-neural probabiliy measure and le ξ = E [dq/dp]. Then i follows from he Maringale Represenaion Theorem, see e.g. Øksendal 1998, Thm. 4.3.4), ha a process λ = λ ) exiss such ha or, equivalenly using ξ 0 = E[dQ/dP ] = 1), { 5.6) ξ = exp 1 2 dξ = ξ λ dz, 0 λ s 2 ds 0 λ s dz s }. According o Girsanov s Theorem, i.e. Theorem 3.1, he process z Q = z Q ) defined by 5.7) dz Q = dz + λ d, z Q 0 = 0, is hen a sandard Brownian moion under he Q-measure. Subsiuing his ino he dynamics of discouned prices, we obain 5.8) d P = diag P ) [ µ r 1 σ λ ) d + σ dz Q If discouned prices are o be Q-maringales, we mus have ha 5.9) σ λ = µ r 1. From hese argumens i follows ha he exisence of a soluion λ o his sysem of equaions is a necessary condiion for he exisence of a risk-neural probabiliy measure. Noe ha he sysem has N equaions one for each asse) in d unknowns, λ 1,..., λ d one for each exogenous shock). On he oher hand, if a soluion λ exiss and saisfies cerain echnical condiions, hen a riskneural probabiliy measure Q is defined by dq/dp = ξ T, where ξ T is obained by leing = T T in 5.6). The echnical condiions are ha ξ T has finie variance and ha exp 0 λ d} 2 has finie expecaion. The laer condiion is Novikov s condiion which ensures ha he process ξ = ξ ) is a maringale.) We summarize hese findings as follows: Theorem 5.4 If a risk-neural probabiliy measure exiss, here mus be a soluion o 5.9) for all. If a soluion λ exiss for all and he process λ = λ ) saisfies echnical condiions, hen a risk-neural probabiliy measure exiss. Any process λ = λ ) solving 5.9) is called a marke price of risk process. To undersand his erminology, noe ha he i h equaion in he sysem 5.9) can be wrien as d σ ij λ j = µ i r. j=1 If he price of he i h asse is only sensiive o he j h exogenous shock, he equaion reduces o σ ij λ j = µ i r, ]. { 1 2

5.2 Asse pricing in coninuous-ime models 107 implying ha λ j = µ i r σ ij. Therefore, λ j is he compensaion in erms of excess expeced reurn per uni of risk semming from he j h exogenous shock. Noe ha any risk-neural probabiliy measure is associaed o a marke price of risk process. Suppose λ is a marke price of risk process and le Q denoe he associaed risk-neural probabiliy measure and z Q he associaed sandard Brownian moion. Then 5.10) d P = diag P )σ dz Q and [ ] dp = diagp ) r 1 d + σ dz Q. So under a risk-neural probabiliy all asse prices have a drif equal o he shor rae. volailiies are no affeced by he change of measure. Nex, le us look a he relaion beween marke prices of risk and sae-price deflaors. Suppose ha λ is a marke price of risk and ξ in 5.6) defines he associaed risk-neural probabiliy measure. From Theorem 5.3 we know ha, under a regulariy condiion, he process ζ defined by { ζ = ξ e 0 rs ds = exp r s ds 1 } λ s 2 ds λ s dz s 2 0 is a sae-price deflaor. Since dξ = ξ λ dz, an easy applicaion of Iô s Lemma implies ha 5.11) dζ = ζ [r d + λ dz ]. As we have already seen, he relaive drif of a sae-price deflaor equals he negaive of he shor-erm ineres rae. Now, we see ha he sensiiviy vecor of a sae-price deflaor equals he negaive of a marke price of risk. Up o echnical condiions, here is a one-o-one relaion beween marke prices of risk and sae-price deflaors. Suppose he rank of σ equals k for all. If k < d, here are several soluions o 5.9). We can wrie one soluion as 5.12) λ = ˆσ ) 1 ˆσ ˆσ ˆµ r 1), where ˆσ is he k d marix obained from σ by removing rows corresponding o redundan asses, i.e. rows ha can be wrien as a linear combinaion of oher rows in he marix. Similarly, ˆµ is he k-dimensional vecor ha is lef afer deleing he elemens corresponding o he redundan asses. In he special case where k = d, we have λ = ˆσ 1 ˆµ r 1). Insead of represening a rading sraegy by he number of unis held of each asse, we can also represen in erms of he fracions of he oal value of he rading sraegy invesed in each asse. Le π denoe he vecor of fracions invesed in he N risky asses. Then he fracion of wealh invesed in he bank accoun mus be 1 π 1. Leing V π rading sraegy, we ge dv π = V π [ ] r + π µ r 1)) d + π σ dz. 0 0 The denoe he value of such a

5.2 Asse pricing in coninuous-ime models 108 ) 1 Consider he rading sraegy given by he fracions of wealh ˆπ = ˆσ ˆσ ˆµ r 1) in he non-redundan asses and he fracion 1 ˆπ 1 in he bank accoun. The dynamics of he value V of his rading sraegy is given by dv = V [ r + ˆπ ˆµ r 1) ) d + ˆπ ˆσ dz ] = V [ r + λ 2) d + λ ) dz ]. I can be shown ha ˆπ is he rading sraegy wih he highes expeced coninuously compounded growh rae, i.e. he rading sraegy maximizing E[ln VT π/v 0 π )], and i is herefore referred o as he growh-opimal rading sraegy. Consequenly, λ defined in 5.12) is he relaive sensiiviy vecor of he value of he growh-opimal rading sraegy. 5.2.7 Complee vs. incomplee markes A financial marke is said o be dynamically) complee if all relevan risks can be hedged by forming porfolios of he raded financial asses. More formally, le L denoe he se of all random variables wih finie variance) whose oucome can be deermined from he exogenous shocks o he economy over he enire period [0, T ]. In mahemaical erms, L is he se of all random variables ha are measurable wih respec o he σ-algebra generaed by he pah of he Brownian moion z over [0, T ]. On he oher hand, le M denoe he se of possible ime T values ha can be generaed by forming self-financing rading sraegies in he financial marke, i.e. { } M = α, θ) self-financing wih V α,θ bounded from below for all [0, T ]. V α,θ T Of course, for any rading sraegy α, θ) he erminal value V α,θ T is a random variable, whose oucome is no deermined unil ime T. Due o he echnical condiions imposed on rading sraegies, he erminal value will have finie variance, so M is always a subse of L. If, in fac, M is equal o L, he financial marke is said o be complee. If no, i is said o be incomplee. In a complee marke, any random variable of ineres o he invesors can be replicaed by a rading sraegy, i.e. for any random variable W we can find a self-financing rading sraegy wih erminal value V α,θ T = W. Consequenly, an invesor can obain exacly her desired exposure o any of he d exogenous shocks. Real financial markes are no complee in a broad sense, since mos invesors face resricions on he rading sraegies hey can inves in, e.g. shor-selling and porfolio mix resricions, and are exposed o risks ha canno be fully hedged by any financial invesmens, e.g. labor income risk. Inuiively, o have a complee marke, sufficienly many financial asses mus be raded. However, he asses mus also be sufficienly differen in erms of heir response o he exogenous shocks. Afer all, we canno hedge more risk wih wo perfecly correlaed asses han wih jus one of hese asses. Marke compleeness is herefore closely relaed o he sensiiviy process σ of he raded asses. The following heorem provides he precise relaion involving he rank of he N d marix σ : Theorem 5.5 Suppose ha he shor-erm ineres rae r is bounded. Also, suppose ha a bounded marke price of risk process λ exiss. Then he financial marke is complee if and only if he rank of σ is equal o d almos everywhere).

5.2 Asse pricing in coninuous-ime models 109 Clearly, a necessary bu no sufficien) condiion for he marke o be complee is ha a leas d risky asse are raded if N < d, he marix σ canno have rank d. If σ has rank d, hen here is exacly one soluion o he sysem of equaions 5.9) and, hence, exacly one marke price of risk process λ and if λ is sufficienly nice) exacly one risk-neural probabiliy measure. If he rank of σ is sricly less han d, here will be muliple soluions o 5.9) and herefore muliple marke prices of risk and muliple risk-neural probabiliy measures. Combining hese observaions wih he previous heorem, we have he following conclusion: Theorem 5.6 Suppose ha he shor-erm ineres rae r is bounded and ha he marke is complee. Then here is a unique marke price of risk process λ and, if λ saisfies echnical condiions, here is a unique risk-neural probabiliy measure. This heorem and Theorem 5.3 ogeher imply ha in a complee marke, under echnical condiions, we have a unique sae-price deflaor. An example of an incomplee marke is a marke where he raded asses are only sensiive o k < d of he d exogenous shocks. Decomposing he d-dimensional sandard Brownian moion z ino Z, Ẑ), where Z is k-dimensional and Ẑ is d k)-dimensional, he dynamics of he raded risky asses can be wrien as dp = diagp ) [ µ d + σ dz ]. For example, he dynamics of r, µ, or σ may be affeced by he non-raded risks Ẑ, represening non-hedgeable risk in ineres raes, expeced reurns, and volailiies and correlaions, respecively. Or oher variables imporan for he invesor, e.g. his labor income, may be sensiive o Ẑ. Le us assume for simpliciy ha k = N and he k k marix σ is non-singular. Then we can define a unique marke price of risk associaed wih he raded risks by he k-dimensional vecor Λ = σ ) 1 µ r 1), bu for any well-behaved d k)-dimensional process ˆΛ, he process λ = Λ, ˆΛ) will be a marke price of risk for all risks. Each choice of ˆΛ generaes a valid marke price of risk process and hence a valid risk-neural probabiliy measure and a valid sae-price deflaor. 5.2.8 Exension o inermediae dividends In he coninuous-ime model discussed so far, we have assumed ha he asses have a final dividend paymen a ime T and no dividend paymens before. Clearly, we need o exend his o he case of dividends a oher daes. We disinguish beween lump-sum dividends and coninuous dividends. A lump-sum dividend is a paymen a a single poin in ime, where as a coninuous dividend is paid over a period of ime. Suppose Q is a risk-neural probabiliy measure. Consider an asse paying only a lump-sum dividend of L a ime < T. If we inves he dividend in he bank accoun over he period [, T ], we end up wih a value of L exp{ T r u du}. Thinking of his as a erminal dividend, he value of he asse a ime < mus be P i = E Q [e T ru du L e )] T r u du = E Q [e ] r u du L.

5.2 Asse pricing in coninuous-ime models 110 Inermediae lump-sum dividends are herefore valued similarly o erminal dividends and he discouned price process of such an asse will be a Q-maringale over he period [0, ] where he asse lives. An imporan example is ha of a zero-coupon bond paying one a some fuure dae. The price a ime < of such a bond is given by 5.13) B = E Q [e r u du ]. In erms of a sae-price deflaor ζ, we have 5.14) B = E [ ζ ζ A coninuous dividend is represened by a dividend rae process D = D ), which means ha he oal dividend paid over any period [, ] is equal o D u du. Over a very shor inerval [s, s + ds] he oal dividend paid is approximaely D s ds. provides a ime T value of e T s he period [, T ], we ge a erminal value of T he ime value of such a erminal paymen is [ P i = E Q e T ]. Invesing his in he bank accoun ru du D s ds. Inegraing up he ime T values of all he dividends in ru du T e T s This implies ha for any < < T, we have [ 5.15) P i = E Q e T s ru du D s ds e r u du P i + ru du D s ds. According o he previous secions ] = E Q [ T e s ru du D s ds e s ru du D s ds and he process wih ime value given by P i exp{ 0 r u du} + 0 exp{ s 0 r u du}d s ds is a Q-maringale. In erms of a sae-price deflaor ζ we have ha he process wih ime value ζ P i + 0 ζ sd s ds is a P-maringale and [ ] ζ ζ s P i = E P i + D s ds. ζ ζ Pricing expressions for asses ha have boh coninuous and lump-sum dividends can be obained by combining he expressions above appropriaely. ] ]. 5.2.9 Equilibrium and represenaive agens in complee markes We will no formally define he conceps of equilibrium, Pareo-opimaliy, and represenaive agens in he coninuous-ime framework, bu he meaning of hese conceps is exacly he same as in he discree-ime models. Once again, we have he same wo heorems: Theorem 5.7 If he financial marke is complee, hen every equilibrium consumpion allocaion is Pareo-opimal. Theorem 5.8 Suppose all individuals are greedy and risk-averse. If he financial marke is complee, he economy has a represenaive agen.

5.3 Oher probabiliy measures convenien for pricing 111 In a complee marke we have a unique sae-price deflaor ζ = ζ ) ha values any rading sraegy. Furhermore, he value of any reasonably well-behaved process can be replicaed by an appropriae rading sraegy. In paricular, he consumpion process of any given agen can be valued using he sae-price deflaor. The iniial cos of following a consumpion process c = c ) is herefore given by E[ T 0 ζ c d]. Similarly he iniial value of a fuure income process e = e ) is given by E[ T 0 ζ e d]. We can now formulae he consumpion choice problem of he agen as [ ] T sup E e δ uc ) d, c=c ) 0 [ ] [ T ] T s.. E ζ c d W 0 + E ζ e d, 0 c 0. Here W 0 denoes he iniial wealh of he agen and we have assumed a ime-addiive expeced uiliy represenaion of he agen s preferences. In a complee marke he decision problem of he agen can be divided ino wo seps. Firs, he opimal consumpion process is found by solving he above problem. Then, he agen mus find a rading sraegy ha finances his consumpion process. Such a rading sraegy will exis because he marke is complee. Le us derive he firs-order condiion for c = c ) for he above problem. In doing ha we will ignore he non-negaiviy consrain on he consumpion rae. This will indeed be me, whenever he agen has infinie marginal uiliy a zero consumpion. Using he Lagrangian echnique for his consrained opimizaion problem, we need o maximize [ ] [ T ] [ T ]) T E e δ uc ) d + η W 0 + E ζ e d E ζ c d 0 0 0 0 [ ] [ T ]) = E e δ ) T uc ) ηζ c d + η W 0 + E ζ e d, 0 where η is he Lagrange muliplier. We can focus on he firs erm, since he ohers do no involve c. Le us maximize he inegrand e δ uc ) ηζ c for [ each [0, T ] and each sae of he world T ) ] a ime. Doing ha we will surely maximize E 0 e δ uc ) ηζ c d. The inegrand is maximized whenever e δ u c ) = ηζ and since we know ha ζ 0 = 1, we mus have η = u c 0 ). Hence we have shown ha he unique sae-price deflaor in a complee marke is given by 5.16) ζ = e δ u c ) u c 0 ). This is rue for any agen. In paricular, we can represen he unique sae-price deflaor in a complee marke in erms of he marginal uiliy of a represenaive agen. The opimal consumpion of a represenaive agen mus be equal o he aggregae endowmen of he economy. We shall use his relaion in Chaper 6 o link asse prices, ineres raes, and aggregae consumpion. 0 5.3 Oher probabiliy measures convenien for pricing In he previous secion we discussed he idea of compuing prices using a risk-neural probabiliy measure. For some applicaions i urns ou o be convenien o use differen probabiliy measures o which we urn now.

5.3 Oher probabiliy measures convenien for pricing 112 Given a risk-neural probabiliy measure Q, hen he price P of any asse wih a single paymen dae saisfies ha P A = E Q for all s > before he paymen dae of he asse, i.e. he relaive price process P /A ) is a Q- maringale. In a sense, we use he bank accoun as a numeraire. If he asse pays off P T a ime T, we can compue he ime price as P = E Q [ Ps A s ] [e T rs ds P T ]. This involves he simulaneous risk-neural disribuion of T complex. r s ds and P T, which migh be quie For some asses we can simplify he compuaion of P using a differen, appropriaely seleced, numeraire asse. Le S denoe he price process of a paricular raded asse or he value process of a dynamic rading sraegy. We require ha S > 0. Can we find a probabiliy measure Q S so ha he relaive price process P /S ) is a Q S -maringale? Le us wrie he price dynamics of S and P as dp = P [µ P d + σ P dz ], ds = S [µ S d + σ S dz ]. Then by Iô s Lemma, ) P d = P [ µp µ S + σ S 2 ) ] σ S σ P d + σp σ S ) dz. S S Suppose we can find a well-behaved sochasic process λ S such ha 5.17) σ P σ S ) λ S = µ P µ S + σ S 2 σ S σ P. Then we can define a probabiliy measure Q S by he Radon-Nikodym derivaive { dq S dp = exp 1 T } T ) λ S 2 ds λ S dz. 2 0 0 The process z S defined by dz S = dz + λ S, z S 0 = 0 is a sandard Brownian moion under Q S. We hen ge ) P d = P σ P σ S ) dz S, S so ha P /S ) indeed is a Q S -maringale. S How can we find a λ S saisfying 5.17)? As we have seen, under weak condiions a marke price of risk λ will exis wih he propery ha µ P = r + σ P λ and µ S = r + σ S λ. Subsiuing in hese relaions, he righ-hand side of 5.17) simplifies o σ P σ S ) λ σ S ). We can herefore use λ S = λ σ S. In general we refer o such a probabiliy measure Q S as a maringale measure for he asse wih price S = S ). In paricular, a risk-neural probabiliy measure Q is a maringale measure for he bank accoun.

5.3 Oher probabiliy measures convenien for pricing 113 Given a maringale measure Q S for he asse wih price S, he price P of an asse wih a single paymen P T a ime T saisfies 5.18) P = S E QS If cases where he disribuion of P T /S T under he measure Q S is relaively simple, his provides a compuaionally convenien way of saing he price P in erms of S. In he following subsecions we look a some examples. [ PT S T ]. 5.3.1 A zero-coupon bond as he numeraire forward maringale measures For he pricing of derivaive securiies ha only provide a payoff a a single ime T, i is ypically convenien o use he zero-coupon bond mauring a ime T as he numeraire. Recall ha he price a ime T of his bond is denoed by B T and ha BT T = 1. Le σt denoe he sensiiviy vecor of B T so ha db T = B T [ r + σ T λ ) d + σ T dz ], assuming he exisence of a marke price of risk process λ = λ ). We denoe he maringale measure for he zero-coupon bond mauring a T by Q T and refer o Q T as he T -forward maringale measure. This ype of maringale measure was inroduced by Jamshidian 1987) and Geman 1989). The erm comes from he fac ha under his probabiliy measure he forward price for delivery a ime T of any securiy wih no inermediae paymens is a maringale, i.e. he expeced change in he forward price is zero. If he price of he underlying asse is P, he forward price is P /B T, and by definiion his relaive price is a Q T -maringale. The expecaion under he T -forward maringale measure is someimes called he expecaion in a T -forward risk-neural world. The ime price of an asse paying P T a ime T can be compued as 5.19) P = B T E QT [P T ]. Under he probabiliy measure Q T, he process z T defined by 5.20) dz T = dz + λ σ T ) d is a sandard Brownian moion. To compue he price from 5.19) we only have o know 1) he curren price of he zero-coupon bond ha maures a he paymen dae of he asse and 2) he disribuion of he random paymen of he asse under he T -forward maringale measure Q T. We shall apply his pricing echnique o derive prices of European opions on zero-coupon bonds. The forward maringale measures are also imporan in he analysis of he so-called marke models sudied in Chaper 11. Noe ha if he yield curve is consan and herefore fla) as in he Black-Scholes-Meron model for sock opions, he bond price volailiy σ T is zero and, consequenly, here is no difference beween he risk-neural probabiliy measure and he T -forward maringale measure. The wo measures differ only when ineres raes are sochasic. To emphasize he difference beween hese measures, he risk-neural probabiliy measure, which is associaed wih he shor rae or spo rae bank accoun, is someimes referred o as he spo maringale measure.

5.3 Oher probabiliy measures convenien for pricing 114 5.3.2 An annuiy as he numeraire swap maringale measures As discussed above i is ofen compuaionally advanageous for he pricing of European opions o use as a numeraire he zero-coupon bond mauring a he same ime as he opion. In his subsecion we show ha for he pricing of European swapions i is compuaionally convenien o use anoher numeraire and hence anoher probabiliy measure. A European payer swapion wih expiraion ime T 0 and an exercise rae of K gives he righ o ener ino a payer swap wih some face value H a ime T 0. If he righ is exercised, he holder mus a given fuure poins in ime, T i = T 0 + iδ where i = 1,..., n, pay HδK, bu will receive paymens Hδl Ti T i δ ha are sill unknown. From 2.33) on page 39 we have ha he ime T 0 value of he payoff of such a swapion can be expressed as P T0 = n i=1 B Ti T 0 ) ) Hδ max lδ K, 0 T0, where l δ T 0 is he equilibrium) swap rae prevailing a ime T 0. If we were o use he zero-coupon bond mauring a T 0 as he numeraire, we would have o find he expecaion of he payoff P T0 under he T 0 -forward maringale measure Q T0. Bu since he payoff depends on several differen bond prices, he disribuion of P T0 under Q T0 is raher complicaed. I is more convenien o use anoher numeraire, namely he annuiy bond, which a each of he daes T 1,..., T n provides a paymen of 1 dollar. The value of his annuiy a ime T 0 equals G = n. In paricular, he payoff of he swapion can be resaed as i=1 BTi P T0 ) = G T0 Hδ max lδ K, 0 T0, ) and he payoff expressed in unis of he annuiy bond is simply Hδ max lδ K, 0 T0. The maringale measure corresponding o he annuiy being he numeraire is called he swap maringale measure and will be denoed by Q G in he following. This ype of maringale measure was inroduced by Jamshidian 1997). The price of he European payer swapion can now be wrien as 5.21) P = G E QG [ PT0 G T0 ] = G Hδ E QG [ )] max lδ K, 0 T0, so we only need o know he disribuion of he swap rae l T δ 0 under he swap maringale measure. In Chaper 11 we will look a a model ha is based on he assumpion ha l T δ 0 is lognormally disribued under he swap maringale measure. This resuls in a Black-Scholes-Meron-ype formula for he price of he European swapion very similar o he pricing formula applied by many praciioners, cf. Equaion 5.74) on page 132. 5.3.3 A general pricing formula for European opions We can use he idea of changing he numeraire and he probabiliy measure o obain a general characerizaion of he price of a European call opion. Of course, a similar resul is valid for European pu opions. Le T be he expiry dae and K he exercise price of he opion, so ha he opion payoff a ime T is of he form C T = maxh T K, 0).

5.3 Oher probabiliy measures convenien for pricing 115 For an opion on a raded asse, h T is he price of he underlying asse a he expiry dae. For an opion on a given ineres rae, h T denoes he value of his ineres rae a he expiry dae. We can rewrie he payoff as C T = h T K) 1 {ht )>K}, where 1 {ht >K} is he indicaor funcion for he even h T > K. The value of his indicaor funcion is 1 if h T > K and 0 oherwise. According o 5.19) he ime price of he opion is 1 5.22) C = B T E QT [maxh T ) K, 0)] = B T [ E QT ht K)1 {ht >K}] = B T = B T E QT E QT [ ht 1 {ht >K}] K E Q [ T 1{hT >K}] ) [ ] ht 1 {ht >K} KQ T h T > K)) = B T E QT [ ] ht 1 {ht >K} KB T Q T h T > K). Here Q T h T > K) denoes he probabiliy using he probabiliy measure Q T ) of h T > K given he informaion known a ime. This can be inerpreed as he probabiliy of he opion finishing in-he-money, compued in a hypoheical forward-risk-neural world. For an opion on a raded asse we can rewrie he firs erm in he above pricing formula, since h is hen a valid numeraire wih a corresponding probabiliy measure Q h. Applying 5.18) for boh he numeraires B T The call price is herefore and h, we ge B T E QT [ ht 1 {ht >K}] = h E Qh [ 1{hT >K}] = h Q h h T > K). 5.23) C = h Q h h T > K) KB T Q T h T > K). Boh probabiliies in his formula show he probabiliy of he opion finishing in-he-money, bu under wo differen probabiliy measures. To compue he price of he European call opion in a concree model we jus have o compue hese probabiliies. In paricular, for a call opion on a zero-coupon bond mauring a ime S > T we ge ha 5.24) C = B S Q S B S T > K) KB T Q T B S T > K), where Q S denoes he S-forward maringale measure and Q T, as before, is he T -forward maringale measure. 1 In he compuaion we use he fac ha he expeced value of he indicaor funcion of an even is equal o he probabiliy of ha even. This follows from he general definiion of an expeced value, E[gω)] = ω Ω gω)fω) dω, where fω) is he probabiliy densiy funcion of he sae ω and he inegraion is over all possible saes. The se of possible saes can be divided ino wo ses, namely he se of saes ω for which h T > K and he se of ω for which h T K. Consequenly, E[1 {ht >K}] = 1 {ht >K}fω) dω ω Ω = 1 fω) dω + 0 fω) dω ω:h T >K ω:h T K = fω) dω, ω:h T >K which is exacly he probabiliy of h T > K.

5.4 Forward prices and fuures prices 116 5.4 Forward prices and fuures prices As we noed in Secion 2.5, i is relaively easy o show ha he forward price and he fuures price for he same selemen dae and he same underlying asse are idenical if here is no uncerainy abou fuure ineres raes. The original proof of his resul was given by Cox, Ingersoll, and Ross 1981b). In his secion will characerize forward and fuures prices under less resricive assumpions, in paricular we allow for sochasic ineres raes. 5.4.1 Forward prices A forward wih mauriy dae T and delivery price K provides a payoff of S T K a ime T, where S is he underlying variable, ypically he price of an asse or a specific ineres rae. For forwards conraced upon a ime, K is se so ha he value of he forward a ime is zero. This value of K is called he forward price a ime for he delivery dae T ) and is denoed by F T. A he selemen dae he forward price equals he value of he underlying variable, i.e. FT T = S T. I follows from he resuls on he pricing under he spo maringale measure ha he forward price F T T is fixed so ha 0 = E Q [e T ru du S T F T ) ] = E Q [e T ru du S T ] F T E Q [e T ru du]. Applying 5.13), we ge 5.25) F T = E Q [ e T ru du S T ] B T. If he underlying variable is he price of a raded asse wih no paymens in he period [, T ], we have E Q so ha he forward price can be wrien as [e T ru du S T ] = S, F T = S B T. This expression is consisen wih he resuls for bond forwards given in Secion 2.3 on page 23. Applying a well-known propery of covariances, we have ha E Q [e T ru du S T ] = Cov Q e T ru du, S T ) + E Q [e T ru du] E Q [S T ] = Cov Q e T ru du, S T ) + B T E Q [S T ]. Upon subsiuion of his ino 5.25) we ge he following expression for he forward price Cov Q 5.26) F T = E Q e ) T ru du, S T [S T ] +. We can also characerize he forward price in erms of he T -forward maringale measure. The forward price process in dollars) for conracs wih delivery dae T is a maringale under he B T

5.4 Forward prices and fuures prices 117 T -forward maringale measure. This is clear from he following consideraions. Wih B T numeraire, we have ha he forward price F T is se so ha [ 0 ST F T ] = E QT B T B T T as he and hence F T = E QT [S T ] = E QT [FT T ], which implies ha he forward price F T is a Q T -maringale. 5.4.2 Fuures prices Consider a fuures wih final selemen a ime T and coninuous marking-o-marke. Le Φ T be he fuures price a ime. The fuures price a he selemen ime is Φ T T = S T. We assume ha he fuures is coninuously marked-o-marke so ha over any infiniesimal inerval [, + d] i provides a paymen of dφ T. The following heorem characerizes he fuures price: Theorem 5.9 The fuures price Φ T in paricular is a maringale under he spo maringale measure Q, so ha 5.27) Φ T = E Q [S T ]. Proof: We will prove he heorem by firs considering a discree-ime seing in which posiions can be changed and he fuures conracs marked-o-marke a imes, +, +2,..., +N T. This proof is originally due o Cox, Ingersoll, and Ross 1981b). A proof based on he same idea, bu formulaed direcly in coninuous-ime, was given by Duffie and Sanon 1992). The idea is o se up a self-financing sraegy ha requires an iniial invesmen a ime equal o he fuures price Φ T. Hence, a ime, Φ T is invesed in he bank accoun. In addiion, e r fuures conracs are acquired a a price of zero). A ime +, he deposi a he bank accoun has grown o e r Φ T. The marking-omarke of he fuures posiion yields a payoff of e r Φ T + ) ΦT, which is deposied a he bank accoun, so ha he balance of he accoun becomes e r Φ T +. The posiion in fuures is increased a no exra coss) o a oal of e r+ +r) conracs. A ime +2, he deposi has grown o e r+ +r) Φ T +, which ogeher wih he markingo-marke paymen of e r+ +r) Φ T +2 + ) ΦT gives a oal of e r + +r ) Φ T +2. Coninuing his way, he balance of he bank accoun a ime T = + N will be e r +N 1) + +r ) Φ T +N = e r +N 1) + +r ) Φ T T = e r +N 1) + +r ) S T. The coninuous-ime limi of his is e T ru du S T. The ime value of his paymen is Φ T, since his is he ime invesmen required o obain ha erminal paymen. On he oher hand, we can value he ime T paymen by discouning by e T Hence, Φ T = E Q [e T ru du and aking he risk-neural expecaion. ru du e T ru du S T ] = E Q [S T ],

5.4 Forward prices and fuures prices 118 as was o be shown. The heorem implies ha he ime fuures price for a fuures on a zero-coupon bond mauring a ime S > T is given by Φ T,S = E Q [ B S T ]. For a fuures on a coupon bond wih paymens Y i a ime T i he final selemen is based on he bond price B T = T i>t Y ib Ti T 5.28) Φ T,cpn = E Q and hence he fuures price is [ T i>t Y i B Ti T ] = [ Y i E Q T i>t B Ti T ] = T i>t Y i Φ T,Ti, so ha he he fuures price on a coupon bond is a paymen-weighed average of fuures prices of he zero-coupon bonds mauring a he paymen daes of he coupon bond. Finally, consider Eurodollar fuures conracs. As discussed in Secion 2.6 on page 26 he marking-o-marke paymens of a Eurodollar fuures are based on he changes in he acual fuures price ) E T = 100 0.25 100 Ẽ T = 100 25ϕ T, where Ẽ T is he quoed fuures price for he final selemen ime T, and ϕ T is he corresponding LIBOR fuures rae. A he final selemen ime T he quoed fuures price is fixed a he value ẼT T = 100 1 l T +0.25 ) [ T = 100 1 4 B T +0.25 T ) 1 1 ]), so ha he final selemen is based on he erminal acual fuures price ) ET T 100 0.25 100 Ẽ T T = 100 0.25 400 [ B T +0.25 T ) 1 1 ]) = 100 [ 2 B T +0.25 T ) 1]. I follows from he analysis above ha he acual fuures price a any earlier poin in ime can be compued as E T = E Q The quoed fuures price is herefore [ E T T ] = 100 2 E Q [ B T +0.25 T ) 1]). 5.29) Ẽ T T = 4E T 300 = 500 400 E Q [ B T +0.25 T ) 1]. 5.4.3 A comparison of forward prices and fuures prices From 5.26) and 5.27) we ge ha he difference beween he forward price F T price Φ T is given by Cov Q 5.30) F T Φ T = e T ru du, S T ) B T. and he fuures The forward price and he fuures price will only be idenical if he wo random variables S T and exp ) T r u du are uncorrelaed under he risk-neural probabiliy measure. Of course, his is rue if he shor rae r is consan or deerminisic, in which case we recover he sandard resul.

5.5 American-syle derivaives 119 The forward price is larger [smaller] han he fuures price if he variables exp ) T r u du and S T are posiively [negaively] correlaed under he risk-neural probabiliy measure. An inuiive, heurisic argumen for his goes as follows. If he forward price and he fuures price are idenical, he oal undiscouned paymens from he fuures conrac will be equal o he erminal paymen of he forward. Suppose he ineres rae and he spo price of he underlying asse are posiively correlaed, which ough o be he case whenever exp ) T r u du and S T are negaively correlaed. Then he marking-o-marke paymens of he fuures end o be posiive when he ineres rae is high and negaive when he ineres rae is low. So posiive paymens can be reinvesed a a high ineres rae, whereas negaive paymens can be financed a a low ineres rae. Wih such a correlaion, he fuures conrac is clearly more aracive han a forward conrac when he fuures price and he forward price are idenical. To mainain a zero iniial value of boh conracs, he fuures price has o be larger han he forward price. Conversely, if he sign of he correlaion is reversed. 5.5 American-syle derivaives Consider an American-syle derivaive where he holder can choose o exercise he derivaive a he expiraion dae T or a any ime before T. Le H τ denoe he payoff if he derivaive is exercised a ime τ T. In general, H τ may depend on he evoluion of he economy up o and including ime τ, bu i is usually a simple funcion of he ime τ price of an underlying securiy or he ime τ value of a paricular ineres rae. A each poin in ime he holder of he derivaive mus decide wheher or no he will exercise. Of course, his decision mus be based on he available informaion, so we are seeking an enire exercise sraegy ha ell us exacly in wha saes of he world we should exercise he derivaive. We can represen an exercise sraegy by an indicaor funcion Iω, ), which for any given sae of he economy ω a ime eiher has he value 1 or 0, where he value 1 indicaes exercise and 0 indicaes non-exercise. For a given exercise sraegy I, he derivaive will be exercised he firs ime Iω, ) akes on he value 1. We can wrie his poin in ime as τ I = min{s [, T ] Iω, s) = 1}. This is called a sopping ime in he lieraure on sochasic processes. [ By our earlier analysis, he value of geing he payoff H τi a ime τ I is given by E Q e τ I r u du H τi ]. If we le I[, T ] denoe he se of all possible exercise sraegies over he ime period [, T ], he ime value of he American-syle derivaive mus herefore be 5.31) V = sup E Q I I[,T ] An opimal exercise sraegy I is such ha V = E Q [e τ I [ e τ I r u du H τi ]. r u du H τi ]. 5.6 Diffusion models and he fundamenal parial differenial equaion Many financial models assume he exisence of one or several so-called sae variables, i.e. variables whose curren values conain all he relevan informaion abou he economy. Of course,

5.6 Diffusion models and he fundamenal parial differenial equaion 120 he relevance of informaion depends on he purpose of he model. The sae variables of erm srucure models should be informaive for he erm srucure. In models wih a single sae variable we denoe he ime value of he sae variable by x, while in models wih several sae variables we gaher heir ime values in he vecor x. By assumpion, he curren values of he sae variables are sufficien informaion for he pricing and hedging of fixed income securiies. In paricular, hisorical values of he sae variables, x s for s <, are irrelevan. I is herefore naural o model he evoluion of x by a diffusion process since we know ha such processes have he Markov propery, cf. Secion 3.4 on page 51. We will refer o models of his ype as diffusion models. We will firs consider diffusion models wih a single sae variable, which are naurally ermed one-facor diffusion models. Aferwards, we shall briefly discuss how he resuls obained for one-facor models can be exended o muli-facor models, i.e. models wih several sae variables. 5.6.1 One-facor diffusion models We assume ha a single, one-dimensional, sae variable conains all he relevan informaion, i.e. ha he possible values of x lie in a se S R. We assume ha x = x ) 0 is a diffusion process wih dynamics given by he sochasic differenial equaion 5.32) dx = αx, ) d + βx, ) dz, where z is a one-dimensional sandard Brownian moion, and α and β are well-behaved funcions wih values in R. Given a marke price of risk λ = λx, ), we can wrie he dynamics of he sae variable under he risk-neural probabiliy measure as 5.33) dx = [αx, ) βx, )λx, )] d + βx, ) dz Q. We also assume ha he shor ineres rae depends a mos on x and, i.e. r = rx, ). Consider a securiy wih [ a single paymen of H T a ime T. We know ha he price of he securiy saisfies V = E Q e ] T ru du H T. Assuming ha H T = Hx T, T ), we can rewrie he price as V = V x, ), where V x, ) = E Q x, [e ] T rxu,u) du Hx T, T ) and we have exploied he Markov propery of x ) o wrie he expecaion as a funcion of he curren value of he process. Here E Q x, denoes he expecaion given ha x = x. I follows from Iô s Lemma see Theorem 3.4 on page 56) ha V = V x, ) is also a diffusion process wih dynamics 5.34) dv = V [µx, ) d + σx, ) dz ], where he funcions µ and σ are defined by 5.35) 5.36) µx, )V x, ) = V V x, ) + x x, )αx, ) + 1 2 V 2 x 2 x, )βx, )2, σx, )V x, ) = V x, )βx, ). x We also know ha for a marke price of risk λx, ), we have µx, ) = rx, ) + σx, )λx, )

5.6 Diffusion models and he fundamenal parial differenial equaion 121 for all possible values of x and hence µx, )V x, ) = rx, )V x, ) + σx, )V x, )λx, ) for all x, ). Subsiuing in µ and σ and rearranging, we arrive a a parial differenial equaion PDE) as saed in he following heorem. Theorem 5.10 The funcion V defined by 5.37) V x, ) = E Q x, [e ] T rxu,u) du Hx T, T ) saisfies he parial differenial equaion 5.38) V V x, ) + αx, ) βx, )λx, )) x, ) x ogeher wih he erminal condiion 5.39) V x, T ) = Hx), x S. + 1 2 βx, )2 2 V x, ) rx, )V x, ) = 0, x, ) S [0, T ), x2 The relaion beween expecaions and parial differenial equaions is generally known as he Feynman-Kac heorem, cf. Øksendal 1998, Thm. 8.2.1). Noe ha he coefficien of he V/ x in he PDE is idenical o he risk-neural drif of he sae variable, cf. 5.33). Also noe ha he prices of all securiies wih no paymens before T solve he same PDE. However, he erminal condiions and hereby also he soluions depend on he payoff characerisics of he securiies. When he sae variable iself is he price of a raded asse, he marke price of risk disappears from he pricing PDE. The expeced rae of reurn corresponding o µ) of his asse is αx, )/x, and he volailiy corresponding o σ) is βx, )/x. Since Equaion 5.9) in paricular mus hold for his asse, we have ha or λx, ) = αx,) x rx, ) αx, ) rx, )x = βx,) βx, ) x 5.40) αx, ) βx, )λx, ) = rx, )x. By inserion of his expression, he PDE 5.38) reduces o 5.41) V x, ) + rx, ) x V ) x, ) V x, ) x + 1 2 βx, )2 2 V x, ) = 0, x, ) S [0, T ). x2 Since no knowledge of he marke price of risk is necessary, asses wih price of he form V x, ) are in his case priced by pure no-arbirage argumens. The securiies which can be priced in his way are exacly he redundan securiies. This approach has proven successful in he pricing of sock opions wih he Black-Scholes-Meron model as he prime example. However, i is inappropriae o price ineres rae derivaives jus by modeling he dynamics of he underlying securiy. Consisen pricing of fixed income securiies mus be based on he evoluion of he enire erm srucure of ineres raes. Broadly speaking, he enire erm srucure is he underlying asse for all fixed

5.6 Diffusion models and he fundamenal parial differenial equaion 122 income securiies. Jus as he relevan marke price of risk in he Black-Scholes-Meron seing can be exraced from he curren marke price of he underlying sock, he marke prices of ineres rae risk can be exraced from he enire curren marke yield curve. Given he curren yield curve i.e. curren bond prices) and an assumpion on he volailiy of he curve, all yield curve derivaives bond opions and fuures, caps, floors, swapions, ec.) can be priced by he no-arbirage principle, i.e. wihou precise knowledge of preferences, ec. In a sense, such models have infiniely many sae variables, namely bond prices or yields of all mauriies. We will consider such erm srucure models in Chapers 10 and 11. For he erm srucure models based on one or a few sae variables we need some knowledge of he marke price of risk, eiher from an equilibrium model or by a reasonable assumpion. In many models he PDE 5.38) wih he appropriae erminal condiion can be solved explicily for a large number of ineresing securiies. The soluion is ofen found by coming up wih a qualified guess and verifying ha he guess is correc by subsiuion ino he equaion. Alernaively, he PDE can in some cases be resaed as anoher parial differenial equaion which has a soluion ha is already known. In oher models explici soluions for he imporan securiies canno be found, so ha i is necessary o resor o numerical soluion echniques as hose inroduced in Chaper 16. Hedging In a model wih a single one-dimensional sae variable a locally riskless porfolio can be consruced from any wo securiies. In oher words, he bank accoun can be replicaed by a suiable rading sraegy of any wo securiies. Conversely, i is possible o replicae any risky asse by a suiable rading sraegy of he bank accoun and any oher risky asse. To replicae asse 1 by a porfolio of he bank accoun and asse 2, he porfolio mus a any poin in ime consis of unis of asse 2, plus θ = α = V 1 x x, ) V 2 x x, ) = σ 1x, )V 1 x, ) σ 2 x, )V 2 x, ) 1 σ ) 1x, ) V 1 x, ) σ 2 x, ) invesed in he bank accoun. Then indeed he ime value of he porfolio is Π α + θ V 2 x, ) = 1 σ ) 1x, ) V 1 x, ) + σ 1x, ) σ 2 x, ) σ 2 x, ) V 1x, ) = V 1 x, ),

5.6 Diffusion models and he fundamenal parial differenial equaion 123 and he dynamics of he porfolio value is dπ = α rx, ) d + θ dv 2 x, ) = rx, ) 1 σ 1x, ) σ 2 x, ) = ) V 1 x, ) d + σ 1x, )V 1 x, ) rx, ) + σ 1x, ) σ 2 x, ) µ 2x, ) rx, )) = rx, ) + σ 1 x, )λx, )) V 1 x, ) d + σ 1 x, )V 1 x, ) dz = µ 1 x, )V 1 x, ) d + σ 1 x, )V 1 x, ) dz = dv 1, σ 2 x, )V 2 x, ) µ 2x, )V 2 x, ) d + σ 2 x, )V 2 x, ) dz ) ) V 1 x, ) d + σ 1 x, )V 1 x, ) dz so ha he rading sraegy replicaes asse 1. In paricular, in one-facor erm srucure models any fixed income securiy can be replicaed by a porfolio of he bank accoun and any oher fixed income securiy. We will discuss hedging issues in more deail in Chaper 12. If he sae variable x iself is he price of a raded asse, he consideraions above imply ha any asse can be replicaed by a rading sraegy ha a ime consiss of V x x, ) unis of he underlying asse and an appropriae posiion in he bank accoun. Securiies wih several paymen daes Many financial securiies have more han one paymen dae, e.g. coupon bonds, swaps, caps, and floors. Theorem 5.10 does no direcly apply o such securiies. In he exension o securiies wih several paymens, we mus disinguish beween securiies wih discree lump-sum paymens, i.e. a finie number of paymens a separaed poins in ime, and securiies wih a coninuous sream of paymens. Firs consider a securiy wih discree lump-sum paymens, which are eiher deerminisic or depend on he value of he sae variable a he paymen dae. Then Theorem 5.10 can be applied in order o separaely find he curren value of each of hese paymens afer which he value of he securiy follows from a simple summaion. Suppose ha he securiy provides paymens H j x Tj ) a ime T j for j = 1, 2,..., N, where T 1 < T 2 < < T N. Then he price of he securiy a ime < T 1 is given by where V j x, ) solves he PDE V x, ) = V j x, ) + αx, ) βx, )λx, )) V j x, ) x wih he erminal condiion N V j x, ), j=1 + 1 2 βx, )2 2 V j x 2 x, ) rx, )V jx, ) = 0, x, ) S [0, T j ), V j x, T j ) = H j x), x S. Here is an alernaive pricing approach. Clearly, a he ime of a paymen he value of he securiy will drop exacly by he paymen. The ex-paymen value will equal he cum-paymen value minus he size of he paymen. Leing + denoe immediaely afer ime, we can express his relaion as V x, T j +) = V x, T j ) H j x).

5.6 Diffusion models and he fundamenal parial differenial equaion 124 If he drop in he price [V x, T j +) V x, T j )] was less han he paymen H j x), an arbirage profi could be locked in by buying he securiy immediaely before he ime of paymen and selling i again immediaely afer he paymen was received. Beween paymen daes, i.e. in he inervals T j, T j+1 ), he price of he securiy will saisfy he PDE 5.38). If he price of he securiy is o be priced by numerical echniques, his alernaive approach is simpler han ha explained in he previous paragraphs, since only one PDE needs o be solved raher han one PDE for each paymen dae. Nex consider a securiy providing coninuous paymens a he rae h = hx, ) hroughou [0, T ] and a erminal lump-sum paymen of H T = Hx T, T ). From 5.15) we know ha he price of such a securiy in our diffusion seing is given by [ V x, ) = E Q e T rxu,u) du Hx T, T ) + T e s rxu,u) du h s ds Theorem 5.10 can be exended o show ha he funcion V in his case will solve he PDE V V x, ) + αx, ) βx, )λx, )) x, ) x + 1 2 βx, )2 2 V x, ) rx, )V x, ) + hx, ) = 0, x, ) S [0, T ), x2 wih he erminal condiion V x, T ) = Hx, T ) for all x S. The only change in he PDE relaive o he case wih no inermediae dividends is he addiion of he erm hx, ) on he lef-hand side of he equaion. In he special case where he paymen rae is proporional o he value of he securiy, i.e. hx, ) = qx, )V x, ), he PDE can be wrien as 5.42) V V x, ) + αx, ) βx, )λx, )) x, ) x + 1 2 βx, )2 2 V x 2 x, ) rx, ) qx, ) ) V x, ) = 0, x, ) S [0, T ). In his case he price can be wrien as 5.43) V x, ) = E Q [e ] T [rxu,u) qxu,u)] du Hx T, T ) Fuures prices In Secion 5.4.2 we showed ha he fuures price of an asse is generally given by he risk-neural expecaion of he price of he underlying asse a he final selemen dae. In our diffusion seing, he fuures price is Φ T = Φ T x, ) and we can wrie he pricing relaion as Φ T x, ) = E Q [Sx T, T )], where S denoes he price of he underlying asse. Comparing his wih 5.43) we see ha we can hink of he fuures price as being he price of an asse ha has a erminal paymen of Sx T, T ) and a coninuous dividend rae equal o rx, )Sx, ). From 5.42) we ge ha he fuures price funcion Φ T x, ) mus saisfy he PDE ]. 5.44) Φ T x, ) + αx, ) βx, )λx, )) ΦT x x, ) + 1 2 βx, )2 2 Φ T x, ) = 0, x, ) S [0, T ). x2

5.6 Diffusion models and he fundamenal parial differenial equaion 125 American-syle derivaives In a diffusion model wih a sae variable x, we can wrie he indicaor funcion represening he exercise sraegy of an American-syle derivaive as Ix, ), so ha Ix, ) = 1 if and only if he derivaive is exercised a ime when x = x. An exercise sraegy divides he space S [0, T ] of poins x, ) ino an exercise region and a coninuaion region. The coninuaion region corresponding o a given exercise sraegy I is he se C I = {x, ) S [0, T ] Ix, ) = 0} and he exercise region is hen he remaining par E I = {x, ) S [0, T ] Ix, ) = 1}, which can also be wrien as E I = S [0, T ]) \ C I. To an opimal exercise sraegy I x, ) corresponds opimal coninuaion and exercise regions C and E. I is inuiively clear ha he price funcion V x, ) for an American-syle derivaive mus saisfy he PDE 5.38) in he coninuaion region corresponding o he opimal exercise sraegy, i.e. for x, ) C. Bu since he coninuaion region is no known, bu is par of he soluion, i is much harder o solve he PDE for American-syle derivaives han for European-syle derivaives. Explici soluions have only been obained in rivial cases where premaure exercise is known o be inopimal, and he American-syle derivaive is herefore effecively a European-syle securiy. This is for example he case for an American call opion on a zero-coupon bond. However, numerical soluion echniques for PDEs can, wih some modificaions, also be applied o he case of Americansyle derivaives; see Chaper 16. 5.6.2 Muli-facor diffusion models Assume now ha he shor-erm ineres rae and he securiies we wan o price depend on n sae variables x 1,..., x n and ha he vecor x = x 1,..., x n ) follows he sochasic process 5.45) dx = αx, ) d + βx, ) dz, where z is a vecor of n independen sandard Brownian moions. Here, αx, ) = α 1 x, ),..., α n x, )) is he vecor of expeced changes in each of he n sae variables, and βx, ) is an n n-dimensional marix, which conains informaion abou he variances and covariances of changes in he differen sae variables. To be more precise, he insananeous variance-covariance marix of changes in he sae variables is given by βx, )βx, ) d. We can wrie 5.45) componenwise as n dx i = α i x, ) d + β i x, ) dz = α i x, ) d + β ij x, ) dz j. As discussed in Secion 3.4, he covariance beween changes in he i h and he j h sae variable over he nex infiniesimal ime period is given by n Cov dx i, dx j ) = β ik x, )β jk x, ) d, k=1 j=1

5.6 Diffusion models and he fundamenal parial differenial equaion 126 while he variance of he change in he i h sae variable is n Var dx i ) = β ik x, ) 2 d. In paricular, he volailiy of he i h sae variable is he sandard deviaion β i x, ) = n β ik x, ) 2, k=1 where denoes he lengh of he vecor. The insananeous correlaion beween changes in he i h and he j h sae variable is ρ ij x, ) = k=1 n Cov dx i, dx j ) Var dx i ) Var dx j ) = k=1 β ikx, )β jk x, ) β i x, ) β j x, ). Consider again a securiy wih a single paymen of H T V = V x, ), where V x, ) = E Q x, [e ] T rxu,u) du Hx T, T ). = Hx T, T ) a ime T. Is price is I follows from he muli-dimensional version of Iô s Lemma see Theorem 3.5 on page 66) ha he dynamics of V is 5.46) dv V = µx, ) d + where he funcions µ and σ j are defined as 5.47) 5.48) n σ j x, ) dz j, j=1 µx, )V x, ) = V n x, ) + V x, )α j x, ) x j σ j x, )V x, ) = + 1 2 n k=1 n j=1 n j=1 k=1 V x k x, )β kj x, ). We also know ha for a marke price of risk λx, ), we have 5.49) µx, ) = rx, ) + σx, )λx, ) = rx, ) + 2 V x j x k x, )ρ jk x, ) β j x, ) β k x, ), n σ j x, )λ j x, ). Subsiuing in µ and σ, we arrive a a PDE as summarized in he following muli-dimensional version of Theorem 5.10: Theorem 5.11 The funcion V defined by 5.50) V x, ) = E Q x, [e ] T rxu,u) du Hx T, T ) saisfies he parial differenial equaion n ) V 5.51) x, ) + n V α j x, ) β jk x, )λ k x, ) x, ) x j + 1 2 n j=1 k=1 j=1 k=1 n ρ jk x, ) β j x, ) β k x, ) 2 V x, ) rx, )V x, ) = 0, x, ) S [0, T ), x j x k j=1

5.7 The Black-Scholes-Meron model and Black s varian 127 and he erminal condiion V x, T ) = Hx), x S. Using marix noaion he PDE can be wrien more compacly as 5.52) V ) x, ) + V αx, ) βx, )λx, ) x, ) x + 1 ) 2 r βx, )βx, ) 2 V x, ) rx, )V x, ) = 0, x, ) S [0, T ), x2 where V/ x is he vecor of firs-order derivaives V/ x j, 2 V/ x 2 is he n n marix of secondorder derivaives 2 V/ x i x j, and rm) denoes he race of he marix M, which is defined as he sum of he diagonal elemens, rm) = j M jj. In a model wih n sae variables he bank accoun can be replicaed by a suiably consruced rading sraegy in n+1 sufficienly differen) securiies. Conversely, any securiy can be replicaed by a suiably consruced rading sraegy in he bank accoun and n oher sufficienly differen) securiies. For securiies wih more han one paymen dae he analysis mus be modified similarly o he one-dimensional case. In Chaper 8 we will sudy specific muli-facor diffusion models of he erm srucure of ineres raes. In some of hese models explici soluions o he PDE can be found for some imporan securiies. 5.7 The Black-Scholes-Meron model and Black s varian The bes known model for derivaive pricing is he Black-Scholes-Meron model developed by Black and Scholes 1973) and Meron 1973) for he pricing of European opions on socks. This model also serves as an example of a diffusion model. Praciioners ofen apply slighly modified versions of he Black-Scholes-Meron model and opion pricing formula o price oher derivaives han sock opions, including many fixed-income securiies. These modificaions are ofen based on Black 1976) who adaped he Black-Scholes-Meron seing o he pricing of European opions on commodiy fuures. However, as we shall discuss a he end of he secion, he use of he Black-76 approach o fixed-income securiies is no heoreically jusified. 5.7.1 The Black-Scholes-Meron model The criical assumpions underlying he Black-Scholes-Meron opion pricing model are ha he riskless ineres rae r coninuously compounded) is consan over ime and ha he price S of he underlying asse follows a coninuous sochasic process wih a consan relaive volailiy, i.e. 5.53) ds = µs, ) d + σs dz, where z is a sandard Brownian moion, σ is a consan, and µ is a nice funcion. I is ofen assumed ha µs, ) = µs for a consan parameer µ, bu ha is no necessary. However, we mus require ha he funcion µ is such ha he value space for he price process will be S = R +. Furhermore, we assume ha he underlying asse has no paymens in he life of he derivaive securiy.

5.7 The Black-Scholes-Meron model and Black s varian 128 We seek o deermine he price of a derivaive securiy ha a ime T pays off HS T ), which depends on he price of he underlying asse S T and on no oher uncerain variables. The ime price V of he derivaive asse is hen given by V = V S, ) where V S, ) = E Q S, [e ] T r du HS T ) = e r[t ] E Q [HS T )] and he risk-neural dynamics of he underlying asse price is ds = rs d + σs dz, so ha S T is lognormally disribued. The funcion V S, ) solves he PDE 5.54) V V S, ) + rs S S, ) + 1 2 σ2 S 2 2 V S, ) = rv S, ), S, ) S [0, T ). S2 In addiion, he price funcion mus saisfy he erminal condiion of he form 5.55) V S, T ) = HS), for all S S, where S is he se of all possible erminal prices S T of he underlying asse. For a European call opion wih an exercise price of K he payoff funcion is given by HS) = maxs K, 0). The price C = CS, ) can hen be found eiher by solving he PDE 5.54) wih he relevan erminal condiion or by calculaing he discouned risk-neural expeced payoff, i.e. CS, ) = e r[t ] E Q S, [maxs T K, 0)]. Applying Theorem A.4 in Appendix A, he laer approach immediaely gives he famous Black- Scholes-Meron formula for he price of a European call opion on a sock: 2 5.56) CS, ) = S N d 1 S, )) Ke r[t ] N d 2 S, )), where 5.57) 5.58) d 1 S, ) = lns /K) + r[t ] σ T d 2 S, ) = lns /K) + r[t ] σ T + 1 2 σ T, 1 2 σ T = d 1 S, ) σ T. 2 According o Abramowiz and Segun 1972), he cumulaive disribuion funcion N ) of he sandard normal disribuion can be approximaed wih six-digi accuracy as follows: Nx) 1 nx) a 1 bx) + a 2 bx) 2 + a 3 bx) 3 + a 4 bx) 4 + a 5 bx) 5), x 0, where nx) = e x2 /2 / 2π is he probabiliy densiy funcion, and bx) = 1 1 + cx, c = 0.2316419, a 1 = 0.31938153, a 2 = 0.356563782, a 3 = 1.781477937, a 4 = 1.821255978, a 5 = 1.330274429. For x < 0, we can use he relaion Nx) = 1 N x), where N x) can be compued using he approximaion above.

5.7 The Black-Scholes-Meron model and Black s varian 129 I can be verified ha he funcion CS, ) defined in 5.56) solves he PDE 5.54) wih he relevan erminal condiion. Applying he pu-call pariy for European sock opions, we ge he price formula for a European pu opion: 5.59) πs, ) = Ke r[t ] N d 2 S, )) S N d 1 S, )). Sricly speaking, o derive he opion pricing formulas above, he only assumpion needed on he price of he underlying asse is ha S T given S ) is lognormally disribued in he risk-neural world. Leing σ T denoe he sandard deviaion of ln S T, he formulas above will hold. However, if we wan o use he pricing formulas for opions on he same underlying asse, bu wih differen mauriies, we mus have ha he uncerain prices of he underlying asse a differen fuure poins in ime mus all be lognormally disribued in he risk-neural world. This is ensured by he assumpion 5.53). For he case where he underlying asse has paymens in he life of he derivaive asse, he same procedure applies wih some minor correcions, which can be found in Hull 2003). 5.7.2 Black s model Black 1976) inroduced a varian of he Black-Scholes-Meron model, which he applied for he pricing of European opions on commodiy fuures. Le us consider a European call opion ha expires a T, has an exercise price K, and is wrien on a commodiy fuures expiring a T, where T T. The fuures price a ime is denoed by Φ T. The payoff of he opion a ime T is maxφ T T K, 0). To price his opion, Black assumed a consan riskless ineres rae and ha he fuures price Φ T T a expiry of he opion is lognormally disribued in he risk-neural world. The sandard deviaion of ln Φ T T wih he informaion available a ime ) is denoed by σ T. I is furher assumed ha he expecaion of he ime T fuures price equals he curren fuures price, i.e. E Q [Φ T T ] = ΦT. We can see ha his is correc when he riskless ineres rae is consan and he asse underlying he fuures conrac has a price S ha follows a process of he form ds = S [r d + σ dz Q ] in a risk-neural world. As discussed in Secion 2.5, he fuures price is hen idenical o he forward price, which is given by F T ha = S e r[t ]. From Iô s Lemma Theorem 3.4 on page 56) we ge df T = σf T dz Q, from which i follows ha he expeced change in he forward price and hence he fuures price) is equal o zero, so ha E Q [Φ T T 5.60) ln Φ T T ] = ΦT, and Φ T T = F T T is lognormally disribued wih ) N ln Φ T 1 2 σ2 [T ], σ 2 [T ] Furhermore, under hese assumpions he volailiy of he fuures price equals he volailiy of he price of he underlying asse. According o he risk-neural pricing principle, he opion price can be wrien as [ )] C = e r[t ] E Q max Φ T T K, 0..

5.7 The Black-Scholes-Meron model and Black s varian 130 Applying 5.60) and Theorem A.4 of Appendix A, we can compue he price as [ ) )] 5.61) C = e r[t ] Φ T N ˆd1 Φ T, ) KN ˆd2 Φ T, ), where 5.62) 5.63) ˆd 1 Φ T ˆd 2 Φ T, ) = lnφt /K) σ T, ) = lnφt /K) σ T + 1 2 σ T, 1 2 σ T = ˆd 1 Φ T, ) σ T. The expression 5.61) is called Black s formula. For European pu opions we similarly have ha [ π = e r[t ] KN ˆd ) 2 Φ T, ) Φ T N ˆd )] 1 Φ T, ). Analogously o he Black-Scholes-Meron model i is no sricly necessary o assume ha he fuures price Φ T Φ T T follows a geomeric Brownian moion. I suffices ha he fuure fuures price is lognormally disribued in a hypoheical risk-neural world and ha he expeced change in he fuures price equals zero. The parameer σ is hen jus a measure for he sandard deviaion over he life of he opion, bu i is sill ofen referred o as he volailiy of he fuures price. Since he original developmen, Black s model has been adaped by marke paricipans for he pricing of various fixed income securiies, such as bond opions, caps/floors, and swapions. Since he payoff of hese securiies depends on fuure ineres raes, he original Black assumpion of consan ineres raes is of course inappropriae. The pricing formulas are derived by firs compuing he expeced payoff in a risk-neural world and hen discouning by he curren riskless discoun facor. The uncerainy of fuure ineres raes is aken ino accoun when he expeced payoff is compued, bu no in he discouning. Le us look a some examples: Bond opions As in Secion 2.7, we consider a European call opion wih expiraion ime T and exercise price K, wrien on a bond wih price B. Black s model applied o bond opions involves he forward price F T,cpn of he underlying bond wih delivery a expiraion of he opion. According o Theorem 2.2 on page 25, he forward price is 5.64) F T,cpn T = Y i>t ib Ti B T = B <T Y i<t ib Ti B T, where T 1 < T 2 < < T n are he paymen daes and Y i denoes he paymen of he bond a ime T i. To apply Black s model, he following assumpions mus be made: a) he fuures price equals he forward price; b) he forward price a he expiraion ime of he opion, F T,cpn T = B T, is lognormally disribued in he risk-neural world wih he sandard deviaion of ln F T,cpn T given by σ T ; c) he expeced change in he forward price of he bond beween ime and T equals zero in he risk-neural world; d) he opion price can be compued as he risk-neural expeced payoff muliplied by he curren riskless discoun facor.

5.7 The Black-Scholes-Meron model and Black s varian 131 The disribuional assumpion is saisfied if he forward price F T,cpn follows a sochasic process wih a consan relaive volailiy σ and a drif of zero. The expeced payoff in a risk-neural world is hen ) ) E Q [max B T K, 0)] = F T,cpn N ˆd1 F T,cpn, ) KN ˆd2 F T,cpn, ), where he funcions ˆd 1 and ˆd 2 are as in 5.62) 5.63). By muliplying he expeced payoff wih he riskless discoun facor, i.e. he zero-coupon bond price B T, we arrive a Black s formula for a European call opion on a bond: [ C K,T,cpn = B T 5.65) = B ) N ˆd1 F T,cpn, ) KN ) Y i B Ti N F T,cpn <T i<t Similarly, he price of a European pu opion on a bond is 5.66) π K,T,cpn = KB T N ˆd ) 2 F T,cpn, ) B )] ˆd2 F T,cpn, ), ) ) ˆd1 F T,cpn, ) KB T N ˆd2 F T,cpn, ). <T i<t Y i B Ti ) N ˆd ) 1 F T,cpn, ). Caps and floors As discussed in Secion 2.8, a cap can be seen as a porfolio of caples. The i h caple gives a payoff a ime T i of ) 5.67) CT i i = Hδ max l Ti T K, 0 i δ, cf. 2.11). Under he assumpions ha a) seen from ime, he risk-neural disribuion of l Ti T i δ sandard deviaion of ln l Ti T given by σ i δ i Ti δ ; = LTi δ,ti T i δ is lognormal wih he b) he expeced change in he forward rae L Ti δ,ti in a risk-neural world; beween ime and T i δ is equal o zero c) we can discoun he risk-neural expeced payoff wih he curren discoun facor; we obain Black s formula for he caple price [ 5.68) C i = HδB Ti L Ti δ,ti N where he funcions ˆd i 1 and ˆd i 2 are given by ) ˆdi 1 L Ti δ,ti, ) KN ˆdi 2 L Ti δ,ti, )) ], < T i δ, 5.69) 5.70) ˆd i 1L Ti δ,ti, ) = lnlti δ,t i /K) σ i Ti δ + 1 2 σ i Ti δ, ˆd i 2L Ti δ,ti, ) = ˆd i 1L Ti δ,ti, ) σ i Ti δ. The assumpions a)-b) are saisfied if he forward rae L Ti δ,ti he process in he risk-neural world follows 5.71) dl Ti δ,ti = σ i L Ti δ,ti dz Q

5.7 The Black-Scholes-Meron model and Black s varian 132 wih a consan volailiy σ i. The price for he enire cap is obained by summaion: 5.72) C = Hδ n i=1 B Ti [ L Ti δ,ti N ) ˆdi 1 L Ti δ,ti, ) KN ˆdi 2 L Ti δ,ti, )) ], T 0. As discussed in Secion 2.8 he price mus be adjused slighly if he firs paymen is already known. For a floor he corresponding formula is 5.73) F = Hδ n i=1 B Ti [ KN ˆd ) i 2L Ti δ,ti, ) L Ti δ,ti N ˆd i 1L Ti δ,ti, )) ], T 0. Swapions Le us look a a European payer swapion which was inroduced in Secion 2.9.2. From 2.33) on page 39 we have ha he payoff of a payer swapion a he expiraion ime T 0 can be expressed as P T0 = n i=1 B Ti T 0 ) ) Hδ max lδ K, 0 T0, where l T δ 0 is he equilibrium) swap rae, and K is he exercise rae. Black s formula for he price of a European payer swapion is n 5.74) P = Hδ i=1 B Ti ) [ Lδ,T ) 0 N ˆd1 L δ,t0, ) KN ˆd2 L δ,t0, )) ], < T 0, where he funcions ˆd 1 and ˆd 2 are as in 5.62) and 5.63) wih T = T 0. As in Secion 2.9, he forward swap rae. By analogy, he following expression for he price of a European receiver swapion is obained: n 5.75) R = Hδ i=1 B Ti ) [ KN ˆd ) 2 L δ,t0, ) δ,t0 L N ˆd 1 L δ,t0, )) ], < T 0, The assumpions underlying he formula are ha he swap rae l T δ δ,t0 0 = L T 0 a he expiraion dae of he swapion is lognormally disribued, or more precisely ha ln l T δ δ,t0 0 = ln L T 0 is risk-neurally normally disribued wih variance σ 2 [T 0 ], and ha he risk-neural expecaion of he change in he forward swap rae is zero. These assumpions are saisfied if he forward swap rae he risk-neural world follows he sochasic process 5.76) d L δ,t0 δ,t0 = σ L dz Q. L δ,t0 L δ,t0 is in The prices of sock opions are ofen expressed in erms of implici volailiies. The implici volailiy for a given European opion on a sock is ha value of σ, which by subsiuion ino he Black-Scholes-Meron formula, ogeher wih he observable variables S, r, K, and T, yields a price equal o he observed marke price. Similarly, prices of caps, floors, and swapions are expressed in erms of implici ineres rae volailiies compued wih reference o he Black pricing formula. According o 5.72) differen σ-values mus be applied for each caple in a cap. For a cap wih more han one remaining paymen dae, many combinaions of he σ i s will resul in he same

5.7 The Black-Scholes-Meron model and Black s varian 133 cap price. If we require ha all he σ i s mus be equal, only one common value will resul in he marke price. This value is called he implici fla volailiy of he cap. If caps wih differen mauriies, bu he same frequency and overlapping paymen daes, are raded, a erm srucure of volailiies, σ 1, σ 2,..., σ n, can be derived. For example, if a one-year and a wo-year cap on he one-year LIBOR rae are raded, he unique value of σ 1 ha makes Black s price equal o he marke price of he one-year cap can be deermined. Nex, by applying his value of σ 1, a unique value of σ 2 can be deermined so ha he Black price and he marke price of he wo-year cap are idenical. The volailiies σ i deermined by his procedure are called implici spo volailiies. A graph of he spo volailiies as a funcion of he mauriy, i.e. σ i as a funcion of T i δ, will usually be a humped curve, ha is an increasing curve for mauriies up o 2-3 years and hen a decreasing curve for longer mauriies. 3 A similar, hough slighly flaer, curve is obained by depicing he fla volailiies as a funcion of he mauriy of he cap, since fla volailiies are averages of spo volailiies. The picure is he same wheher implici or hisorical forward rae volailiies are used. If we consider formula 2.27) and assume as an approximaion ha he weighs w i are consan over ime, he variance of he fuure swap rae can be wrien as [ n ] n n Var [ l T δ 0 ] = Var w i L Ti δ,ti T 0 = w i w j σ i σ j ρ ij, i=1 i=1 j=1 where σ i denoes he sandard deviaion of he forward rae L Ti δ,ti T 0, and ρ ij denoes he correlaion beween he forward raes L Ti δ,ti T 0 and L Tj δ,tj T 0. The prices of swapions will herefore depend on boh he volailiies of he relevan forward raes and heir correlaions. 4 If implici forward rae volailiies have already been deermined from he marke prices of caples and caps, implici forward rae correlaions can be deermined from he marke prices of swapions by an applicaion of Black s formula for swapions, cf. formula 5.74). 5.7.3 Problems in applying Black s model o fixed income securiies As already hined upon above, he assumpions underlying he applicaion of Black s formula on ineres rae dependen securiies are highly problemaic. Le us ake a closer look a he criical poins. Firsly, he lognormaliy assumpion for bond prices and ineres raes is doubful. For several reasons he price of a bond canno follow a geomeric Brownian moion hroughou is life. We know ha he price converges o he erminal paymen of he bond as he mauriy dae approaches. Furhermore, he bond price is limied from above by he sum of he fuure bond paymens under he appropriae assumpion ha all forward raes are non-negaive. When he bond price approaches is upper limi or he mauriy dae approaches, he volailiy of he bond price has o go o zero. The volailiy of he bond price will herefore depend on boh he level of he price and he ime o mauriy. A lognormaliy assumpion can a mos be an approximaion o he rue disribuion. In addiion, he forward price and he fuures price on a bond are no necessarily equal when he ineres rae uncerainy is aken ino accoun. I is less clear wheher 3 See for example he discussion in Hull 2003, Ch. 22). 4 These consideraions are aken from Rebonao 1996, Sec. 1.4).

5.8 An overview of coninuous-ime erm srucure models 134 i is reasonable o assume ha fuure ineres raes are lognormally disribued, and ha he expeced changes in he forward raes and he forward swap raes are zero in a risk-neural world. We will discuss his furher in laer chapers. Secondly, he muliplicaion of he curren discoun facor and he risk-neural expecaion of he payoff does no lead o he correc price. In fac, as we have seen in Secion 5.3.1, his is rue if we ake he expecaion under he appropriae forward maringale measure insead of he risk-neural measure. Thirdly, simulaneous applicaions of Black s formula o differen derivaive securiies are inconsisen. If for example we apply Black s formula for he pricing of a European opion on zero-coupon bond, we mus assume ha he price of he zero-coupon bond is lognormally disribued. If we also apply Black s formula for he pricing of a European opion on a coupon bond, we mus assume ha he price of he coupon bond is lognormally disribued. Since he price of he coupon bond is a weighed average of he prices of zero-coupon bonds, cf. 1.2) on page 6, and a sum of lognormally disribued random variables is no lognormally disribued, he assumpions are inconsisen. 5 Similarly, he swap rae is a linear combinaion of forward raes according o 2.27) on page 38. When Black s formula is applied for he pricing of caples, i is implicily assumed ha he relevan forward raes are lognormally disribued. Then he swap rae will no be lognormally disribued, so ha i is inconsisen o use Black s formula for swapions also. Furhermore, lognormaliy assumpions for boh ineres raes and bond prices are inconsisen. Several research papers sugges oher models for bond opion pricing ha are also based on specific assumpions on he evoluion of he price of he underlying bond. The mos prominen examples are Ball and Torous 1983) and Schaefer and Schwarz 1987). A criical analysis of such models can be seen in Rady and Sandmann 1994). A problem in applying hese models is ha he assumpions on he price dynamics for differen bonds may be inconsisen, and hence he opion pricing formula obained in he model will only be valid for opions on one paricular bond. To ensure consisen pricing of differen fixed income securiies we mus model he evoluion of he enire erm srucure of ineres raes. In many of he consisen erm srucure models we shall discuss in he following chapers, we will obain relaively simple and inernally consisen pricing formulas for many of he popular fixed income securiies. As we shall see in Chaper 11, i is in fac possible o consruc consisen erm srucure models in which Black s formula is he correc pricing formula for some securiies, bu, even in hose models, applicaions of Black s formula for differen classes of securiies are inconsisen. 5.8 An overview of coninuous-ime erm srucure models Economiss and financial analyss apply erm srucure models in order o improve heir undersanding of he way he erm srucure of ineres raes is se by he marke and how i evolves over ime, 5 A similar problem is presen when he Black-Scholes-Meron formula is used boh for he pricing of opions on a sock index and opions on he individual socks. If he prices of he individual socks are lognormally disribued, he value of he index will no be lognormally disribued. However, i can be shown ha he disribuion of a sum of many lognormally disribued random variables is very accuraely approximaed by a lognormal disribuion wih carefully seleced parameers, cf. Turnbull and Wakeman 1991).

5.8 An overview of coninuous-ime erm srucure models 135 price fixed-income securiies in a consisen way, faciliae he managemen of he ineres rae risk ha affecs he valuaion of individual securiies, financial invesmen porfolios, and real invesmen projecs. As we shall see in he following chapers, a large number of differen erm srucure models has been suggesed in he las hree decades. All he models have boh desirable and undesirable properies so ha he choice of model will depend on how one weighs he pros and he cons. Ideally, we seek a model which has as many as possible of he following characerisics: 6 a) flexible: he model should be able o handle mos siuaions of pracical ineres, i.e. i should apply o mos fixed income securiies and under all likely saes of he world; b) simple: he model should be so simple ha i can deliver answers e.g. prices and hedge raios) in a very shor ime; c) well-specified: he necessary inpu for applying he model mus be relaively easy o observe or esimae; d) realisic: he model should no have clearly unreasonable properies; e) empirically accepable: he model should be able o describe acual daa wih sufficien precision; f) heoreically sound: he model should be consisen wih he broadly acceped principles for he behavior of individual invesors and he financial marke equilibrium. No model can compleely comply wih all hese objecives. A realisic, empirically accepable, and heoreically sound model is bound o be quie complex and will probably no be able o deliver prices and hedge raios wih he speed requesed by many praciioners. On he oher hand, simpler models will have inappropriae heoreical and/or empirical properies. 5.8.1 Overall caegorizaion We can spli he many erm srucure models ino wo caegories: absolue pricing models and relaive pricing models. An absolue pricing model of he erm srucure of ineres raes aims a pricing all fixed-income securiies, boh he basic securiies, i.e. bonds and bond-like conracs such as swaps, and he derivaive securiies such as bond opions and swapions. In conras, a relaive pricing model of he erm srucure akes he currenly observed erm srucure of ineres raes, i.e. he prices of bonds, as given and aims a pricing derivaive securiies relaive o he observed erm srucure. The same disincion can be used for oher asse classes. For example, he Black-Scholes-Meron model is a relaive pricing model since i prices sock opions relaive o he price of he underlying sock, which is aken as given. An absolue sock opion pricing model would derive prices of boh he underlying sock and he sock opion. Absolue pricing models are someimes referred o as equilibrium models, while relaive pricing models are called pure no-arbirage models. In his conex he erm equilibrium model does no 6 The presenaion is in par based on Rogers 1995).

5.8 An overview of coninuous-ime erm srucure models 136 necessarily imply ha he model is based on explici assumpions on he preferences and endowmens of all marke paricipans including he bond issuers, e.g. he governmen) which in he end deermine he supply and demand for bonds and herefore bond prices and ineres raes. Indeed, many absolue pricing models of he erm srucure are based on an assumpion on he dynamics of one or several sae variables and sipulaed relaions beween he shor rae and he sae variables and beween he marke prices of risk and he sae variables. These assumpions deermine boh he curren erm srucure and he dynamics of ineres raes and prices of fixed income securiies. These models do no explain how hese assumpions are produced by he acions of marke paricipans. Neverheless, i is ypically possible o jusify he assumpions of hese models by some more basic assumpions on preferences, endowmens, ec., so ha he model assumpions are compaible wih marke equilibrium; see he discussion and he examples in Secion 6.4. The pure no-arbirage models offer no explanaion o why he curren erm srucure is as observed. We can also divide he erm srucure models ino diffusion models and non-diffusion models. Again, by a diffusion model we mean a model in which all relevan prices and quaniies are funcions of a sae variable of a finie preferably low) dimension and ha his sae variable follows a Markov diffusion model. A non-diffusion model is a model which does no mee his definiion of a diffusion model. While he risk-neural pricing echniques are valid boh in diffusion and non-diffusion models, he PDE approach can only be applied in diffusion models. All wellknown absolue pricing models of he erm srucure are diffusion models. In conras, relaive pricing models are ypically formulaed as non-diffusion models a he ouse, which is naural since he evoluion of he enire erm srucure mus be modeled and he erm srucure consiss of, in principle, infiniely many values. In order o enjoy he benefis of diffusion models, non-diffusion models are someimes successfully reformulaed as diffusion models. We will consider examples of his idea in Chaper 10. 5.8.2 Some frequenly applied models Apparenly, he firs dynamic model of he erm srucure of ineres raes was inroduced by Meron 1970). His model is a diffusion model wih a single sae variable, which is he shor-erm ineres r iself. I makes good sense o use an ineres rae as he sae variable in models of bond prices and oher ineres rae derivaives. There is an obvious pracical advanage in using he shor rae as he sae variable. As we have seen above i is necessary o specify how he shor rae depends on he sae variable chosen, which is eviden when he shor rae iself is he sae variable. Furhermore, if anoher sae variable x is used and r is a monoonic funcion of x, we can express all relevan funcions in erms of r and insead of x and. Therefore we migh as well use r as he sae variable from he beginning. Meron s assumpions imply ha he shor rae follows a generalized Brownian moion under he risk-neural probabiliy measure, i.e. 5.77) dr = ˆϕ d + β dz Q, where ˆϕ and β are consans. Following Meron s idea, many oher one-facor diffusion models wih r as he single sae variable have been suggesed in he lieraure. They all ake a cerain risk-neural dynamics of he shor rae of he form dr = ˆαr ) d + βr ) dz Q,

5.8 An overview of coninuous-ime erm srucure models 137 where ˆα and β are well-behaved funcions. Eiher he funcional form of ˆα is assumed direcly or i is derived from assumpions on he real-world drif α and he marke price of risk λ. In any case, i is necessary o know α and λ, if he model should be used for more han jus compuing prices a a given dae. The pricing differences beween he models sem from differences in he specificaion of he funcions ˆα and β. I urns ou ha models in which ˆα and β are affine funcions of he curren value of r are paricularly racable and allow many closed-form pricing equaions o be derived. 7 Such models are called affine models. The wo mos famous erm srucure models are he one-facor affine diffusion models inroduced by Vasicek 1977) and Cox, Ingersoll, and Ross 1985b). In he Vasicek model he basic assumpion is ha he shor rae follows an Ornsein-Uhlenbeck process, cf. Secion 3.8.2, and ha he marke price of risk is consan, i.e. 5.78) dr = κ[θ r ] d + β dz, λ = λ consan. The risk-neural drif of he shor rae is hen κ[θ r ] βλ = κθ βλ) κr, which is affine in r. The variance rae is simply β 2, which is consan and hence a degenerae) affine funcion of r. In he Cox-Ingersoll-Ross or CIR) model he assumpion is ha he shor rae follows a square-roo process, cf. Secion 3.8.3, and ha he marke price of risk is proporional o he square-roo of he shor rae, i.e. 5.79) dr = κ[θ r ] d + β r dz, λ = λ r. The risk-neural drif of he shor rae is hen κ[θ r ] βλr = κθ κ + βλ)r, which is affine in r. The variance rae is β 2 r, which is also affine in r. Chaper 7 provides a deailed analysis of hese mos imporan one-facor diffusion models and menions a number of oher, non-affine one-facor models. A common problem for he one-facor models is ha because all bond prices are assumed o be affeced by a single exogenous variable, he price changes in any wo bonds over an infiniesimal ime period will be perfecly correlaed, which conflics wih empirical evidence. Empirical sudies by for example Lierman and Scheinkman 1991) and Sambaugh 1988) conclude ha a leas wo, and perhaps hree or four, sae variables are necessary for he model o give a reasonable descripion of acual yield curve movemens. This moivaes he sudy of muli-facor diffusion models. In hese models, he shor rae is assumed o be a funcion of several sae variables and each of hese sae variables are assumed o follow some diffusion process. Again i is common o disinguish beween affine models and non-affine models, where an affine model is a model in which he risk-neural drif raes and he variance and covariance raes of he sae variables are affine funcions of he curren value of he sae variables. Beaglehole and Tenney 1991) and Longsaff and Schwarz 1992a) have suggesed wo-facor affine models ha exend he Vasicek model and he CIR model, respecively. We will review hese and oher muli-facor models in Chaper 8. The one- and muli-facor diffusion models are absolue pricing models. They involve a small number of sae variables and consan parameers. The derived prices and ineres raes will also be funcions of he sae variables and hese few parameers. Consequenly, he resuling erm srucure of ineres raes canno ypically fi he currenly observed erm srucure perfecly. If 7 A funcion of he form fx) = a 0 + a 1 x is affine in x. The funcion is only linear in a sric mahemaical erminology if a 0 = 0.

5.9 Exercises 138 he main applicaion of he model is o price derivaive securiies, his mismach is roublesome. If he model is no able o price he underlying securiies i.e. he zero-coupon bonds) correcly, why rus he model prices for derivaive securiies? To compleely avoid his mismach one mus apply relaive pricing models for he derivaive securiies. We divide he relaive pricing models of he erm srucure ino hree subclasses: calibraed diffusion models, Heah-Jarrow-Moron HJM) models, and marke models. The common saring poin of all hese models is o ake he curren erm srucure as given and hen model he riskneural dynamics of he enire erm srucure. This is done very direcly in he HJM models and he marke models. The HJM models are based on assumpions abou he dynamics of he enire curve of insananeous, coninuously compounded forward raes, T f T. I urns ou ha only he volailiy srucure of he forward rae curve needs o specified in order o price erm srucure derivaives. We will discuss he general HJM model and various concree models in Chaper 10. The marke models are closely relaed o he HJM models, bu focus on he pricing of money marke producs such as caps, floors, and swapions. These producs involve LIBOR raes ha are se for specific periods, e.g. 3 monhs, 6 monhs, and 12 monhs, wih a similar compounding period. The marke models are all based on as assumpion abou a number of forward LIBOR raes or swap raes. Again, only he volailiy srucure of hese raes needs o be specified. Marke models are sudied in Chaper 11. The hird subclass of relaive pricing models consiss of so-called calibraed diffusion models. These models can be seen as exensions of absolue pricing models of he diffusion ype. The basic idea is o replace one of he consan parameers in a diffusion model by a suiable deerminisic funcion of ime ha will make he erm srucure of he model exacly mach he currenly observed erm srucure in he marke. These calibraed diffusion models can be reformulaed as HJM models, bu since hey are developed in a special way we rea hem separaely in Chaper 9. 5.9 Exercises EXERCISE 5.1 Show Equaion 5.5).

Chaper 6 The Economics of he Term Srucure of Ineres Raes 6.1 Inroducion A bond is nohing bu a sandardized and ransferable loan agreemen beween wo paries. The issuer of he bond is borrowing money from he holder of he bond and promises o pay back he loan according o a predefined paymen scheme. The presence of he bond marke allows individuals o rade consumpion opporuniies a differen poins in ime among each oher. An individual who has a clear preference for curren capial o finance invesmens or curren consumpion can borrow by issuing a bond o an individual who has a clear preference for fuure consumpion opporuniies. The price of a bond of a given mauriy is, of course, se o align he demand and supply of ha bond, and will consequenly depend on he araciveness of he real invesmen opporuniies and on he individuals preferences for consumpion over he mauriy of he bond. The erm srucure of ineres raes will reflec hese dependencies. In Secions 6.2 and 6.3 we derive relaions beween equilibrium ineres raes and aggregae consumpion and producion in seings wih a represenaive agen. In Secion 6.4 we give some examples of equilibrium erm srucure models ha are derived from he basic relaions beween ineres raes, consumpion, and producion. Since agens are concerned wih he number of unis of goods hey consume and no he dollar value of hese goods, he relaions found in hose secion apply o real ineres raes. However, mos raded bonds are nominal, i.e. hey promise he delivery of cerain dollar amouns, no he delivery of a cerain number of consumpion goods. The real value of a nominal bond depends on he evoluion of he price of he consumpion good. In Secion 6.5 we explore he relaions beween real raes, nominal raes, and inflaion. We consider boh he case where money has no real effecs on he economy and he case where money does affec he real economy. The developmen of arbirage-free dynamic models of he erm srucure was iniiaed in he 1970s. Unil hen, he discussions among economiss abou he shape of he erm srucure were based on some relaively loose hypoheses. The mos well-known of hese is he expecaion hypohesis, which posulaes a close relaion beween curren ineres raes or bond reurns and expeced fuure ineres raes or bond reurns. Many economiss sill seem o rely on he validiy of his hypohesis, and a lo of man power has been spend on esing he hypohesis empirically. In 139

6.2 Real ineres raes and aggregae consumpion 140 Secion 6.6, we review several versions of he expecaion hypohesis and discuss he consisency of hese versions. We argue ha neiher of hese versions will hold for any reasonable dynamic erm srucure model. Some alernaive radiional hypohesis are briefly reviewed in Secion 6.7. 6.2 Real ineres raes and aggregae consumpion In order o sudy he link beween ineres raes and aggregae consumpion, we assume he exisence of a represenaive agen maximizing an expeced ime-addiive uiliy funcion, E[ T 0 e δ uc ) d]. As discussed in Secion 5.2.9, a represenaive agen will exis in a complee marke. The parameer δ is he subjecive ime preference rae wih higher δ represening a more impaien agen. C is he consumpion rae of he agen, which is hen also he aggregae consumpion level in he economy. In erms of he uiliy and ime preference of he represenaive agen he sae price deflaor is herefore characerized by cf. Equaion 5.16) on page 111. ζ = 1 u C 0 ) e δ u C ), Assume ha C = C ) follows a sochasic process of he form dc = C [µ C d + σ C dz ], where z = z ) is a possibly muli-dimensional) sandard Brownian moion. By an applicaion of Iô s Lemma we obain C u 6.1) dζ = ζ [ δ ) C ) u µ C + 1 u ) C ) C ) 2 C2 u C ) σ C 2 d C u C ) u C ) ) σ C dz ]. Recalling from Secion 5.2.3 ha he equilibrium shor-erm ineres rae equals minus he relaive drif of he sae-price deflaor, we can wrie he shor rae as 6.2) r = δ + C u C ) u µ C 1 u C ) C ) 2 C2 u C ) σ C 2. This is he ineres rae a which he marke for shor-erm borrowing and lending will clear. The equaion relaes he equilibrium shor-erm ineres rae o he ime preference rae and he expeced growh rae µ C and he variance rae σ C 2 of aggregae consumpion growh over he nex insan. We can observe he following relaions: There is a posiive relaion beween he ime preference rae and he equilibrium ineres rae. The inuiion behind his is ha when he agens of he economy are impaien and has a high demand for curren consumpion, he equilibrium ineres rae mus be high in order o encourage he agens o save now and pospone consumpion. The muliplier of µ C in 6.2) is he relaive risk aversion of he represenaive agen, which is posiive. Hence, here is a posiive relaion beween he expeced growh in aggregae consumpion and he equilibrium ineres rae. This can be explained as follows: We expec higher fuure consumpion and hence lower fuure marginal uiliy, so posponed paymens due o saving have lower value. Consequenly, a higher reurn on saving is needed o mainain marke clearing.

6.2 Real ineres raes and aggregae consumpion 141 If u is posiive, here will be a negaive relaion beween he variance of aggregae consumpion and he equilibrium ineres rae. If he represenaive agen has decreasing absolue risk aversion, which is cerainly a reasonable assumpion, u has o be posiive. The inuiion is ha he greaer he uncerainy abou fuure consumpion, he more will he agens appreciae he sure paymens from he riskless asse and hence he lower a reurn is necessary o clear he marke for borrowing and lending. In he special case of consan relaive risk aversion, uc) = c 1 γ /1 γ), Equaion 6.2) simplifies o 6.3) r = δ + γµ C 1 2 γ1 + γ) σ C 2. In paricular, we see ha if he drif and variance raes of aggregae consumpion are consan, i.e. aggregae consumpion follows a geomeric Brownian moion, hen he shor-erm ineres rae will be consan over ime. Consequenly, he yield curve will be fla and consan over ime. This is clearly an unrealisic case. To obain ineresing models we mus eiher allow for variaions in he expecaion and he variance of aggregae consumpion growh or allow for non-consan relaive risk aversion or boh). We can also characerize he equilibrium erm srucure of ineres raes in erms of he expecaions and uncerainy abou fuure aggregae consumpion. 1 zero-coupon bond paying one consumpion uni a ime T is given by 6.4) B T = e δt ) E [u C T )] u, C ) The equilibrium ime price of a where C T is he uncerain fuure aggregae consumpion level. We can wrie he lef-hand side of he equaion above in erms of he yield y T of he bond as B T = e yt T ) 1 y T T ), using a firs order Taylor expansion. Turning o he righ-hand side of he equaion, we will use a second-order Taylor expansion of u C T ) around C : u C T ) u C ) + u C )C T C ) + 1 2 u C )C T C ) 2. This approximaion is reasonable when C T says relaively close o C, which is he case for fairly low and smooh consumpion growh and fairly shor ime horizons. Applying he approximaion, he righ-hand side of 6.4) becomes e δt ) E [u C T )] u C ) e δt ) 1 + u C ) u C ) E [C T C ] + 1 u C ) 2 u C ) Var [C T C ] 1 δt ) + e δt ) C u [ ] C ) CT E + 1 2 e δt ) C 2 u C ) u C ) u C ) Var C 1 where Var [ ] denoes he variance condiional on he informaion available a ime, and we have used he approximaion e δt ) 1 δt ). Subsiuing he approximaions of boh sides 1 The presenaion is adaped from Breeden 1986). [ CT C ], )

6.3 Real ineres raes and aggregae producion 142 ino 6.4) and rearranging, we find he following approximae expression for he zero-coupon yield: 6.5) y T δ + e δt ) C u ) C ) E [C T /C 1] u 1 C ) T 2 e δt ) C 2 u C ) Var [C T /C ] u. C ) T Again assuming u > 0, u < 0, and u > 0, we can sae he following conclusions. The equilibrium yield is increasing in he subjecive rae of ime preference. The equilibrium yield for he period [, T ] is posiively relaed o he expeced growh rae of aggregae consumpion over he period and negaively relaed o he uncerainy abou he growh rae of consumpion over he period. The inuiion for hese resuls is he same as for shor-erm ineres rae discussed above. We see ha he shape of he equilibrium ime yield curve T y T is deermined by how expecaions and variances of consumpion growh raes depend on he lengh of he forecas period. For example, if he economy is expeced o ener a shor period of high growh raes, real shor-erm ineres raes end o be high and he yield curve downward-sloping. 6.3 Real ineres raes and aggregae producion In order o sudy he relaion beween ineres raes and producion, we will look a a slighly simplified version of he general equilibrium model of Cox, Ingersoll, and Ross 1985a). Consider an economy wih a single physical good ha can be used eiher for consumpion or invesmen. All values are expressed in unis of his good. The insananeous rae of reurn on an invesmen in he producion of he good is 6.6) dη η = gx ) d + ξx ) dz 1, where z 1 is a sandard one-dimensional Brownian moion and g and ξ are well-behaved real-valued funcions given by Moher Naure) of some sae variable x. To be more specific, η 0 goods invesed in he producion process a ime 0 will grow o η goods a ime if he oupu of he producion process is coninuously reinvesed in his period. We can inerpre g as he expeced real growh rae of he economy and he volailiy ξ assumed posiive for all x) as a measure of he uncerainy abou he growh rae of he economy. The producion process has consan reurns o scale in he sense ha he disribuion of he rae of reurn is independen of he scale of he invesmen. There is free enry o he producion process. We can hink of individuals invesing in producion direcly by forming heir own firm or indirecly be invesing in socks of producion firms. For simpliciy we ake he firs inerpreaion. All producers, individuals and firms, ac compeiively so ha firms have zero profis and jus passes producion reurns on o heir owners. All individuals and firms ac as price akers. We assume ha he sae variable is one-dimensional and evolves according o he sochasic differenial equaion 6.7) dx = mx ) d + v 1 x ) dz 1 + v 2 x ) dz 2, where z 2 is anoher sandard one-dimensional Brownian moion independen of z 1, and m, v 1, and v 2 are well-behaved real-valued funcions. The insananeous variance rae of he sae variable is v 1 x) 2 + v 2 x) 2, he covariance rae of he sae variable and he real growh rae is ξx)v 1 x) so ha he correlaion beween he sae and he growh rae is ξx)v 1 x)/[v 1 x) 2 + v 2 x) 2 ]. Unless v 2 0, he sae variable is imperfecly correlaed wih he real producion reurns. If v 1 is posiive

6.3 Real ineres raes and aggregae producion 143 [negaive], hen he sae variable is posiively [negaively] correlaed wih he growh rae of he economy since ξ is assumed posiive). Since he sae deermines he expeced reurns and he variance of reurns on real invesmens, we may hink of x as a produciviy or echnology variable. In addiion o he invesmen in he producion process, we assume ha he agens have access o a financial asse wih a price P wih dynamics of he form 6.8) dp P = µ d + σ 1 dz 1 + σ 2 dz 2. As a par of he equilibrium we will deermine he relaion beween he expeced reurn µ and he volailiy coefficiens σ 1 and σ 2. Finally, he agens can borrow and lend funds a an insananeously riskless ineres rae r, which is also deermined in equilibrium. The marke is herefore complee. Oher financial asses affeced by z 1 and z 2 may be raded, bu hey will be redundan. We will ge he same equilibrium relaion beween expeced reurns and volailiy coefficiens for hese oher asses as for he one modeled explicily. For simpliciy we sick o he case wih a single financial asse. If an agen a each ime consumes a a rae of c 0, invess a fracion α of his wealh in he producion process, invess a fracion π of wealh in he financial asse, and invess he remaining fracion 1 α π of wealh in he riskless asse, his wealh W will evolve as 6.9) dw = {r W + W α gx ) r ) + W π µ r ) c } d + W α ξx ) dz 1 + W π σ 1 dz 1 + W π σ 2 dz 2. Since a negaive real invesmen is physically impossible, we should resric α o he non-negaive numbers. However, we will assume ha his consrain is no binding. Le us look a an agen maximizing expeced uiliy of fuure consumpion. The indirec uiliy funcion is defined as [ ] T JW, x, ) = sup E e δs ) uc s ) ds, α s,π s,c s) s [,T ] i.e. he maximal expeced uiliy he agen can obain given his curren wealh and he curren value of he sae variable. Applying dynamic programming echniques, i can be shown ha he opimal choice of α and π saisfies α = J [ W g r) σ2 1 + σ2 2 W J W W ξ 2 σ2 2 µ r) σ 1 6.10) π = J [ W σ 1 W J W W ξσ2 2 g r) + 1 ] 6.11) σ2 2 µ r) ξσ 2 2 ] + J W x W J W W σ 2 v 1 σ 1 v 2 ξσ 2, + J W x W J W W v 2 σ 2. In equilibrium, prices and ineres raes are such ha a) all agens ac opimally and b) all markes clear. In paricular, summing up he posiions of all agens in he financial asse we should ge zero, and he oal amoun borrowed by agens on a shor-erm basis should equal he oal amoun lend by agens. Since he available producion apparaus is o be held by some invesors, summing he opimal α s over invesors we should ge 1. Since we have assumed a complee marke, we can consruc a represenaive agen, i.e. an agen wih a given uiliy funcion so ha he equilibrium ineres raes and price processes are he same in he single agen economy as in he larger muli agen economy. Alernaively, we may hink of he case where all agens in he economy are idenical so ha hey will have he same indirec uiliy funcion and always make he same consumpion and invesmen choice.

6.3 Real ineres raes and aggregae producion 144 In an equilibrium, π = 0, and hence 6.11) implies ha ) 6.12) µ r = σ JW x 1 W J W W g r) )σ 2 v 2. ξ JW W J W W Subsiuing his ino he expression for α and using he fac ha α = 1 in equilibrium, we ge ha ) JW 1 = g r) σ2 1 + σ2 2 W J W W ξ 2 σ2 2 ) JW x σ2 v 1 σ 1 v 2 + W J W W ξσ 2 ) JW g r JW x = W J W W ξ 2 + W J W W σ 1 ξ σ 1 ξσ2 2 ) v1 ξ. g r) + ) J W x W J W W JW Consequenly, he equilibrium shor-erm ineres rae can be wrien as ) W JW W 6.13) r = g ξ 2 + J W x ξv 1. J W J W W J W W )σ 2 v 2 This equaion ies he equilibrium real shor-erm ineres rae o he producion side of he economy. Le us address each of he hree righ-hand side erms: The equilibrium real ineres rae r is posiively relaed o he expeced real growh rae g of he economy. The inuiion is ha for higher expeced growh raes, he producive invesmens are more aracive relaive o he riskless invesmen, so o mainain marke clearing he ineres rae has o be higher. The erm W J W W /J W σ 1 ξσ2 2 is he relaive risk aversion of he represenaive agen s indirec uiliy, which is assumed o be posiive. Hence, we see ha he equilibrium real ineres rae r is negaively relaed o he uncerainy abou he growh rae of he economy, represened by he insananeous variance ξ 2. For a higher uncerainy, he safe reurns of a riskless invesmen is relaively more aracive, so o esablish marke clearing he ineres rae has o decrease. The las erm in 6.13) is due o he presence of he sae variable. The covariance rae of he sae variable and he real growh rae of he economy is equal o ξv 1. Suppose ha high values of he sae variable represens good saes of he economy, where he wealh of he agen is high. Then he marginal uiliy J W will be decreasing in x, i.e. J W x < 0. If he sae variable and he growh rae of he economy are posiively correlaed, we see from 6.10) ha he hedge demand of he producive invesmen is decreasing, and hence he demand for deposiing money a he shor rae increasing, in he magniude of he correlaion boh J W x and J W W are negaive). To mainain marke clearing, he ineres rae mus be decreasing in he magniude of he correlaion as refleced by 6.13). We see from 6.12) ha he marke prices of risk are given by ) JW x W J W W, λ 2 = )v 2 = J W W J W W 6.14) λ 1 = g r ξ J W x J W v 2.

6.4 Equilibrium ineres rae models 145 Applying he relaion W JW W g r = J W we can rewrie λ 1 as W JW W 6.15) λ 1 J W 6.4 Equilibrium ineres rae models 6.4.1 Producion-based models ) ξ 2 J W x J W ξv 1, ) ξ J W x J W v 1. As a special case of heir general equilibrium model wih producion, Cox, Ingersoll, and Ross 1985b) consider a model where he represenaive agen is assumed o have logarihmic uiliy. Consequenly, he relaive risk aversion of he direc uiliy funcion is 1. I can be shown ha he indirec uiliy funcion of he agen will be of he form JW, x) = A ln W + Bx). In paricular, J W x = 0 and he relaive risk aversion of he indirec uiliy funcion is also 1. I follows from 6.13) ha he equilibrium real shor-erm ineres rae is equal o rx ) = gx ) ξx ) 2. The auhors furher assume ha he expeced rae of reurn and he variance rae of he reurn of he producive invesmen are boh proporional o he sae, i.e. gx) = k 1 x, ξx) 2 = k 2 x, where k 1 > k 2. Then he equilibrium shor-rae becomes rx) = k 1 k 2 )x kx. Assume now ha he sae variable follows a square-roo process dx = x κx ) d + ρσ x x dz 1 + 1 ρ 2 σ x x dz 2 = x κx ) d + σ x x d z, where z is a sandard Brownian moion wih correlaion ρ wih he sandard Brownian moion z 1. Then he dynamics of he real shor rae is 6.16) dr = r κr ) d + σ r r d z, where r = k x and σ r = kσ x. The marke prices of risk given in 6.14) and 6.15) simplify o λ 1 = ξx) = k 2 x = k 2 /k r, λ 2 = 0. We see ha his producion economy generaes he CIR erm srucure model summarized in 5.79) and sudied in deail in Secion 7.5. Longsaff and Schwarz 1992a) sudy a wo-facor version of he model. They assume ha he producion reurns are given by dη η = gx 1, x 2 ) d + ξx 2 ) dz 1, where gx 1, x 2 ) = k 1 x 1 + k 2 x 2, ξx 2 ) 2 = k 3 x 2,

6.4 Equilibrium ineres rae models 146 so ha he sae variable x 2 affecs boh expeced reurns and uncerainy of producion, while he sae variable x 1 only affecs he expeced reurn. Wih log uiliy he shor rae is again equal o he expeced reurn minus he variance, rx 1, x 2 ) = gx 1, x 2 ) ξx 2 ) 2 = k 1 x 1 + k 2 k 3 )x 2. The sae variables are assumed o follow independen square-roo processes, dx 1 = ϕ 1 κ 1 x 1 ) d + β 1 x1 dz 2, dx 2 = ϕ 2 κ 2 x 2 ) d + β 2 x2 dz 3, where z 2 are independen of z 1 and z 3, bu z 1 and z 3 may be correlaed. The marke prices of risk associaed wih he Brownian moions are λ 1 x 2 ) = ξx 2 ) = k 2 x2, λ 2 = λ 3 = 0. We will discuss he implicaions of his model in much more deail in Chaper 8 6.4.2 Consumpion-based models Oher auhors ake a consumpion-based approach for developing models of he erm srucure of ineres raes. For example, Goldsein and Zapaero 1996) presen a simple model in which he equilibrium shor-erm ineres rae is consisen wih he erm srucure model of Vasicek 1977). They assume ha aggregae consumpion evolves as dc = C [µ C d + σ C dz ], where z is a one-dimensional sandard Brownian moion, σ C is a consan, and he expeced consumpion growh rae µ C follows an Ornsein-Uhlenbeck process dµ C = κ µ C µ C ) + θ dz. The represenaive agen is assumed o have a consan relaive risk aversion of γ. from 6.3) ha he equilibrium real shor-erm ineres rae is I follows r = δ + γµ C 1 2 γ1 + γ)σ2 C wih dynamics 6.17) dr = κ r r ) d + σ r dz, where σ r = γθ and r = γ µ C + δ 1 2 γ1 + γ)σ2 C. The marke price of risk is given by λ = γσ C, which is consan. This is exacly he Vasicek model, which we inroduced in Secion 5.8. We will give a horough reamen of he model in Secion 7.4. In fac, we can generae any affine erm srucure model in his way. Assume ha he expeced growh rae and he variance rae of aggregae consumpion are linear in some sae variables, i.e. µ C = a 0 + n a i x i, σ C 2 = b 0 + i=1 n b i x i, i=1

6.5 Real and nominal ineres raes and erm srucures 147 hen he equilibrium shor rae will be r = δ + γa 0 1 ) 2 γ1 + γ)b 0 + γ n i=1 a i 1 ) 2 1 + γ)b i x i. Of course, we should have b 0 + n i=1 b ix i 0 for all values of he sae variables. The marke price of risk is λ = γσ C. If he sae variables x i follow processes of he affine ype, we have an affine erm srucure model. We will reurn o he affine models boh in Chaper 7 and Chaper 8. For oher erm srucure models developed wih he consumpion-based approach, see e.g. Bakshi and Chen 1997). 6.5 Real and nominal ineres raes and erm srucures In his secion we discuss he difference and relaion beween real ineres raes and nominal ineres raes. Nominal ineres raes are relaed o invesmens in nominal bonds, which are bonds ha promise given paymens in a given currency, say dollars. The purchasing power of hese paymens are uncerain, however, since he fuure price level of consumer goods is uncerain. Real ineres raes are relaed o invesmens in real bonds, which are bonds whose dollar paymens are adjused by he evoluion in he consumer price index and effecively provide a given purchasing power a he paymen daes. 2 Alhough mos bond issuers and invesors would probably reduce relevan risks by using real bonds raher han nominal bonds, he vas majoriy of bonds issued and raded a all exchanges is nominal bonds. Surprisingly few real bonds are raded. To he exen ha people have preferences for consumpion unis only and no for heir moneary holdings) hey should base heir consumpion and invesmen decisions on real ineres raes raher han nominal ineres raes. The relaions beween ineres raes and consumpion and producion discussed in he previous secions apply o real ineres raes. In a world where raded bonds are nominal we can quie easily ge a good picure of he erm srucure of nominal ineres raes. Bu wha abou real ineres raes? Tradiionally, economiss hink of nominal raes as he sum of real raes and he expeced consumer price) inflaion rae. This relaion is ofen referred o as he Fisher hypohesis or Fisher relaion in honor of Fisher 1907). However, neiher empirical sudies nor modern financial economics heories as we shall see below) suppor he Fisher hypohesis. In he following we shall firs derive some generally valid relaions beween real raes, nominal raes, and inflaion and invesigae he differences beween real and nominal asse prices. Then we will discuss wo differen ypes of models in which we can say more abou real and nominal raes. The firs seing follows he neoclassical radiion in assuming ha moneary holdings do no affec he preferences of he agens so ha he presence of money has no effecs on real raes and real asse reurns. Hence, he relaions derived earlier in his chaper sill applies. However, several empirical findings indicae ha he exisence of money does have real effecs. For example, real sock reurns are negaively correlaed wih inflaion and posiively correlaed wih money growh. Also, asses ha are posiively correlaed wih inflaion have a lower expeced reurn. 3 In 2 Since no all consumers will wan he same composiion of differen consumpion goods as ha refleced by he consumer price index, real bonds will no necessarily provide a perfecly cerain purchasing power for each invesor. 3 Such resuls are repored by, e.g., Fama 1981), Fama and Gibbons 1982), Chen, Roll, and Ross 1986), and Marshall 1992).

6.5 Real and nominal ineres raes and erm srucures 148 he second seing we consider below, money is allowed o have real effecs. Economies wih his propery are called moneary economies. 6.5.1 Real and nominal asse pricing As before, le ζ = ζ ) denoe a sae-price deflaor, which evolves over ime according o dζ = ζ [r d + λ dz ], where r = r ) is he shor-erm real ineres rae and λ = λ ) is he marke price of risk. From Equaion 5.14), we know ha he ime real price of a real zero-coupon bond mauring a ime T is given by B T [ ] ζt = E. ζ If he real price S = S ) of an asse follows he sochasic process hen we know ha ds = S [µ S d + σ S dz ], 6.18) µ S r = σ Sλ mus hold in equilibrium. From Chaper 5 we also know ha we can characerize real prices in erms of he risk-neural probabiliy measure Q, which is formally defined by he change-of-measure process [ ] { dq ξ E = exp 1 T } T λ s 2 ds λ s dz s. dp 2 The real price of an asse can hen be wrien as [ ] ζt S = E S T = E Q ζ [e T rs ds S T ]. In paricular, he ime real price of a real zero-coupon bond mauring a T is B T = E Q [e T rs ds]. In order o sudy nominal prices and ineres raes, we inroduce he consumer price index I, which is inerpreed as he dollar price I of a uni of consumpion. We wrie he dynamics of I = I ) as 6.19) di = I [i d + σ I dz ]. We can inerpre di /I as he realized inflaion rae over he nex insan, i as he expeced inflaion rae, and σ I as he percenage volailiy vecor of he inflaion rae. Consider now a nominal bank accoun which over he nex insan promises a riskless moneary reurn represened by he nominal shor-erm ineres rae r. If we le N denoe he ime dollar value of such an accoun, we have ha dn = r N d.

6.5 Real and nominal ineres raes and erm srucures 149 The real price of his accoun is Ñ = N /I, since his is he number of unis of he consumpion good ha has he same value as he accoun. An applicaion of Iô s Lemma implies a real price dynamics of 6.20) dñ = [ r Ñ i + σ I 2) ] d σ I dz. Noe ha he real reurn on his insananeously nominally riskless asse, dñ/ñ, is risky. Since he percenage volailiy vecor is given by σ I, he expeced reurn is given by he real shor rae plus σ I λ. Comparing his wih he drif erm in he equaion above, we have ha r i + σ I 2 = r σ Iλ. Consequenly he nominal shor-erm ineres rae is given by 6.21) r = r + i σ I 2 σ Iλ, i.e. he nominal shor rae is equal o he real shor rae plus he expeced inflaion rae minus he variance of he inflaion rae minus a risk premium. The presence of he las wo erms invalidaes he Fisher relaion, which says ha he nominal ineres rae is equal o he sum of he real ineres rae and he expeced inflaion rae. The Fisher hypohesis will hold if and only if he inflaion rae is insananeously riskless. Since mos raded asses are nominal, i would be nice o have a relaion beween expeced nominal reurns and volailiy of nominal prices. For his purpose, le S denoe he dollar price of a financial asse and assume ha he price dynamics can be described by d S = S [ µ S d + σ S dz ]. The real price of his asse is given by S = S /I and by Iô s Lemma ds = S [ µs i σ Sσ I + σ I 2) d + σ S σ I ) dz ]. The expeced excess real rae of reurn on he asse is herefore µ S r = µ S i σ Sσ I + σ I 2 r = µ S r σ Sσ I σ Iλ, where we have inroduced he nominal shor rae r by applying 6.21). The volailiy vecor of he real reurn on he asse is σ S = σ S σ I. Subsiuing he expressions for µ S r and σ S ino he relaion 6.18), we obain 6.22) µ S r = σ S λ, where λ is he nominal marke price of risk vecor defined by 6.23) λ = σ I + λ. In erms of expecaions, we know ha S = E I [ ζ T ζ ] ST, I T

6.5 Real and nominal ineres raes and erm srucures 150 from which i follows ha S = E [ ζt ζ ] [ ] I ST = E ST, I T ζt ζ where ζ = ζ /I for any. Since he lef-hand side is he curren nominal price and he righhand side involves he fuure nominal price or payoff, i is reasonable o call ζ = ζ ) a nominal sae-price deflaor. Is dynamics is given by 6.24) d ζ = ζ [ r d + λ dz ] so he drif rae is minus) he nominal shor rae and he volailiy vecor is minus) he nominal marke price of risk, compleely analogous o he real counerpars. We can also inroduce a nominal risk-neural measure Q by he change-of-measure process [ d ξ E Q ] { = exp 1 T } T λ s 2 ds λ s dz s. dp 2 Then he nominal price of a non-dividend paying asse can be wrien as [ ] S = E ST = E Q [e ζt ζ ] T r s ds ST. In paricular, he ime nominal price of a nominal zero-coupon bond mauring a T is [ ] B T = E = E Q [e ζt ζ ] T r s ds. To sum up, he prices of nominal bonds are relaed o he nominal shor rae and he nominal marke price of risk in exacly he same way as he prices of real bonds are relaed o he real shor rae and he real marke price of risk. Models ha are based on specific exogenous assumpions abou he shor rae dynamics and he marke price of risk can be applied boh o real erm srucures and o nominal erm srucures. This is indeed he case for mos popular erm srucure models. The auhors ha offer heoreical argumens for heir proposed erm srucure model, cf. Secion 6.4, ypically rely on relaions beween real raes and real producion or consumpion. Neverheless, hese models are ofen applied on nominal bonds and erm srucures. This is generally no suppored by economic heory, as we can see from he relaion 6.21) beween he nominal shor rae and he real shor rae and he relaion 6.23) beween he real and he nominal marke price of risk. Above we derived an equilibrium relaion beween real and nominal shor-erm ineres raes. Wha can we say abou he relaion beween longer-erm real and nominal ineres raes? Applying he well-known relaion Covx, y) = Exy) Ex) Ey), we can wrie [ ] B T ζt I = E ζ I T 6.25) = E [ ζt ζ = B T E [ I I T ] E [ I I T ] + Cov ζt ζ, I ] + Cov ζt ζ, I I T ). I T )

6.5 Real and nominal ineres raes and erm srucures 151 From he dynamics of he sae-price deflaor and he price index, we ge { ζ T T = exp r s + 12 ) } T ζ λ s 2 ds λ s dz s, { I T = exp i s 12 ) } T I σ Is 2 ds σ Is dz s, T which can be subsiued ino he above relaion beween prices on real and nominal bonds. However, he covariance-erm on he righ-hand side can only be explicily compued under very special assumpions abou he variaions over ime in r, i, λ, and σ I. 6.5.2 No real effecs of inflaion In his subsecion we will ake as given some process for he consumer price index and assume ha moneary holdings do no affec he uiliy of he agens direcly. As before he aggregae consumpion level is assumed o follow he process dc = C [µ C d + σ C dz ] so ha he dynamics of he real sae-price densiy is The shor-erm real rae is given by dζ = ζ [r d + λ dz ]. 6.26) r = δ C u C ) u C ) µ C 1 u C ) 2 C2 u C ) σ C 2 and he marke price of risk vecor is given by 6.27) λ = C u ) C ) u σ C. C ) rae as By subsiuing he expression 6.27) for λ ino 6.21), we can wrie he shor-erm nominal r = r + i σ I 2 C u ) C ) u σ C ) Iσ C. In he special case where he represenaive agen has consan relaive risk aversion, i.e. uc) = C 1 γ /1 γ), and boh he aggregae consumpion and he price index follow geomeric Brownian moions, we ge consan raes 6.28) 6.29) r = δ + γµ C 1 2 γ1 + γ) σ C 2, r = r + i σ I 2 γσ I σ C. Breeden 1986) considers he relaions beween ineres raes, inflaion, and aggregae consumpion and producion in an economy wih muliple consumpion goods. In general he presence of several consumpion goods complicaes he analysis considerably. Breeden shows ha he equilibrium nominal shor rae will depend on boh an inflaion rae compued using he average weighs of he differen consumpion goods and an inflaion rae compued using he marginal weighs of he differen goods, which are deermined by he opimal allocaion o he differen goods of

6.5 Real and nominal ineres raes and erm srucures 152 an exra dollar of oal consumpion expendiure. The average and he marginal consumpion weighs will generally be differen since he represenaive agen may shif o oher consumpion goods as his wealh increases. However, in he special probably unrealisic) case of Cobb-Douglas ype uiliy funcion, he relaive expendiure weighs of he differen consumpion goods will be consan. For ha case Breeden obains resuls similar o our one-good conclusions. 6.5.3 A model wih real effecs of money In he nex model we consider, cash holdings ener he direc uiliy funcion of he agens). This may be raionalized by he fac ha cash holdings faciliae frequen consumpion ransacions. In such a model he price of he consumpion good is deermined as a par of he equilibrium of he economy, in conras o he models sudied above where we ook an exogenous process for he consumer price index. We follow he se-up of Bakshi and Chen 1996) closely. The general model We assume he exisence of a represenaive agen who chooses a consumpion process C = C ) and a cash process M = M ), where M is he dollar amoun held a ime. As before, le I be he uni dollar price of he consumpion good. Assume ha he represenaive agen has an addiively ime-separable uiliy of consumpion and he real value of he moneary holdings, i.e. M = M /I. A ime he agen have he opporuniy o inves in a nominally riskless bank accoun wih a nominal rae of reurn of r. When he agen chooses o hold M dollars in cash over he period [, + d], she herefore gives up a dollar reurn of M r d, which is equivalen o a consumpion of M r d/i unis of he good. Given a real) sae-price deflaor ζ = ζ ), he oal cos of choosing C and M is hus E [ ζ 0 C + M r /I ) d ]. In sum, he opimizaion problem of he agen can be wrien as follows: [ ] sup E e δ U C, M /I ) d C,M ) [ s.. E 0 0 ζ C + M I r where W 0 is he iniial real) wealh of he agen. The firs order condiions are ) ] d W 0, 6.30) 6.31) e δ U C C, M /I ) = ψζ, e δ U M C, M /I ) = ψζ r, where U C and U M are he firs-order derivaives of U wih respec o he firs and second argumen, respecively. ψ is a Lagrange muliplier, which is se so ha he budge condiion holds as an equaliy. Again, we see ha he sae-price deflaor is given in erms of he marginal uiliy wih respec o consumpion. Imposing he iniial value ζ 0 = 1 and recalling he definiion of have 6.32) ζ = e δ U CC, M ) U C C 0, M 0 ). M, we We can apply he sae-price deflaor o value all paymen sreams. For example, an invesmen of one dollar a ime in he nominal bank accoun generaes a coninuous paymen sream a he

6.5 Real and nominal ineres raes and erm srucures 153 rae of r s dollars o he end of all ime. The corresponding real invesmen a ime is 1/I and he real dividend a ime s is r s /I s. Hence, we have he relaion [ 1 ] ζ s r s = E ds, I ζ I s or, equivalenly, 6.33) [ 1 = E I e δs ) U CC s, M ] s ) r s U C C, M ds. ) I s Subsiuing he firs opimaliy condiion 6.30) ino he second 6.31), we see ha he nominal shor rae is given by 6.34) r = U MC, M /I ) U C C, M /I ). The inuiion behind can be explained in he following way. If you have an exra dollar now you can eiher keep i in cash or inves i in he nominally riskless bank accoun. If you keep i in cash your uiliy grows by U M C, M /I )/I. If you inves i in he bank accoun you will earn a dollar ineres of r ha can be used for consuming r /I exra unis of consumpion, which will increase your uiliy by U C C, M /I ) r /I. A he opimum, hese uiliy incremens mus be idenical. Combining 6.33) and 6.34), we ge ha he price index mus saisfy he recursive relaion [ 1 6.35) = E e δs ) U M C s, M ] s ) 1 I U C C, M ds. ) I s Le us find expressions for he equilibrium real shor rae and he marke price of risk in his seing. As always, he real shor rae equals minus he percenage drif of he sae-price deflaor, while he marke price of risk equals minus he percenage volailiy vecor of he sae-price deflaor. In an equilibrium, he represenaive agen mus consume he aggregae consumpion and hold he oal money supply in he economy. Suppose ha he aggregae consumpion and he money supply follow exogenous processes of he form dc = C [µ C d + σ C dz ], dm = M [µ M d + σ M dz ]. Assuming ha he endogenously deermined price index will follow a similar process, he dynamics of M = M /I will be di = I [i d + σ I dz ], d M = M [ µ M d + σ M dz ], where µ M = µ M i + σ I 2 σ Mσ I, σ M = σ M σ I. Given hese equaions and he relaion 6.32), we can find he drif and he volailiy vecor of he sae-price deflaor by an applicaion of Iô s Lemma. We find ha he equilibrium real shor-erm

6.5 Real and nominal ineres raes and erm srucures 154 ineres rae can be wrien as 6.36) C U CC C, r = δ + M ) ) U C C, M µ C + M U CM C, M ) ) ) U C C, M ) 1 C 2 U CCC C, M ) 2 U C C, M σ C 2 1 M 2 U CMM C, M ) ) 2 U C C, M σ M 2 C M U CCM C, M ) ) U C C, M σ C ) σ M, while he marke price of risk vecor is 6.37) λ = = C u C ) u C ) C u C ) u C ) ) σ C + ) σ C + µ M M U CM C, M ) ) σ M U C C, M ) M U CM C, M ) ) U C C, M σ M σ I ). ) Wih U CM < 0, we see ha asses ha are posiively correlaed wih he inflaion rae will have a lower expeced real reurn, oher hings equal. inflaion risk so ha hey do no have o offer as high an expeced reurn. Inuiively such asses are useful for hedging The relaion 6.21) is also valid in he presen seing. Subsiuing he expression 6.37) for he marke price of risk ino 6.21), we obain 6.38) r r i + σ I 2 = C u ) C ) u σ C ) Iσ C An example M U CM C, M ) ) U C C, M σ I ) σ M. To obain more concree resuls, we mus specify he uiliy funcion and he exogenous processes C and M. Assume a uiliy funcion of he Cobb-Douglas ype, ) 1 γ C ϕ 1 ϕ M uc, M) = 1 γ, where ϕ is a consan beween zero and one, and γ is a posiive consan. The limiing case for γ = 1 is log uiliy, uc, M) = ϕ ln C + 1 ϕ) ln M. By insering he relevan derivaives ino 6.36), we see ha he real shor rae becomes 6.39) r = δ + [1 ϕ1 γ)]µ C 1 ϕ)1 γ) µ M 1 2 [1 ϕ1 γ)][2 ϕ1 γ)] σ C 2 + 1 2 1 ϕ)1 γ)[1 1 ϕ)1 γ)] σ M 2 + 1 ϕ)1 γ)[1 ϕ1 γ)]σ C σ M, which for γ = 1 simplifies o 6.40) r = δ + µ C σ C 2. We see ha wih log uiliy, he real shor rae will be consan if aggregae consumpion C = C ) follows a geomeric Brownian moion. From 6.34), he nominal shor rae is 6.41) r = 1 ϕ ϕ C M.

6.5 Real and nominal ineres raes and erm srucures 155 The raio C / M is called he velociy of money. If he velociy of money is consan, he nominal shor rae will be consan. Since M = M /I and I is endogenously deermined, he velociy of money will also be endogenously deermined. To idenify he nominal shor rae for any γ and he real shor rae for γ 1, we have o deermine he price level in he economy, which is given recursively in 6.35). This is possible under he assumpion ha boh C and M follow geomeric Brownian moions. We conjecure ha I = km /C for some consan k. From 6.35), we ge 1 k = 1 ϕ ϕ Insering he relaions C s = exp {µ C 12 ) C σ C 2 M s M = exp [ Cs ) 1 γ ) ] 1 e δs ) Ms E ds. C M } s ) + σcz s z ) {µ M 12 σ M 2 ) s ) + σ M z s z ) and applying a sandard rule for expecaions of lognormal variables, we ge 1 k = which implies ha he conjecure is rue wih k = ϕ 1 ϕ, }, { exp δ + 1 γ)µ C 1 2 σ C 2 ) µ M + σ M 2 + 1 } 2 1 γ)2 σ C 2 1 γ)σcσ M )s ) ds, δ 1 γ)µ C 12 σ C 2 ) + µ M σ M 2 12 1 γ)2 σ C 2 + 1 γ)σ Cσ M ). From an applicaion of Iô s Lemma, i follows ha he price index also follows a geomeric Brownian moion 6.42) di = I [i d + σ I dz ], where i = µ M µ C + σ C 2 σ M σ C, σ I = σ M σ C. Wih I = km /C, we have M = C /k, so ha he velociy of money C / M = k is consan, and he nominal shor rae becomes 6.43) r = 1 ϕ ϕ k = δ 1 γ)µ C 1 2 σ C 2 ) + µ M σ M 2 1 2 1 γ)2 σ C 2 + 1 γ)σ Cσ M, which is also a consan. Wih log uiliy, he nominal rae simplifies o δ + µ M σ M 2. In order o obain he real shor rae in he non-log case, we have o deermine µ M and σ M and plug ino 6.39). We ge µ M = µ C + 1 2 σ C 2 + σ M σ C and σ M = σ C and hence 6.44) r = δ + γµ C γ σ C 2 [ 1 2 1 + γ) + ϕ1 γ) ], which is also a consan. In comparison wih 6.28) for he case where money has no real effecs, he las erm in he equaion above is new.

6.5 Real and nominal ineres raes and erm srucures 156 Anoher example Bakshi and Chen 1996) also sudies anoher model specificaion in which boh nominal and real shor raes are ime-varying, bu evolve independenly of each oher. To obain sochasic ineres raes we have o specify more general processes for aggregae consumpion and money supply han he geomeric Brownian moions used above. which case we have already seen ha r = δ + µ C σ C 2, r = 1 ϕ ϕ The dynamics of aggregae consumpion is assumed o be C They assume log-uiliy γ = 1) in = 1 ϕ M ϕ dc = C [α C + κ C x ) d + σ C x dz 1 ], C I M. where x can be inerpreed as a echnology variable and is assumed o follow he process dx = κ x θ x x ) d + σ x x dz 1. The money supply is assumed o be M = M 0 e µ M g /g 0, where [ )] dg = g κ g θ g g ) d + σ g g ρ CM dz 1 + 1 ρ 2 CM dz 2, and where z 1 and z 2 are independen one-dimensional Brownian moions. Following he same basic procedure as in he previous model specificaion, he auhors show ha he real shor rae is 6.45) r = δ + α C + κ C σ 2 C)x, while he nominal shor rae is 6.46) r = δ + µ M )δ + µ M + κ gθ g ) δ + µ M + κ g + σ 2 g)g. Boh raes are ime-varying. The real rae is driven by he echnology variable x, while he nominal rae is driven by he moneary shock process g. In his se-up, shocks o he real economy have opposie effecs of he same magniude on real raes and inflaion so ha nominal raes are unaffeced. The real price of a real zero-coupon bond mauring a ime T is of he form B T = e at ) bt )x, while he nominal price of a nominal zero-coupon bond mauring a T is B T = ãt ) + bt )g δ + µ M + κ g + σ 2 g)g, where a, b, ã, and b are deerminisic funcions of ime for which Bakshi and Chen provide closedform expressions. In he very special case where hese processes are uncorrelaed, i.e. ρ CM = 0, he real and nominal erm srucures of ineres raes are independen of each oher! Alhough his is an exreme resul, i does poin ou ha real and nominal erm srucures in general may have quie differen properies.

6.6 The expecaion hypohesis 157 6.6 The expecaion hypohesis The expecaion hypohesis relaes curren long-erm raes and expeced fuure shor raes. This basic issue was discussed already by Fisher 1896) and furher developed and concreized by Hicks 1939) and Luz 1940). The presenaion below follows Cox, Ingersoll, and Ross 1981a) closely. There are various versions of he expecaion hypohesis. The mos exreme version of he hypohesis posulaes ha ineres raes are se so ha expeced holding period reurns on all possible invesmen sraegies in bonds and deposis are idenical. I is easy o demonsrae ha such a claim canno hold in an equilibrium unless ineres raes and bond prices are non-sochasic; cf. Exercise 6.1. 6.6.1 Versions of he pure expecaion hypohesis A less exreme version says ha he expeced reurn over he nex ime period is he same for all invesmens in bonds and deposis. In oher words here is no difference beween expeced reurns on long-mauriy and shor-mauriy bonds. In he coninuous-ime limi we consider reurns over he nex insan. hypohesis claims ha The riskless reurn over [, + d] is r d, so for any zero-coupon bond, he 6.47) E [ db T B T ] = r d, for all T >, or, equivalenly, ha B T = E [ e T rs ds], for all T >. This is he local pure expecaions hypohesis. Anoher inerpreaion says ha he reurn from holding a zero-coupon bond o mauriy should equal he expeced reurn from rolling over shor-erm bonds over he same ime period, i.e. 1 [ 6.48) = E e T rs ds], for all T > B T or, equivalenly, B T = E [e T rs ds]) 1, for all T >. This is he reurn-o-mauriy pure expecaion hypohesis. A relaed claim is ha he yield on any zero-coupon bond should equal he expeced yield on a roll-over sraegy in shor bonds. Since an invesmen of one a ime in he bank accoun generaes e T rs ds 1 a ime T, he ex-pos realized yield is T expecaion hypohesis says ha 6.49) y T = 1 T ln BT T [ 1 = E T r s ds. Hence, his yield-o-mauriy pure T r s ds ], or, equivalenly, B T = e E[ T rs ds], for all T >. Finally, he unbiased pure expecaion hypohesis saes ha he forward rae for ime T prevailing a ime < T is equal o he ime expecaion of he shor rae a ime T, i.e. ha

6.6 The expecaion hypohesis 158 forward raes are unbiased esimaes of fuure spo raes. In symbols, This implies ha ln B T = f T = E [r T ], for all T >. T f s ds = T [ ] T E [r s ] ds = E r s ds, from which we see ha he unbiased version of he pure expecaion hypohesis is indisinguishable from he yield-o-mauriy version. This is no he case in a discree-ime seing where hese wo versions can be shown o be inconsisen wih each oher.) We will firs show ha he hree versions are inconsisen when fuure raes are uncerain. This follows from an applicaion of Jensen s inequaliy which saes ha if X is a random variable and f is a convex funcion, i.e. f > 0, hen fe[x]) < E[fX)]. Since fx) = e x is a convex funcion, we have E[e X ] > e E[X] for any random variable X. In paricular for X = T r s ds, we ge [ E e T rs ds] > e E[ T rs ds] e E[ T rs ds] > E [e T rs ds]) 1. For X = T r s ds, we ge [ E e T rs ds] > e E[ T rs ds] = e E[ T rs ds]. We can conclude ha a mos one of he versions of he pure expecaions hypohesis can hold. 6.6.2 The pure expecaion hypohesis and equilibrium Nex, le us see wheher he differen versions can be consisen wih any equilibrium. Assume ha ineres raes and bond prices are generaed by a d-dimensional sandard Brownian moion z. Assuming absence of arbirage here exiss a marke price of risk process λ so ha for any mauriy T, he zero-coupon bond price dynamics is of he form [ 6.50) db T = B T r + σ T ) ) λ d + σ T ) ] dz, where σ T denoes he d-dimensional sensiiviy vecor of he bond price. Recall ha he same λ applies o all zero-coupon bonds so ha λ is independen of he mauriy of he bond. Comparing wih 6.47), we see ha he local expecaion hypohesis will hold if and only if ) σ T λ = 0 for all T. This is rue if eiher invesors are risk-neural or ineres rae risk is uncorrelaed wih consumpion. Neiher of hese condiions hold in real life. To evaluae he reurn-o-mauriy version, firs noe ha an applicaion of Iô s Lemma on 6.50) show ha ) 1 d B T = 1 B T [ r σ T ) λ + σ T 2) d σ T ) ] dz. On he oher hand, according o he hypohesis 6.48) he percenage drif of 1/B T equals r. 4 To mach he wo expressions for he drif, we mus have 6.51) σ T ) λ = σ T 2, for all T. 4 This can be seen as follows. The absolue drif of 1/B T equals he limi of 1 E[ 1 B + T 1 B T ] as 0.

6.6 The expecaion hypohesis 159 Is his possible? Cox, Ingersoll, and Ross 1981a) conclude ha i is impossible. If he exogenous shock z and herefore σ T and λ are one-dimensional, hey are righ, since λ mus hen equal σ T, and his mus hold for all T. Since λ is independen of T and he volailiy σ T approaches zero for T, his canno hold when ineres raes are sochasic. However, as poined ou by McCulloch 1993) and Fisher and Gilles 1998), in muli-dimensional cases he key condiion 6.51) may indeed hold, a leas in very special cases. Le ϕ be a d-dimensional funcion wih he propery ha ϕτ) 2 is independen of τ. Define λ = 2ϕ0) and σ T = ϕ0) ϕt ). Then 6.51) is indeed saisfied. However, all such funcions ϕ seem o generae very srange bond price dynamics. The examples given in he wo papers menioned above are ) ) 2e τ e 2τ cosk2 τ) ϕτ) = k, ϕτ) = k 1 e τ 1, sink 2 τ) where k, k 1, k 2 are consans. We approach he unbiased or yield-o-mauriy version in a similar manner. From 6.50), Iô s Lemma implies ha 6.52) d ln B T ) = r + σ T ) λ 1 ) 2 σt 2 d + σ T ) dz. On he oher hand, he hypohesis 6.49) implies ha he absolue) drif of ln B T is equal o r. 5 Hence, he hypohesis will hold if and only if σ T ) λ = 1 2 σt 2, for all T. Again, i is possible ha he condiion holds. Jus le ϕ and σ T be as for he reurn-o-mauriy hypohesis and le λ = ϕ0). Bu such specificaions are no likely o represen real life erm srucures. The conclusion o be drawn from his analysis is ha neiher of he differen versions of he pure expecaion hypohesis seem o be consisen wih any reasonable descripion of he erm srucure of ineres raes. 6.6.3 The weak expecaion hypohesis Above we looked a versions of he pure expecaion hypohesis ha all aligns an expeced reurn or yield wih a curren ineres rae or yield. However, as poined ou by Campbell 1986), Applying he hypohesis 6.48), we ge [ ] 1 1 E B+ T 1 B T = 1 [ [e ]) T+ [ E r E s ds + E ])] T r e s ds = 1 T+ [e ] E r s ds T e r s ds T = E [e r s ds 1 e + As 0, he raio in he las expression will approach r, so ha he enire expression will approach r E [e T r s ds ] = r /B T. The percenage drif of 1/BT is herefore r according o he hypohesis. 5 This is analogous o he previous foonoe. According o he hypohesis, 1 [ ] E ln B+ T ln BT = 1 [ T ] [ T ]] [ E E + r s ds + E r s ds = 1 [ + ] E r s ds, + which approaches r as 0. r s ds ].

6.7 Liquidiy preference, marke segmenaion, and preferred habias 160 here is also a weak expecaion hypohesis ha allows for a difference beween he relevan expeced reurn/yield and he curren rae/yield, bu resrics his difference o be consan over ime. The local weak expecaion hypohesis says ha [ ] db T E B T = r + gt )) d for some deerminisic funcion g. In he pure version g is idenically zero. For a given ime-omauriy here is a consan insananeous holding erm premium. Comparing wih 6.50), we see ha his hypohesis will hold when he marke price of risk λ is consan and he bond price sensiiviy vecor σ T is a deerminisic funcion of ime-o-mauriy. These condiions are saisfied in he Vasicek 1977) model, which we will discuss in deail in Secion 7.4, and in oher models of he Gaussian class. Similarly, he weak yield-o-mauriy expecaion hypohesis says ha f T = E [r T ] + ht ) for some deerminisic funcion h wih h0) = 0, i.e. ha here is a consan insananeous forward erm premium. The pure version requires h o be idenically equal o zero. I can be shown ha his condiion implies ha he drif of ln B T ha also his hypohesis will hold when λ is consan and σ T as is he case in he Gaussian models. equals r + ht ). 6 Comparing wih 6.52), we see is a deerminisic funcion of T The class of Gaussian models have several unrealisic properies. For example, such models allow negaive ineres raes and requires bond and ineres rae volailiies o be independen of he level of ineres raes. So far, he validiy of even weak versions of he expecaion hypohesis has no been shown in more realisic erm srucure models. Considering our discussion of boh he pure and he weak expecaion hypohesis, analysis of he shape of he yield curve and models of erm srucure dynamics should no be based on his hypohesis. Hence, i is surprising, maybe even disappoining, ha empirical ess of he expecaion hypohesis have generaed such a huge lieraure in he pas. 6.7 Liquidiy preference, marke segmenaion, and preferred habias Anoher radiional explanaion of he shape of he yield curve is given by he liquidiy preference hypohesis inroduced by Hicks 1939). He realized ha he expecaion hypohesis basically ignores invesors aversion owards risk and argued ha expeced reurns on long bonds T 6 We proceed as in foonoe 5. From he weak yield-o-mauriy hypohesis, i follows ha ln B T = E [r s] + hs )) ds. Hence, 1 [ ] E ln B+ T ln BT = 1 [ T T E E + [r s] + hs + ))) ds + + ] T The limi of = 1 [ + E r s ds 1 1 T + hs + )) ds ) T hs ) ds hs + )) ds + ] E [r s] + hs )) ds T ) hs ) ds. as 0 is exacly he derivaive of T wih respec o. Applying Leibniz rule and h0) = 0, his derivaive equals T sum, he drif rae of ln B T becomes r + ht ) according o he hypohesis. hs ) ds h s ) ds = ht ). In

6.8 Concluding remarks 161 should exceed he expeced reurns on shor bonds o compensae for he higher price flucuaions of long bonds. According o his view he yield curve should end o be increasing. Noe ha he word liquidiy in he name of he hypohesis is no used in he usual sense of he word. Shor bonds are no necessarily more liquid han long bonds. A beer name would be he mauriy preference hypohesis. In conras he marke segmenaion hypohesis inroduced by Culberson 1957) claims ha invesors will ypically prefer o inves in bonds wih ime-o-mauriy in a cerain inerval, a mauriy segmen, perhaps in an aemp o mach liabiliies wih similar mauriies. For example, a pension fund wih liabiliies due in 20-30 years can reduce risk by invesing in bonds of similar mauriy. On he oher hand, cenral banks ypically operae in he shor end of he marke. Hence, separaed marke segmens can exis wihou any relaion beween he bond prices and he ineres raes in differen mauriy segmens. If his is really he case, we canno expec o see coninuous or smooh yield curves and discoun funcions across he differen segmens. A more realisic version of his hypohesis is he preferred habias hypohesis pu forward by Modigliani and Such 1966). An invesor may prefer bonds wih a cerain mauriy, bu should be willing o move away from ha mauriy if she is sufficienly compensaed in erms of a higher yield. 7 The differen segmens are herefore no compleely independen of each oher, and yields and discoun facors should depend on mauriy in a smooh way. I is really no possible o quanify he marke segmenaion or he preferred habias hypohesis wihou seing up an economy wih agens having differen favorie mauriies. The resuling equilibrium yield curve will depend heavily on he degree of risk aversion of he various agens as illusraed by an analysis of Cox, Ingersoll, and Ross 1981a). 6.8 Concluding remarks The equilibrium erm srucure of ineres raes wih invesor heerogeneiy or more general uiliy funcions han sudied in his chaper, see, e.g., Duffie and Epsein 1992), Wang 1996), Riedel 1999, 2000), Wacher 2002). The effecs of cenral banks on he erm srucure are discussed and modeled by, e.g., Babbs and Webber 1994), Balduzzi, Berola, and Foresi 1997), and Piazzesi 2001). 6.9 Exercises EXERCISE 6.1 The expecaion hypohesis) An exreme version of he expecaion hypohesis says ha prices are se so ha he expeced raes of reurn on any invesmens over a given period are idenical. The purpose of his exercise is o show ha his saemen canno hold. In he following we consider ime poins 0 < 1 < 2. a) Show ha if he expecaion hypohesis holds, hen 1 B 1 B 2 0 0 = 1 E 0 [ B 2 1 ]. Hin: Compare wo invesmen sraegies over he period [ 0, 1]. The firs sraegy is o buy a ime 0 zero-coupon bonds mauring a ime 1. The second sraegy is o buy a ime 0 zero-coupon bonds mauring a ime 2 and o sell hem again a ime 1. 7 In a sense he liquidiy preference hypohesis simply says ha all invesors prefer shor bonds.

6.9 Exercises 162 b) Show ha if he expecaion hypohesis holds, hen 1 B 2 0 = 1 B 2 1 B E 0 1 0 c) Show from he wo previous quesions ha he hypohesis implies ha [ ] 1 1 *) E 0 = [ E 0 B 2 ]. 1 d) Show ha *) can only hold under full cerainy. Hin: Use Jensen s inequaliy. [ 1 B 2 1 ]. EXERCISE 6.2 Go hrough he derivaions in Secion 6.5.3. EXERCISE 6.3 Consaninides 1992) develops a heory of he nominal erm srucure of ineres raes by specifying exogenously he nominal sae-price deflaor ζ. In a slighly simplified version, his assumpion is ha ζ = e g+x α) 2, where g and α are consans, and x = x ) follows he Ornsein-Uhlenbeck process dx = κx d + σ dz, where κ and σ are posiive consans wih σ 2 < κ and z = z ) is a sandard one-dimensional Brownian moion. a) Derive he dynamics of he nominal sae-price deflaor. Express he nominal shor-erm ineres rae, r, and he nominal marke price of risk, λ, in erms of he variable x. b) Find he dynamics of he nominal shor rae. c) Find parameer consrains ha ensure ha he shor rae says posiive? Hin: The shor rae is a quadraic funcion of x. Find he minimum value of his funcion. d) Wha is he disribuion of x T given x? e) Le Y be a normally disribued random variable wih mean µ and variance v 2. Show ha [ E e γy 2] } = 1 + 2γv 2 ) 1/2 exp { γµ2. 1 + 2γv 2 f) Use he resuls of he wo previous quesions o derive he ime price of a nominal zero-coupon bond wih mauriy T, i.e. B T. I will be an exponenial-quadraic funcion of x. Wha is he yield on his bond? g) Find he percenage volailiy σ T of he price of he zero-coupon bond mauring a T. h) The insananeous expeced excess rae of reurn on he zero-coupon bond mauring a T is ofen called he erm premium for mauriy T. Explain why he erm premium is given by σ T λ and show ha he erm premium can be wrien as 4σ 2 α 2 x ) 1 F T )) α 1 x α 1 F T ) )eκt ), 1 F T ) where 1 F τ) = ). σ 2 + 1 σ2 e κ κ 2κτ For which values of x will he erm premium for mauriy T be posiive/negaive? For a given sae x, is i possible ha he he erm premium is posiive for some mauriies and negaive for ohers?

Chaper 7 One-facor diffusion models 7.1 Inroducion This chaper is devoed o he sudy of one-facor diffusion models of he erm srucure of ineres raes. They all ake he shor rae as he sole sae variable and, hence, implicily assume ha he shor rae conains all he informaion abou he erm srucure ha is relevan for pricing and hedging ineres rae dependen claims. All he models assume ha he shor rae is a diffusion process 7.1) dr = αr, ) d + βr, ) dz, where z = z ) 0 is a sandard Brownian moion under he real-world probabiliy measure P. The marke price of risk a ime is of he form λr, ). The shor rae dynamics under he risk-neural probabiliy measure Q i.e. he spo maringale measure) is herefore 7.2) dr = ˆαr, ) d + βr, ) dz Q, where z Q = z Q ) is a sandard Brownian moion under Q, and ˆαr, ) = αr, ) βr, )λr, ). We le S R denoe he value space for he shor rae, i.e. he se of values which he shor rae can have wih sricly posiive probabiliy. 1 A model of he ype 7.2) is called ime homogeneous if ˆα and β are funcions of he ineres rae only and no of ime. Oherwise i is called ime inhomogeneous. In he ime homogeneous models he disribuion of a given variable a a fuure dae depends only on he curren shor rae and how far ino he fuure we are looking. For example, he disribuion of r +τ given r = r is he same for all values of he disribuion depends only on he horizon τ and he iniial value r. For he same reason asse prices only depend on he curren shor rae and he ime o mauriy of he asse. For example, he price of a zero-coupon bond B T = B T r, ) only depends on r and he ime o mauriy T, cf. Theorem 7.1 below. In ime inhomogeneous models, hese consideraions are no valid, which renders he analysis of such models more complicaed. Furhermore, ime homogeneiy seems o be a realisic propery: why should he drif and he 1 Recall ha since he real-world and he risk-neural probabiliy measures are equivalen, he process can have exacly he same values under he differen probabiliy measures. 163

7.2 Affine models 164 volailiy of he shor rae depend on he calendar dae? Surely, he drif and he volailiy change over ime, bu his is due o changes in fundamenal economic variables, no jus he passage of ime. However, ime inhomogeneous models have some pracical advanages, which makes hem worhwhile looking a. We will do ha in Chaper 9. In he presen chaper we consider only ime homogeneous models. We will focus on he pricing of bonds, forwards and fuures on bonds, and European opions on bonds wihin he differen models. As discussed in Chaper 2, hese opion prices lead o prices of oher imporan asses such as caps, floors, and European swapions. The pricing echniques applied are hose developed in Chaper 5: soluion of a parial differenial equaion PDE) or compuaion of he expeced payoff under a suiable maringale measure. In Secion 7.2 we will consider some general aspecs of he so-called affine models. Then in Secions 7.3 7.5 we will look a hree specific affine models, namely he classical models of Meron 1970), Vasicek 1977), and Cox, Ingersoll, and Ross 1985b). Some non-affine models are oulined and discussed in Secion 7.6. Secion 7.7 gives a shor inroducion o he issues of esimaing he parameers of he models and esing o wha exen he models are suppored by he daa. Finally, Secion 7.8 offers some concluding remarks. 7.2 Affine models In a ime homogeneous one-facor model, he dynamics of he shor rae is of he form dr = ˆαr ) d + βr ) dz Q under he spo maringale risk-neural) measure. page 121 is hen The fundamenal PDE of Theorem 5.10 on 7.3) V V r, ) + ˆαr) r r, ) + 1 2 βr)2 2 V r, ) rv r, ) = 0, r, ) S [0, T ), r2 wih he erminal condiion 7.4) V r, T ) = Hr), r S, where he funcion H denoes he ineres rae dependen payoff of he asse. In his secion we will sudy a subse of his class of models, namely he so-called affine models. An affine model is a model where he risk-adjused drif rae ˆαr) and he variance rae βr) 2 are affine funcions of he shor rae, i.e. of he form 7.5) ˆαr) = ˆϕ ˆκr, βr) 2 = δ 1 + δ 2 r, where ˆϕ, ˆκ, δ 1, and δ 2 are consans. We require ha δ 1 + δ 2 r 0 for all he values of r which he process for he shor rae can have, i.e. for r S, so ha he variance is well-defined. The dynamics of he shor rae under he spo maringale measure is herefore given by he sochasic differenial equaion 7.6) dr = ˆϕ ˆκr ) d + δ 1 + δ 2 r dz Q. This subclass of models is racable and resuls in nice, explici pricing formulas for bonds and forwards on bonds and, in mos cases, also for bond fuures and European opions on bonds.

7.2 Affine models 165 7.2.1 Bond prices, zero-coupon raes, and forward raes As before, B T denoes he price a ime of a zero-coupon bond giving a paymen of 1 uni of accoun wih cerainy a ime T and nohing a all oher poins in ime. We know ha in a one-facor model, his price can be wrien as a funcion of ime and he curren shor rae, B T = B T r, ). The following heorem shows ha, in a model of he ype 7.6), B T r, ) is an exponenial-affine funcion of he curren shor rae. The proof of his resul is based only on he fac ha B T r, ) saisfies he parial differenial equaion 7.3) wih he erminal condiion B T r, T ) = 1. Theorem 7.1 In he model 7.6) he ime price of a zero-coupon bond mauring a ime T is given as 7.7) B T r, ) = e at ) bt )r, where he funcions aτ) and bτ) saisfy he following sysem of ordinary differenial equaions: 7.8) 7.9) ogeher wih he condiions a0) = b0) = 0. 1 2 δ 2bτ) 2 + ˆκbτ) + b τ) 1 = 0, τ 0, T ), a τ) ˆϕbτ) + 1 2 δ 1bτ) 2 = 0, τ 0, T ), Proof: We will show ha he price B T r, ) in 7.7) is a soluion o he parial differenial equaion 7.3). Since a0) = b0) = 0, he erminal condiion B T r, T ) = 1 is saisfied for all r S. The relevan derivaives are 7.10) B T r, ) = BT r, ) a T ) + b T )r), B T r r, ) = BT r, )bt ), 2 B T r 2 r, ) = BT r, )bt ) 2. Afer subsiuing hese derivaives ino 7.3) and dividing hrough by B T r, ), we ge 7.11) a T ) + b T )r bt )ˆαr) + 1 2 bt )2 βr) 2 r = 0, r, ) S [0, T ). Subsiuing 7.5) ino 7.11) and gahering erms involving r, we find ha he funcions a and b mus saisfy he equaion a T ) ˆϕbT ) + 12 δ 1bT ) 2 ) ) 1 + 2 δ 2bT ) 2 + ˆκbT ) + b T ) 1 r = 0, r, ) S [0, T ). This can only be rue if 7.8) and 7.9) hold. 2 Conversely, i can be shown ha he zero-coupon bond prices B T r, ) are only of he exponenialaffine form 7.7), if he drif rae and he variance rae are affine funcions of he shor rae as 2 Suppose A + Br = 0 for all r S. Given r 1, r 2 S, where r 1 r 2. Then, A + Br 1 = 0 and A + Br 2 = 0. Subracing one of hese equaions from he oher, we ge B[r 1 r 2 ] = 0, which implies ha B = 0. I follows immediaely ha A mus also equal zero.

7.2 Affine models 166 in 7.5). 3 The differenial equaions 7.8) 7.9) are called Ricai equaions. The funcions a and b are deermined by firs solving 7.8) wih he condiion b0) = 0 o obain he b-funcion. The soluion o 7.9) wih he condiion a0) = 0 can be wrien in erms of he b-funcion as τ 7.12) aτ) = ˆϕ bu) du 1 τ 0 2 δ 1 bu) 2 du, 0 since aτ) = aτ) a0) = τ 0 a u) du. For many frequenly applied specificaions of ˆϕ, ˆκ, δ 1, and δ 2, explici expressions for a and b can be obained in his way. For oher specificaions he Ricai equaions can be solved numerically by very efficien mehods. In all he models we will consider, he funcion bτ) is posiive for all τ. Consequenly, bond prices will be decreasing in he shor rae consisen wih he radiional relaion beween bond prices and ineres raes. Nex, we sudy he yield curves in he affine models 7.6). The zero-coupon rae a ime for he period up o ime T is denoed by y T and is also a funcion of he curren shor rae, = y T r, ). Wih coninuous compounding we have y T cf. 1.10) on page 9. I follows from 7.7) ha 7.13) y T r, ) = ln BT r, ) T B T r, ) = e yt r,)t ), = at ) T + bt ) r, T i.e. any zero-coupon rae is an affine funcion of he shor rae. If b is posiive, all zero-coupon raes are increasing in he shor rae. An increase in he shor rae will induce an upward shif of he enire yield curve T y T r, ). However, unless bτ) is proporional o τ, he shif is no a parallel shif since he coefficien bt )/T ) is hen mauriy dependen. In he imporan models, his coefficien is decreasing in mauriy T, so ha a shif in he shor rae affecs zero-coupon raes of shor mauriies more han zero-coupon raes of long mauriies, which seems o be a reasonable propery. Noe ha he zero-coupon rae for a fixed ime o mauriy of τ can be wrien as 7.14) y +τ r, ) = aτ) τ + bτ) τ r, which is independen of, which again sems from he ime homogeneiy of he model. The forward rae f T a ime for a loan over an infiniesimally shor period beginning a ime T is also given by a funcion of he curren shor rae, f T we have cf. 1.14) on page 9. From 7.7) we ge ha B T f T T r, ) r, ) = B T r, ), 7.15) f T r, ) = a T ) + b T )r. = f T r, ). Wih coninuous compounding Hence, he forward raes are also affine in he shor rae r. For a fixed ime o mauriy τ, he forward rae is 3 For deails see Duffie 2001, Sec. 7E). f +τ r, ) = a τ) + b τ)r.

7.2 Affine models 167 Le us consider he dynamics of he price B T = B T r, ) of a zero-coupon bond wih a fixed mauriy dae T. Noe ha we are mosly ineresed in he evoluion of prices and ineres raes in he real world he maringale measures are only used for deriving he pricing formulas. In any one-facor model of he ype 7.1), Iôs Lemma implies ha B db T T = r, ) + αr, ) BT r r, ) + 1 2 βr, ) 2 2 B T ) r 2 r, ) d + βr, ) BT r r, ) dz under he real-world probabiliy measure. Since he funcion r, ) B T r, ) solves he parial differenial equaion 7.3), he drif can be rewrien as B T r, ) + αr, ) BT r r, ) + 1 2 βr, ) 2 2 B T r 2 r, ) The volailiy of he zero-coupon bond price is = r B T r, ) + βr, )λr, ) BT r r, ). σ T r, ) = B T r r, ) B T βr, ), r, ) so ha we can wrie he dynamics of he price as 7.16) db T = B T [ r + λr, )σ T r, ) ) d + σ T r, ) dz ]. In he ime homogeneous affine models i follows from 7.10) ha he volailiy is 7.17) σ T r, ) = bt )βr). Wih bt ) and βr) being posiive, σ T r, ) will be negaive. To be precise, he volailiy of he zero-coupon bond price is herefore he absolue value of σ T r, ), i.e. bt )βr). In equilibrium, risky asses will normally have an expeced rae of reurn ha exceeds he locally riskfree ineres rae. This can only be he case if he marke price of risk λr, ) is negaive. When we look a he dynamics of zero-coupon raes, we are ofen more ineresed in he evoluion of a rae wih a fixed ime o mauriy τ = T say, he 5 year ineres rae) raher han a rae wih a fixed mauriy dae T. Hence, we sudy he dynamics of ȳ τ = y +τ = y +τ r, ) for a fixed τ. Iô s Lemma and 7.13) imply ha 7.18) dȳ τ = bτ) τ αr ) d + bτ) τ βr ) dz under he real-world probabiliy measure. Here we have used ha 2 y/ r 2 = 0 and assumed ha he marke price of risk and, herefore, he drif of he shor rae under he real-world measure αr ) = ˆαr ) + λr )βr ) is ime homogeneous. Similarly, he forward rae wih fixed ime o mauriy τ is f τ = f +τ = f +τ r, ), which by Iô s Lemma and 7.15) evolves as d f τ = b τ)αr ) d + b τ)βr ) dz. 7.2.2 Forwards and fuures Theorem 2.1 on page 24 gives a general characerizaion of forward prices on zero-coupon bonds. Leing F T,S r, ) denoe he forward price a ime wih a curren shor rae of r for delivery a

7.2 Affine models 168 ime T of a zero-coupon bond mauring a ime S, we have ha F T,S r, ) = B S r, )/B T r, ). In he affine models where he zero-coupon price is given by 7.7), he forward price becomes 7.19) F T,S r, ) = exp { as ) at )) bs ) bt )) r}, where he funcions a and b are he same as in Theorem 7.1. For a fuures on a zero-coupon bond we le Φ T,S r, ) denoe he fuures price. From Secion 5.4.2 we have ha he fuures price is given by Φ T,S r, ) = E Q [ r, B S r T, T ) ] and can be found by solving he parial differenial equaion 5.44), which in a ime homogeneous one-facor model is 7.20) Φ T,S r, ) + αr) βr)λr)) ΦT,S r, ) r ogeher wih he erminal condiion Φ T,S r, T ) = B S r, T ). + 1 2 βr)2 2 Φ T,S r 2 r, ) = 0, r, ) S [0, T ), Theorem 7.2 Assume an affine model of he ype 7.6). For a fuures conrac wih final selemen dae T and a zero-coupon bond mauring a ime S as he underlying asse, he fuures price a ime wih a shor rae of r is given by 7.21) Φ T,S r, ) = e AT ) BT )r, where he funcions Aτ) and Bτ) saisfy he following sysem of ordinary differenial equaions: 7.22) 7.23) 1 2 δ 2Bτ) 2 + ˆκBτ) + B τ) = 0, τ 0, T ), A τ) ˆϕBτ) + 1 2 δ 1Bτ) 2 = 0, τ 0, T ), wih he condiions A0) = as T ) and B0) = bs T ), where a and b are as in Theorem 7.1. If δ 2 = 0, we have Bτ) = bτ + S T ) bτ). The soluion o 7.23) wih A0) = as T ) can generally be wrien as τ 7.24) Aτ) = as T ) + ˆϕ Bu) du 1 τ 0 2 δ 1 Bu) 2 du. 0 The proof of his heorem is analogous o he proof of Theorem 7.1, since he PDE 7.20) is almos idenical o he PDE 7.3) saisfied by he zero-coupon bond price. The las claim in he heorem above is lef for he reader as Exercise 7.3. The claim implies ha, for δ 2 = 0, he fuures price becomes 7.25) Φ T,S r, ) = e AT ) bs ) bt ))r. Comparing wih he forward price expression 7.19), we see ha, for δ 2 = 0, we have T,S F T,S Φ r r, ) F T,S r, ) = r r, ) Φ T,S r, ),

7.2 Affine models 169 i.e. any change in he erm srucure of ineres raes will generae idenical percenage changes in forward prices and fuures prices wih similar erms. If he underlying bond is a coupon bond wih paymens Y i a ime T i, i follows from Theorem 2.2 on page 25 ha he forward price a ime for delivery a ime T is given by 7.26) F T,cpn r, ) = Y i F T,Ti r, ), T i>t ino which we can inser 7.19) on he righ-hand side. From 5.28) we ge ha he same relaion holds for fuures prices: 7.27) Φ T,cpn r, ) = Y i Φ T,Ti r, ), T i>t ino which we can inser 7.21) on he righ-hand side. For Eurodollar fuures we have from 5.29) on page 118 ha he quoed fuures price is which in an affine model becomes Ẽ T r, ) = 500 400 E Q [ r, B T +0.25 r, T )) 1], Ẽ T r, ) = 500 400 E Q r, [e a0.25)+b0.25)rt ], where a and b are as in Theorem 7.1. Above we concluded ha for a fuures on a zero-coupon bond he fuures price is given by Φ T,S r, ) = E Q [ r, B S r, T ) ] ] = E Q r, [e as T ) bs T )rt = e AT ) BT )r, where A and B solve he differenial equaions 7.22) 7.23) wih A0) = as T ), B0) = bs T ). Analogously, we ge ha E Q r, ] [e a0.25)+b0.25)rt = e ÂT ) ˆBT )r, where  and ˆB solve he same differenial equaions, bu wih he condiions ˆB0) = b0.25). In paricular,  is given as Â0) = a0.25), τ 7.28) Âτ) = a0.25) + ˆϕ 0 The quoed Eurodollar fuures price is herefore ˆBu) du 1 2 δ 1 τ 0 ˆBu) 2 du. 7.29) Ẽ T r, ) = 500 400e ÂT ) ˆBT )r. If δ 2 = 0, we have ˆBτ) = bτ) bτ + 0.25). 7.2.3 European opions on coupon bonds: Jamshidian s rick A reasonable affine one-facor model mus have he propery ha bond prices are decreasing in he shor rae, which is he case if he funcion bτ) is posiive. This is rue in he specific models sudied laer in his chaper. This propery can be used o show ha a European call opion on a coupon bond can be seen as a porfolio of European call opions on zero-coupon bonds. Since his resul was firs derived by Jamshidian 1989), we shall refer o i as Jamshidian s rick.

7.2 Affine models 170 Le us firs recall he noaion of Secion 2.7. Since prices now depend on he shor rae, we le C K,T,S r, ) denoe he price a ime, given he prevailing shor rae r = r, of a European call opion expiring a ime T wih an exercise price of K wrien on a zero-coupon bond paying one uni of accoun a ime S. Here, T < S. We also consider an opion on a coupon bond wih paymens Y i a ime T i i = 1, 2,..., n), where T 1 < T 2 < < T n. The price of his bond is Br, ) = Y i B Ti r, ), T i> where we sum over all he fuure paymen daes. C K,T,cpn r, ) denoes he price a ime of a European call wih expiry dae T, an exercise price of K, and wih he coupon bond as is underlying asse. Jamshidian s resul can hen be formulaed as follows: Theorem 7.3 In an affine one-facor model, where he zero-coupon bond prices are given by 7.7) wih bτ) > 0 for all τ, he price of a European call on a coupon bond is 7.30) C K,T,cpn r, ) = Y i C Ki,T,Ti r, ), T i>t where K i = B Ti r, T ), and r is defined as he soluion o he equaion 7.31) Br, T ) = K. Proof: The payoff of he opion on he coupon bond is ) maxbr T, T ) K, 0) = max Y i B Ti r T, T ) K, 0. Since he zero-coupon bond price B Ti r T, T ) is a monoonically decreasing funcion of he ineres rae r T, he whole sum T Y i>t ib Ti r T, T ) is monoonically decreasing in r T. Therefore, exacly one value r of r T will make he opion finish a he money, i.e. 7.32) Br, T ) = Y i B Ti r, T ) = K. T i>t T i>t Leing K i = B Ti r, T ), we have ha T Y i>t ik i = K. For r T < r, Y i B Ti r T, T ) > Y i B Ti r, T ) = K, T i>t T i>t and B Ti r T, T ) > B Ti r, T ) = K i, so ha ) max Y i B Ti r T, T ) K, 0 = Y i B Ti r T, T ) K T i>t T i>t = Y i B T i ) r T, T ) K i T i>t = Y i max B Ti r T, T ) K i, 0 ). T i>t

7.2 Affine models 171 For r T r, and Y i B Ti r T, T ) Y i B Ti r, T ) = K, T i>t T i>t B Ti r T, T ) B Ti r, T ) = K i, so ha ) max Y i B Ti r T, T ) K, 0 = 0 = Y i max B Ti r T, T ) K i, 0 ). T i>t T i>t Hence, for all possible values of r T we may conclude ha ) max Y i B Ti r T, T ) K, 0 = Y i max B Ti r T, T ) K i, 0 ). T i>t The payoff of he opion on he coupon bond is hus idenical o he payoff of a porfolio of opions on zero-coupon bonds, namely a porfolio consising for each i wih T i > T ) of Y i opions on a zero-coupon bond mauring a ime T i and an exercise price of K i. Consequenly, he value of he opion on he coupon bond a ime T equals he value of ha porfolio of opions on T i>t zero-coupon bonds. The formal derivaion is as follows: C K,T,cpn r, ) = E Q r, [e ] T ru du max Br T, T ) K, 0) [ = E Q r, e T ru du Y i max B Ti r T, T ) K i, 0 )] = Y i E Q r, T i>t T i>t = Y i C Ki,T,Ti r, ), T i>t [e T ru du max B Ti r T, T ) K i, 0 )] which ends he proof. To compue he price of a European call opion on a coupon bond we mus numerically solve one equaion in one unknown o find r ) and calculae n prices of European call opions on zerocoupon bonds, where n is he number of paymen daes of he coupon bond afer expiraion of he opion. In he following secions we shall go hrough hree differen ime homogeneous, affine models in which he price of a European call opion on a zero-coupon bond is given by relaively simple Black-Scholes ype expressions. 4 From 5.24) we ge ha he price of a European call wih expiraion dae T and an exercise price of K i which is wrien on a zero-coupon bond mauring a T i is given by C Ki,T,Ti r, ) = B Ti x, )Q Ti r,b Ti r T, T ) > K i ) K i B T r, )Q T r,b Ti r T, T ) > K i ). In he proof of Theorem 7.3 we found ha B Ti r T, T ) > K i r T < r 4 As discussed by Wei 1997), a very precise approximaion of he price can be obained by compuing he price of jus one European call opion on a paricular zero-coupon bond. However, since he exac price can be compued very quickly by Jamshidian s rick, he approximaion is no ha useful in hese one-facor models, bu more appropriae in muli-facor models. We will discuss he approximaion more closely in Chaper 12.

7.3 Meron s model 172 for all i. Togeher wih Theorem 7.3 hese expressions imply ha he price of a European call on a coupon bond can be wrien as 7.33) C K,T,cpn r, ) = T i>t = T i>t } Y i {B Ti r, )Q Ti r,r T < r ) K i B T r, )Q T r,r T < r ) Y i B Ti r, )Q Ti r,r T < r ) KB T r, )Q T r,r T < r ). Noe ha he probabiliies involved are probabiliies of he opion finishing in he money under differen probabiliy measures. The precise model specificaions will deermine hese probabiliies and, hence, he opion price. 7.3 Meron s model 7.3.1 The shor rae process Apparenly, he firs dynamic, coninuous-ime model of he erm srucure of ineres raes was inroduced by Meron 1970). In his model he shor rae follows a generalized Brownian moion under he spo maringale measure, i.e. 7.34) dr = ˆϕ d + β dz Q, where ˆϕ and β are consans. This is a very simple ime homogeneous affine model wih a consan drif rae and volailiy, which conradics empirical observaions. This assumpion implies ha 7.35) r T = r + ˆϕ[T ] + β[z Q T zq ], < T. Since z Q T zq N0, T ), we see ha, given he shor rae r = r a ime, he fuure shor rae r T is normally disribued under he spo maringale measure wih mean E Q r,[r T ] = r + ˆϕ[T ] and variance Var Q r,[r T ] = β 2 [T ]. If he marke price of risk λr, ) is consan, he drif rae of he shor rae under he real-world probabiliy measure will also be a consan ϕ = ˆϕ + βλ. In his case he fuure shor rae is also normally disribued under he real-world probabiliy measure wih mean r + ϕ[t ] and variance β 2 [T ]. A model like Meron s) where he fuure shor rae is normally disribued is called a Gaussian model. A normally disribued random variable can ake on any real valued number, so he value space S for he ineres rae in a Gaussian model is S = R. 5 In paricular, he shor rae in a Gaussian model can be negaive wih sricly posiive probabiliy, which conflics wih boh economic heory and empirical observaions. If he ineres rae is negaive, a loan is o be repaid wih a lower amoun han he original proceeds. This allows so-called maress arbirage: borrow money and pu hem ino your maress unil he loan is due. The difference beween he proceeds 5 Fuure ineres raes may no have he same disribuion under he real-world probabiliy measure and he maringale measures, bu we know ha he measures are equivalen so ha he value space is measure-independen.

7.3 Meron s model 173 and he repaymen is a riskless profi. Noe, however, ha in a deflaion period he smaller amoun o be repaid may represen a higher purchasing power han he original proceeds, so in such an economic environmen borrowing a negaive nominal raes is no an arbirage. On he oher hand, who would lend money a a negaive nominal rae? I is cerainly advanageous o keep he money in he pocke where hey earn a zero ineres rae. Hence, boh nominal and real ineres raes should say non-negaive. 6 7.3.2 Bond pricing Meron s model is of he affine form 7.6) wih ˆκ = 0, δ 1 = β 2, and δ 2 = 0. Theorem 7.1 implies ha he prices of zero-coupon bonds in Meron s model are exponenially-affine, 7.36) B T r, ) = e at ) bt )r. According o 7.8), he funcion bτ) solves he simple ordinary differenial equaion b τ) = 1 wih b0) = 0, which implies ha 7.37) bτ) = τ. The funcion aτ) can hen be deermined from 7.12): 7.38) aτ) = ˆϕ τ u du 1 τ 2 β2 u 2 du = 1 2 ˆϕτ 2 1 6 β2 τ 3. 0 0 Noe ha since he fuure shor rae is normally disribued, he fuure zero-coupon bond prices are lognormally disribued in Meron s model. 7.3.3 The yield curve Le us see which shapes he yield curve can have in Meron s model. Equaion 7.14) implies ha he τ-mauriy zero-coupon yield is y +τ = r + 1 2 ˆϕτ 1 6 β2 τ 2. Hence, for all values of ˆϕ and β, he yield curve is a parabola wih downward-sloping branches. The maximum zero-coupon yield is obained for a ime o mauriy of τ = 3 ˆϕ/2β 2 ) and equals r + 3 ˆϕ 2 /8β 2 ). Moreover, y +τ is negaive for τ > τ, where τ = 3 ) ˆϕ ˆϕ β 2 2 + 2 4 + 2β2 r 3 From 7.18) we see ha in Meron s model he τ-mauriy zero-coupon rae evolves as. dȳ τ = αr ) d + β dz under he real-world probabiliy measure, where αr ) = ˆϕ + βλr ) is he real-world drif rae of he shor-erm ineres rae. Since dȳ τ is obviously independen of τ, all zero-coupon raes will change by he same. In oher words, he yield curve will only change by parallel shifs. See also 6 Bank accouns ofen provide some services valuable o he cusomer, so ha heir deposi raes ne of fees) may be slighly negaive.

7.3 Meron s model 174 Exercise 7.1.) We can herefore conclude ha Meron s model can only generae a compleely unrealisic form and dynamics of he yield curve. Neverheless, we will sill derive forward prices, fuures prices, and European opion prices, since his illusraes he general procedure in a relaively simple seing. 7.3.4 Forwards and fuures By subsiuing he expressions 7.37) and 7.38) ino 7.19), we ge ha he forward price on a zero-coupon bond under Meron s assumpions is { F T,S r, ) = exp 1 [ S ) 2 T ) 2] + 1 [ 2 6 β2 S ) 3 T ) 3] } S T )r. In Meron s model δ 2 equals 0, so by Theorem 7.2 he B funcion in he fuures price on a zerocoupon bond is given by Bτ) = bτ + S T ) bτ) = S T. Applying 7.24), he fuures price can be wrien as Φ T,S r, ) = exp { 1 2 ˆϕS T )S + T 2) 1 } 6 β2 S T ) 2 2T + S 3) S T )r. Forward and fuures prices on coupon bonds can be found by insering he expressions above ino 7.26) and 7.27). In Eq. 7.29), we ge ˆBτ) = bτ) bτ + 0.25) = 0.25 and from 7.28) we conclude ha Âτ) = a0.25) 0.25 ˆϕτ 1 2 0.25)2 β 2 τ = 1 2 0.25)2 ˆϕ + 1 6 0.25)3 β 2 0.25 ˆϕτ 1 2 0.25)2 β 2 τ. The quoed Eurodollar fuures price in Meron s model is herefore Ẽ T r, ) = 500 400e Âτ)+0.25r. 7.3.5 Opion pricing To price European opions on zero-coupon bonds in Meron s seing, we shall apply he T - forward maringale measure echnique inroduced in Secion 5.3.1 on page 113. The price on a European call opion wih expiry dae T and exercise price K wrien on a zero-coupon bond mauring a ime S is generally given by 7.39) C K,T,S r, ) = B T [ r, ) E QT r, max B S r T, T ) K, 0 )]. To compue he expecaion on he righ-hand side we mus know he disribuion of B S r T, T ) under he T -forward maringale measure given ha r = r. Recall ha he forward price for delivery a ime T of a zero-coupon bond expiring a ime S according o 2.3) on page 24 is F T,S = BS B T In paricular, F T,S T = BT S is deermining he payoff of he opion. We will find he disribuion of BT S = BS r T, T ) by deriving he dynamics of he forward price F T,S. From Secion 5.3.1 we have ha he forward price F T,S is a maringale under he T -forward maringale measure, so he forward price has a drif rae of zero under his probabiliy measure. The forward price is a funcion of he prices of wo zero-coupon bonds. We can herefore.

7.3 Meron s model 175 deermine he volailiy of he forward price by an applicaion of Iô s Lemma for funcions of muliple sochasic processes see Theorem 3.5). Firs noe ha he volailiy of he forward price will depend only on he volailiies of he wo zero-coupon bond prices, so ha we need no worry abou he drif raes of hese prices. Also noe ha volailiies are invarian o changes of probabiliy measure. In Meron s model, he relaive volailiy of he zero-coupon bond mauring a ime S is σ S r, ) = β[s ], cf. 7.17) and 7.37), so ha db S =... d β[s ]B S dz T, db T =... d β[t ]B T dz T. I now follows from Iô s Lemma check i!) ha 7.40) df T,S = σ S r, ) σ T r, ) ) F T,S = β[s T ]F T,S dz T. dz T Hence, he forward price follows a geomeric Brownian moion wih a drif rae of zero under T - forward maringale measure). In paricular, ln BT S T,S = ln FT is normally disribued wih variance [ ] 7.41) v, T, S) 2 Var QT r, ln F T,S T = β 2 S T ) 2 T ) and mean 7.42) m, T, S) E QT r, [ ln F T,S T ] = ln F T,S 1 2 v, T, S)2, cf. he analysis of he geomeric Brownian moion in Secion 3.8.1 on page 57. We can now compue he price of he call opion in 7.39) by an applicaion of Theorem A.4 in Appendix A: 7.43) where we have used he fac ha Here, d 1 and d 2 are given by 7.44) 7.45) d 1 = C K,T,S r, ) = B T [ r, ) E QT max F T,S r T, T ) K, 0 )] m, T, S) ln K v, T, S) d 2 = d 1 v, T, S) = and v, T, S) = β[s T ] T. = B T r, ) r, {E QT r, [ F T,S r T, T ) ] Nd 1 ) KNd 2 )} = B S r, )Nd 1 ) KB T r, )Nd 2 ), [ E QT r, F T,S r T, T ) ] = F T,S r, ) = BS r, ) B T r, ). 1 + v, T, S) = v, T, S) ln 1 B S v, T, S) ln r, ) KB T r, ) B S ) r, ) KB T + 1 v, T, S), r, ) 2 ) 1 v, T, S), 2 Noe ha he pricing formula 7.43) has he same srucure as he Black-Scholes-Meron formula: he price on he underlying asse muliplied by one probabiliy minus he presen value of he exercise price muliplied by anoher probabiliy. In addiion, he relevan uncerainy index is here v, T, S), where v, T, S) 2 [ = Var QT r, ln B S r T, T ) ] [ = Var r, ln B S r T, T ) ],

7.4 Vasicek s model 176 which corresponds o he erm σ 2 [T ] = Var S, [ln S T ] in he Black-Scholes-Meron formula. The price of a European call opion on a coupon bond can be found by combining he pricing formula 7.43) and Jamshidian s rick of Theorem 7.3: C K,T,cpn r, ) = { Y i B T i r, )Nd i 1) K i B T r, )Nd i 2) } 7.46) where T i>t = Y i B Ti r, )Nd i 1) B T r, ) Y i K i Nd i 2) T i>t T i>t = Y i B Ti r, )Nd i 1) KB T r, )Nd i 2), d i 1 = T i>t 1 B T v, T, T i ) ln i ) r, ) K i B T + 1 r, ) 2 v, T, T i), d i 2 = d i 1 v, T, T i ), v, T, T i ) = β[t i T ] T, and we have used he fac ha he d i 2 s are idenical, cf. he discussion a he end of Secion 7.2.3. 7.4 Vasicek s model 7.4.1 The shor rae process One of he inappropriae properies of Meron s model is he consan drif of he shor rae. Wih a consan posiive [negaive] drif he shor rae is expeced o increase [decrease] in all he fuure which is cerainly no realisic. Many empirical sudies find ha ineres raes exhibi mean reversion in he sense ha if an ineres rae is high by hisorical sandards, i will ypically fall in he near fuure. Conversely if he curren ineres rae is low. Vasicek 1977) assumes ha he shor rae follows an Ornsein-Uhlenbeck process: 7.47) dr = κ[θ r ] d + β dz, where κ, θ, and β are posiive consans. Noe ha his is he dynamics under he real-world probabiliy measure. As we saw in Secion 3.8.2 on page 59, his process is mean revering. If r > θ, he drif of he ineres rae is negaive so ha he shor rae ends o fall owards θ. If r < θ, he drif is posiive so ha he shor rae ends o increase owards θ. The shor rae is herefore always drawn owards θ, which we call he long-erm level of he shor rae. The parameer κ deermines he speed of adjusmen. As in Meron s model he volailiy of he shor rae is consan, which conflics wih empirical sudies of ineres raes. See Secion 3.8.2 for simulaed pahs illusraing he impac of he various parameers on he process. I follows from Secion 3.8.2 ha Vasicek s model is a Gaussian model. More precisely, he fuure shor rae in Vasicek s model is normally disribued wih mean and variance given by 7.48) E r, [r T ] = θ + r θ)e κ[t ], 7.49) Var r, [r T ] = β2 2κ 2κ[T 1 e ]) under he real-world probabiliy measure P. As T, he mean approaches θ and he variance approaches β 2 /2κ). As κ, he mean approaches he long-erm level θ and he variance

7.4 Vasicek s model 177 0.5 0.5 probabiliy densiy 1 2 5 probabiliy densiy 1 2 5 100 100-5.0% -2.5% 0.0% 2.5% 5.0% 7.5% 10.0% 12.5% 15.0% fuure shor rae -5.0% -2.5% 0.0% 2.5% 5.0% 7.5% 10.0% 12.5% 15.0% fuure shor rae a) κ = 0.36, θ = 0.05, β = 0.0265 b) κ = 0.36, θ = 0.05, β = 0.05 0.5 0.5 probabiliy densiy 1 2 5 probabiliy densiy 1 2 5 100 100-5.0% -2.5% 0.0% 2.5% 5.0% 7.5% 10.0% 12.5% 15.0% fuure shor rae -5.0% -2.5% 0.0% 2.5% 5.0% 7.5% 10.0% 12.5% 15.0% fuure shor rae c) κ = 0.36, θ = 0.08, β = 0.0265 d) κ = 0.7, θ = 0.05, β = 0.0265 Figure 7.1: The disribuion of r T r = 0.05. for T = 0.5, 1, 2, 5, 100 years given a curren shor rae of approaches zero. approaches β 2 [T ]. As κ 0, he mean approaches he curren shor rae r and he variance The curren difference beween he shor rae and is long-erm level is expeced o be halved over a ime period of lengh T = ln 2)/κ. Like oher Gaussian models, Vasicek s model assigns a posiive probabiliy o negaive values of he fuure shor rae and all oher fuure raes), despie he inappropriaeness of his propery. Figure 7.1 illusraes he disribuion of he shor rae 1 2, 1, 2, 5, and 100 years ino he fuure a a curren shor rae of r = 0.05 and four differen parameer combinaions. Figure 7.2 shows he real-world probabiliy of he fuure shor rae r T being negaive given r ) for he same four parameer consellaions. Since r T E r, [r T ])/ Var r, [r T ] is sandard normally disribued, his probabiliy is easily compued as ) ) r T E r, [r T ] P r, r T < 0) = P r, < E r,[r T ] = N E r,[r T ], Varr, [r T ] Varr, [r T ] Varr, [r T ] ino which we can inser 7.48) and 7.49). Clearly, his probabiliy is increasing in he ineres rae volailiy β and decreasing in he speed of adjusmen κ, in he long-erm level θ, and in he curren level of he shor rae r.

7.4 Vasicek s model 178 20% 15% sandard Probabiliy 10% 5% bea=0.05 hea=0.08 kappa=0.7 0% 0 2 4 6 8 10 12 Time horizon, T- Figure 7.2: The probabiliy ha r T is negaive given r = 0.05 as a funcion of he horizon T. The benchmark parameer values are κ = 0.36, θ = 0.05, and β = 0.0265. For pricing purposes we are ineresed in he dynamics of he shor rae under he spo maringale measure and oher relevan maringale measures. Vasicek assumed wihou any explanaion ha he marke price of r-risk is consan, λr, ) = λ. Of course, he marke price of risk is deermined by he supply and demand of differen invesors, so a heoreical derivaion of marke prices of risk mus be based on specific assumpions abou invesor preferences, endowmens, ec. The marke price of risk as well as he shor rae and is dynamics are he oupu of such a model. Vasicek simply assumed ha he shor rae follows 7.47) and ha he marke price of risk is consan. However, as discussed in Secion 6.4, i is possible o consruc an equilibrium model resuling in Vasicek s assumpions. Since absence of arbirage is necessary for an equilibrium o exis, i may seem odd ha a model allowing negaive ineres raes is consisen wih equilibrium. The reason is ha he model does no allow agens o hold cash, so ha he maress arbirage sraegy canno be implemened. Therefore, he equilibrium model supporing he Vasicek model does no eliminae he criique of he lack of realism of Vasicek s model. Wih λr, ) = λ, he dynamics of he shor rae under spo maringale measure Q becomes 7.50) dr = κ[ˆθ r ] d + β dz Q, where ˆθ = θ λβ/κ. Relaive o he real-world dynamics, he only difference is ha he parameer θ is replaced by wih ˆθ. Hence, he process has he same qualiaive properies under he wo probabiliy measures. 7.4.2 Bond pricing Vasicek s model is an affine model since 7.50) is of he form 7.6) wih ˆκ = κ, ˆϕ = κˆθ, δ 1 = β 2, and δ 2 = 0. I follows from Theorem 7.1 ha he price of a zero-coupon bond is 7.51) B T r, ) = e at ) bt )r, where bτ) saisfies he ordinary differenial equaion κbτ) + b τ) 1 = 0, b0) = 0,

7.4 Vasicek s model 179 which has he soluion 7.52) bτ) = 1 κ and from 7.12) we ge 1 e κτ ), τ 7.53) aτ) = κˆθ bu) du 1 τ 0 2 β2 bu) 2 du = y [τ bτ)] + β2 0 4κ bτ)2. Here we have inroduced he auxiliary parameer and used ha τ 0 bu) du = 1 τ bτ)), κ y = ˆθ β2 2κ 2 τ 0 bu) 2 du = 1 1 τ bτ)) κ2 2κ bτ)2. In Secion 7.4.3 we shall see ha y is he long rae, i.e. he limi of he zero-coupon yields as he mauriy goes o infiniy. yields Le us look a some of he properies of he zero-coupon bond price. Simple differeniaion B T r r, ) = bt )BT r, ), 2 B T r 2 r, ) = bt )2 B T r, ). Since bτ) > 0, he zero-coupon price is a convex, decreasing funcion of he shor rae. The dependence of he zero-coupon bond price on he parameer κ is illusraed in Figure 7.3. A high value of κ implies ha he fuure shor rae is very likely o be close o θ, and hence he zero-coupon bond price will be relaively insensiive o he curren shor rae. For κ, he zero-coupon bond price approaches exp{ θ[t ]}, which is 0.7788 for θ = 0.05 and T = 5 as in he figure. 7 Conversely, he zero-coupon bond price is highly dependen on he shor rae for low values of κ. If he curren shor rae is below he long-erm level, a high κ will imply ha T r u du is expeced o be larger and exp{ T r [ u du} smaller) han for a low value of κ. In his case, he zero-coupon bond price B T r, ) = E Q r, exp )] T r u du is hus decreasing in κ. The converse relaion holds whenever he curren shor rae exceeds he long-erm level. Clearly, he zero-coupon price is decreasing in θ as shown in Figure 7.4 since wih higher θ we expec higher fuure raes and, consequenly, a higher value of T r u du. The prices of long mauriy bonds are more sensiive o changes in θ since in he long run θ is more imporan han he curren shor rae. Figure 7.5 shows he relaion beween zero-coupon bond prices and he ineres rae volailiy β. Obviously, he price is no a monoonic funcion of β. For low values of β he prices decrease in β, while he opposie is he case for high β-values. Long-erm bonds are more sensiive o β han shor-erm bonds. Figure 7.6 illusraes how he zero-coupon bond price depends on he marke price of risk parameer λ. Formula 7.16) on page 167 implies ha he dynamics of he zero-coupon bond price B T = B T r, ) can be wrien as db T = B T [ r + λσ T r, ) ) d + σ T r, ) dz ], 7 Noe ha ˆθ goes o θ for κ.

7.4 Vasicek s model 180 0.9 zero-coupon bond price 0.85 0.8 0.75 0.7 r = 0.02 r = 0.04 r = 0.06 r = 0.08 0.65 0 0.5 1 1.5 2 2.5 3 kappa Figure 7.3: The price of a 5 year zero-coupon bond as a funcion of he speed of adjusmen parameer κ for differen values of he curren shor rae r. The oher parameer values are θ = 0.05, β = 0.03, and λ = 0.15. 1 zero-coupon bond price 0.8 0.6 0.4 T-=2, r=0.02 T-=8, r=0.02 T-=2, r=0.08 T-=8, r=0.08 0.2 0 0.04 0.08 0.12 0.16 0.2 hea Figure 7.4: The price of a zero-coupon bond B T r, ) as a funcion of he long-erm level θ for differen combinaions of he ime o mauriy and he curren shor rae. The oher parameer values are κ = 0.3, β = 0.03, and λ = 0.15.

7.4 Vasicek s model 181 1.4 1.2 r=0.02, T-=1 zero-coupon bond price 1 0.8 0.6 0.4 r=0.08, T-=1 r=0.02, T-=8 r=0.08, T-=8 r=0.02, T-=15 r=0.08, T-=15 0.2 0 0.04 0.08 0.12 0.16 0.2 bea Figure 7.5: The price of a zero-coupon bond B T r, ) as a funcion of he volailiy parameer β for differen combinaions of he ime o mauriy T and he curren shor rae r. The values of he fixed parameers are κ = 0.3, θ = 0.05, and λ = 0.15. where σ T r, ) = bt )β is negaive. The more negaive λ is, he higher is he excess expeced reurn on he bond demanded by he marke paricipans, and hence he lower he curren price. Again he dependence is mos pronounced for long-erm bonds. We can also see ha he price volailiy σ T r, ) = bt )β is independen of he ineres rae level and is concavely, increasing in he ime o mauriy. Also noe ha he price volailiy depends on he parameers κ and β, bu no on θ or λ. Finally, Figure 7.7 depics he discoun funcion, i.e. he zero-coupon bond price as a funcion of he ime o mauriy. Noe ha wih a negaive shor rae, he discoun funcion is no necessarily decreasing. For τ, bτ) will approach 1/κ, whereas aτ) if y < 0, and aτ) + if y > 0. Consequenly, if y > 0, he discoun funcion approaches zero for T, which is a reasonable propery. On he oher hand, if y < 0, he discoun funcion will diverge o infiniy, which is clearly inappropriae. The long rae y can be negaive if he raio β/κ is sufficienly large. 7.4.3 The yield curve Since From 7.13) on page 166 he zero-coupon rae y T r, ) a ime for mauriy T is y T r, ) = at ) T + bt ) r. T 7.54) 7.55) a τ) = y [1 b τ)] + β2 2κ bτ)b τ), b τ) = e κτ,

7.4 Vasicek s model 182 1 0.9 zero-coupon bond price 0.8 0.7 0.6 0.5 r=0.02, T-=2 r=0.02, T-=10 r=0.08, T-=2 r=0.08, T-=10 0.4-0.5-0.4-0.3-0.2-0.1 0 0.1 0.2 0.3 lambda Figure 7.6: The price of zero-coupon bonds B T r, ) as a funcion of λ for differen combinaions of he ime o mauriy T and he curren shor rae r. The values of he fixed parameers are κ = 0.3, θ = 0.05, and β = 0.03. 1.2 1 zero-coupon bond price 0.8 0.6 0.4 0.2 r=-0.02 r=0.02 r=0.06 r=0.10 0 0 2 4 6 8 10 12 14 16 years o mauriy Figure 7.7: The price of zero-coupon bonds B T r, ) as a funcion of he ime o mauriy T. The parameer values are κ = 0.3, θ = 0.05, β = 0.03, and λ = 0.15.

7.4 Vasicek s model 183 an applicaion of l Hôspial s rule implies ha and hus bτ) aτ) lim = 1 and lim = 0, τ 0 τ τ 0 τ lim T yt r, ) = r, i.e. he shor rae is exacly he inercep of he yield curve. Similarly, i can be shown ha so ha bτ) aτ) lim = 0 and lim = y, τ τ τ τ lim T yt r, ) = y. The long rae y is herefore consan and, in paricular, no affeced by changes in he shor rae. The following heorem liss he possible shapes of he zero-coupon yield curve T y T r, ) under he assumpions of Vasicek s model. Theorem 7.4 In he Vasicek model he zero-coupon yield curve T y T r, ) will have one of hree shapes depending on he parameer values and he curren shor rae: i) If r < y β2 4κ 2, he yield curve is increasing; ii) if r > y + β2 2κ 2, he yield curve is decreasing; iii) for inermediae values of r, he yield curve is humped, i.e. increasing in T up o some mauriy T and hen decreasing for longer mauriies. Proof: The zero-coupon rae y T r, ) is given by y T at ) bt ) r, ) = + r T T ) bt ) β 2 = y + T 4κ bt ) + r y, where we have insered 7.53). We are ineresed in he relaion beween he zero-coupon rae and he ime o mauriy T, i.e. he funcion Y τ) = y +τ r, ). Defining hτ) = bτ)/τ, we have ha ) β 2 Y τ) = y + hτ) 4κ bτ) + r y. A sraighforward compuaion gives he derivaive ) β Y τ) = h 2 τ) 4κ bτ) + r y κτ β2 + hτ)e 4κ, where we have applied ha b τ) = e κτ. Inroducing he auxiliary funcion we can rewrie Y τ) as 7.56) Y τ) = h τ) gτ) = bτ) + hτ)e κτ h τ) ) r y + β2 4κ gτ).

7.4 Vasicek s model 184 Below we will argue ha h τ) < 0 for all τ and ha gτ) is a monoonically increasing funcion wih g0) = 2/κ and gτ) 1/κ for τ. This will imply he claims of he heorem as can be seen from he following argumens. If r y + β 2 /4κ 2 ) < 0, hen he parenhesis on he righ-hand side of 7.56) is negaive for all τ. yield curve will be monoonically increasing in he mauriy. In his case Y τ) > 0 for all τ, and hence he Similarly, he yield curve will be monoonically decreasing in mauriy, i.e. Y τ) < 0 for all τ, if r y β 2 /2κ 2 ) > 0. For he remaining values of r he expression in he parenhesis on he righ-hand side of 7.56) will be negaive for τ [0, τ ) and posiive for τ > τ, where τ is uniquely deermined by he equaion In ha case he yield curve is humped. r y + β2 4κ gτ ) = 0. Now le us show ha h τ) < 0 for all τ. Simple differeniaion yields h τ) = e κτ τ bτ))/τ 2, which is negaive if e κτ τ < bτ) or, equivalenly, if 1+κτ < e κτ, which is clearly saisfied compare he graphs of he funcions 1 + x and e x ). Finally, by applicaion of l Hôpial s rule, i can be shown ha g0) = 2/κ and gτ) 1/κ for τ. By differeniaion and edious manipulaions i can be shown ha g is monoonically increasing. Figure 7.8 shows he possible shapes of he yield curve. For any mauriy he zero-coupon rae is an increasing affine funcion of he shor rae. An increase [decrease] in he shor rae will herefore shif he whole yield curve upwards [downwards]. The change in he zero-coupon rae will be decreasing in he mauriy, so ha shifs are no parallel. Twiss of he yield curve where shor raes and long raes move in opposie direcions are no possible. According o 7.15) on page 166, he insananeous forward rae f T r, ) prevailing a ime is given by f T r, ) = a T ) + b T )r. Applying 7.54) and 7.55) his expression can be rewrien as f T r, ) = 1 e κ[t ]) β 2 2κ 2 1 e κ[t ]) ) ˆθ + e κ[t ] r 7.57) = 1 e κ[t ]) ) y + β2 ] e κ[t + e κ[t ] r. 2κ2 Because he shor rae can be negaive, so can he forward raes. 7.4.4 Forwards and fuures The forward price on a zero-coupon bond in Vasicek s model is obained by subsiuing he funcions b and a from 7.52) and 7.53) ino he general expression F T,S r, ) = exp { as ) at )) bs ) bt )) r}, cf. 7.19). In Vasicek s model he δ 2 parameer in he general dynamics 7.6) is zero, so ha he B funcion involved in he fuures price on a zero-coupon bond, Φ T,S r, ) = e AT ) BT )r, according o Theorem 7.2 is Bτ) = bτ + S T ) bτ) = e κτ bs T ).

7.4 Vasicek s model 185 10% 8% zero-coupon yield 6% 4% 2% 0% 0 2 4 6 8 10 12 14 16 18 20 years o mauriy, T- Figure 7.8: The yield curve for differen values of he shor rae. The parameer values are κ = 0.3, θ = 0.05, β = 0.03, and λ = 0.15. The long rae is hen y = 6%. The yield curve is increasing for r < 5.75%, decreasing for r > 6.5%, and humped for inermediae values of r. The curve for r = 6% exhibis a very small hump wih a maximum yield for a ime o mauriy of approximaely 5 years. Subsiuing his ino 7.24) we ge ha he A funcion in he fuures price expression is τ Aτ) = as T ) + κˆθbs T ) e κu du 1 τ 0 2 β2 bs T ) 2 e 2κu du 0 = as T ) + κˆθbs T )bτ) 1 2 β2 bs T ) 2 bτ) 1 ) 2 κbτ)2. Forward and fuures prices on coupon bonds are found by insering he formulas above ino 7.26) and 7.27). For Eurodollar fuures, 7.29) implies ha he quoed price is given by Ẽ T r, ) = 500 400e ÂT ) ˆBT )r, and since δ 2 = 0, we have ˆBτ) = bτ) bτ + 0.25) = b0.25)e κτ. From 7.28) we ge ha τ Âτ) = a0.25) κˆθb0.25) e κu du 1 τ 0 2 β2 b0.25) 2 e 2κu du 0 = a0.25) κˆθb0.25)bτ) 1 2 β2 b0.25) 2 bτ) 1 ) 2 κbτ)2. 7.4.5 Opion pricing To find he price of a European call opion on a zero-coupon bond in Vasicek s model we follow he same procedure as in Meron s model. The price can generally be wrien as C K,T,S r, ) = B T [ r, ) E QT r, max F T,S r T, T ) K, 0 )], where F T,S r T, T ) = B S r T, T ). We seek he disribuion of he forward price F T,S T = F T,S r T, T ) under he T -forward maringale measure Q T. We have already seen ha σ S r, ) = βbs )

7.4 Vasicek s model 186 in Vasicek s model, so ha he dynamics of he forward price becomes 7.58) df T,S = σ S r, ) σ T r, ) ) F T,S dz T = β[bs ) bt )]F T,S dz T. Hence, he forward price follows a geomeric Brownian moion wih a drif rae of zero under he T -forward maringale measure). In paricular, ln BT S T,S = ln FT is normally disribued wih variance 7.59) and mean v, T, S) 2 Var QT r, 7.60) m, T, S) E QT r, [ ln F T,S T ] T = β 2 [bs u) bt u)] 2 du = β2 2κ 3 1 e κ[s T ]) 2 2κ[T 1 e ]) [ ln F T,S T ] = ln F T,S 1 2 v, T, S)2, cf. Secion 3.8.1 on page 57. Theorem A.4 in Appendix A now implies ha 7.61) C K,T,S r, ) = B T [ r, ) E QT max F T,S r T, T ) K, 0 )] = B T r, ) r, {E QT r, [ F T,S r T, T ) ] Nd 1 ) KNd 2 )} = B S r, )Nd 1 ) KB T r, )Nd 2 ), where 7.62) 7.63) d 1 = 1 B S ) v, T, S) ln r, ) KB T + 1 v, T, S), r, ) 2 d 2 = d 1 v, T, S). This resul was firs derived by Jamshidian 1989). Figure 7.9 illusraes how he call price depends on he curren shor rae. An increase in he shor rae has he effec ha he presen value of he exercise price decreases, which leaves he call opion more valuable. This effec is known from he Black-Scholes-Meron sock opion formula. For bond opions here is an addiional effec. When he shor rae increases, he price of he underlying bond decreases, which will lower he call opion value. According o he figure, he laer effec dominaes a leas for he parameers used in generaing he graph. See Exercise 7.2 for more on he relaion beween he call price and he shor rae. The relaion beween he call price and he ineres rae volailiy β is shown in Figure 7.10. An increase in β yields a higher volailiy on he underlying bond, which makes he opion more valuable. However, he price of he underlying bond also depends on β. As shown in Figure 7.5, he bond price will decrease wih β for low values of β, and his effec can be so srong ha he opion can decrease wih β. Because he funcion bτ) is sricly posiive in Vasicek s model, we can apply Jamshidian s rick of Theorem 7.3 for he pricing of a European call opion on a coupon bond: C K,T,cpn r, ) = { Y i B T i r, )Nd i 1) K i B T r, )Nd i 2) } 7.64) T i>t = Y i B Ti r, )Nd i 1) KB T r, )Nd i 2), T i>t

7.4 Vasicek s model 187 0.25 price of call on a zero-coupon bond 0.2 0.15 0.1 0.05 K=0.7 K=0.75 K=0.8 K=0.85 K=0.9 0-0.02 0 0.02 0.04 0.06 0.08 0.1 0.12 he shor rae, r Figure 7.9: The price of a European call opion on a zero-coupon bond as a funcion of he curren shor rae r. The opion expires in T = 0.5 years, while he bond maures in S = 5 years. The prices are compued using Vasicek s model wih parameer values β = 0.03, κ = 0.3, θ = 0.05, and λ = 0.15. 0.35 price of call on a zero-coupon bond 0.3 0.25 0.2 0.15 0.1 0.05 r=0.02, K=0.6 r=0.02, K=0.7 r=0.08, K=0.6 r=0.08, K=0.7 0 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 bea Figure 7.10: The price of a European call opion on a zero-coupon bond as a funcion of he ineres rae volailiy β. The opion expires in T = 0.5 years, while he bond maures in τ = 5 years. The prices are compued using Vasicek s model wih he parameer values κ = 0.3, θ = 0.05, and λ = 0.15.

7.5 The Cox-Ingersoll-Ross model 188 where K i is defined as K i = B Ti r, T ), r is given as he soluion o he equaion Br, T ) = K, and d i 1 = v, T, T i ) = 1 B T v, T, T i ) ln i ) r, ) K i B T + 1 r, ) 2 v, T, T i), d i 2 = d i 1 v, T, T i ), β 2κ 3 1 e κ[ti T ]) 1 e 2κ[T ]) 1/2. Here we have used ha we know ha all he d i 2 s are idenical. 7.5 The Cox-Ingersoll-Ross model 7.5.1 The shor rae process Probably he mos popular one-facor model, boh among academics and praciioners, was suggesed by Cox, Ingersoll, and Ross 1985b). They assume ha he shor rae follows a square roo process 7.65) dr = κ [θ r ] d + β r dz, where κ, θ, and β are posiive consans. We will refer o he model as he CIR model. Some of he key properies of square roo processes were discussed in Secion 3.8.3. Jus as he Vasicek model, he CIR model for he shor rae exhibis mean reversion around a long erm level θ. The only difference relaive o Vasicek s shor rae process is he specificaion of he volailiy, which is no consan, bu an increasing funcion of he ineres rae, so ha he shor rae is less volaile for low levels han for high levels of he rae. This propery seems o be consisen wih observed ineres rae behavior wheher he relaion beween volailiy and shor rae is of he form β r is no so clear, cf. he discussion in Secion 7.7. The shor rae in he CIR model canno become negaive, which is a major advanage relaive o Vasicek s model. The value space of he shor rae in he CIR model is eiher S = [0, ) or S = 0, ) depending on he parameer values; see Secion 3.8.3 for deails. As discussed in Secion 6.4, he CIR model is a special case of a comprehensive general equilibrium model of he financial markes sudied by he same auhors in anoher aricle, Cox, Ingersoll, and Ross 1985a). The shor rae process 7.65) and an expression for he marke price of ineres rae risk, λr, ), is he oupu of he general model under specific assumpions on preferences, endowmens, and he underlying echnology. 8 According o he model he marke price of risk is λr, ) = λ r β, where λ on he righ-hand side is a consan. The drif of he shor rae under he spo maringale measure is herefore αr, ) βr, )λr, ) = κ[θ r] λ r β β r = κθ κ + λ)r. 8 In heir general model r is in fac he real shor erm ineres rae and no he nominal shor erm ineres rae ha we can observe. However, in pracice he model is used for he nominal raes.

7.5 The Cox-Ingersoll-Ross model 189 Defining ˆκ = κ + λ and ˆϕ = κθ, he process for he shor rae under he spo maringale measure can be wrien as 7.66) dr = ˆϕ ˆκr ) d + β r dz Q. Since his is of he form 7.6) wih δ 1 = 0 and δ 2 = β 2, we see ha he CIR model is also an affine model. We can rewrie he dynamics as ] dr = ˆκ [ˆθ r d + β r dz Q, where ˆθ = κθ/κ+λ). Hence, he shor rae also exhibis mean reversion under he spo maringale measure, bu boh he speed of adjusmen and he long erm level are differen han under he real-world probabiliy measure. In Vasicek s model, only he long erm level was changed by he change of measure. In he CIR model he disribuion of he fuure shor rae r T condiional on he curren shor rae r ) is given by he non-cenral χ 2 -disribuion. To be more precise, he probabiliy densiy funcion under he real-world probabiliy measure) for r T given r is where f rt r r) = f χ 2 2q+2,2u 2cr), c = 2κ β 2 1 e κ[t ]), u = cr e κ[t ], q = 2κθ β 2 1, and where f χ 2 a,b ) denoes he probabiliy densiy funcion for a non-cenrally χ 2 -disribued random variable wih a degrees of freedom and non-cenraliy parameer b. The mean and variance of r T are E r, [r T ] = θ + r θ)e κ[t ], Var r, [r T ] = β2 r κ e κ[t ] e 2κ[T ]) + β2 θ 2κ 1 e κ[t ]) 2. Noe ha he mean is jus as in Vasicek s model, cf. 7.48), while he expression for he variance is slighly more complicaed han in he Vasicek model, cf. 7.49). For T, he mean approaches θ and he variance approaches θβ 2 /2κ). For κ, he mean goes o θ and he variance goes o 0. For κ 0, he mean approaches he curren rae r and he variance approaches β 2 r[t ]. The fuure shor rae is also non-cenrally χ 2 -disribued under he spo maringale measure, bu relaive o he expressions above κ is o be replaced by ˆκ = κ + λ and θ by ˆθ = κθ/κ + λ). 7.5.2 Bond pricing Since he CIR model is affine, Theorem 7.1 implies ha he price of a zero-coupon bond mauring a ime T is 7.67) B T r, ) = e at ) bt )r,

7.5 The Cox-Ingersoll-Ross model 190 where he funcions aτ) and bτ) solve he ordinary differenial equaions 7.8) and 7.9), which for he CIR model become 7.68) 7.69) 1 2 β2 bτ) 2 + ˆκbτ) + b τ) 1 = 0, a τ) κθbτ) = 0 wih he condiions a0) = b0) = 0. The soluion o hese equaions is 7.70) 7.71) 2e γτ 1) bτ) = γ + ˆκ)e γτ 1) + 2γ, aτ) = 2κθ β 2 ln2γ) + 1 2 ˆκ + γ)τ ln [γ + ˆκ)eγτ 1) + 2γ] ), where γ = ˆκ 2 + 2β 2, cf. Exercise 7.4. Since B T r r, ) = bt )BT r, ), 2 B T r 2 r, ) = bt )2 B T r, ) and bτ) > 0, he zero-coupon bond price is a convex, decreasing funcion of he shor rae. Furhermore, he price is a decreasing funcion of he ime o mauriy; a concave, increasing funcion of β 2 ; a concave, increasing funcion of λ; and a convex, decreasing funcion of θ. The dependence on κ is deermined by he relaion beween he curren shor rae r and he long erm level θ: if r > θ, he bond price is a concave, increasing funcion of κ; if r < θ, he price is a convex, decreasing funcion of κ. is Manipulaing 7.16) slighly, we ge ha he dynamics of he zero-coupon price B T = B T r, ) db T = B T [ r 1 λbt )) d + σ T r, ) dz ], where σ T r, ) = bt )β r. The volailiy σ T r, ) = bt )β r of he zero-coupon bond price is hus an increasing funcion of he ineres rae level and an increasing funcion of he ime o mauriy, since b τ) > 0 for all τ. Noe ha he volailiy depends on ˆκ = κ + λ and β, bu similar o he Vasicek model) no on θ. 7.5.3 The yield curve Nex we sudy he zero-coupon yield curve T y T r, ). From 7.13) we have ha y T r, ) = I can be shown ha y r, ) = r and ha at ) T + y lim T yt r, ) = bt ) r. T 2κθ ˆκ + γ. Concerning he shape of he yield curve, Kan 1992) has shown he following resul: Theorem 7.5 In he CIR model he shape of he yield curve depends on he parameer values and he curren shor rae as follows: 1) If ˆκ > 0, he yield curve is decreasing for r ˆϕ/ˆκ and increasing for 0 r ˆϕ/γ. For ˆϕ/γ < r < ˆϕ/ˆκ, he yield curve is humped, i.e. firs increasing, hen decreasing.

7.5 The Cox-Ingersoll-Ross model 191 2) If ˆκ 0, he yield curve is increasing for 0 r ˆϕ/γ and humped for r > ˆϕ/γ. The proof of his heorem is raher complicaed and is herefore omied. Esimaions of he model ypically give ˆκ > 0, so ha he firs case applies. See references o esimaions in Secion 7.7.) The erm srucure of forward raes T f T r, ) is given by f T r, ) = a T ) + b T )r, which using 7.68) and 7.69) can be rewrien as ] 7.72) f T r, ) = r + ˆκ [ˆθ r bt ) 1 2 β2 rbt ) 2. 7.5.4 Forwards and fuures The forward price on a zero-coupon bond in he CIR model is found by subsiuing he funcions b and a from 7.70) and 7.71) ino he general expression F T,S r, ) = exp { as ) at )) bs ) bt )) r}, which is known from 7.19). The forward price on a coupon bond follows from 7.26). I is more complicaed o deermine he fuures price on a zero-coupon bond in he CIR model han i was in he models of Meron and Vasicek, due o he fac ha he parameer δ 2 is non-zero in he CIR model. From Theorem 7.2 we have ha he fuures price is of he form Φ T,S r, ) = e AT ) BT )r, where he funcion B is a soluion o he ordinary differenial equaion 7.22), which in he CIR model becomes 1 2 β2 Bτ) 2 + ˆκBτ) + B τ) = 0, τ 0, T ), wih he condiion B0) = bs T ). Then he funcion A can be deermined from 7.24), which in he presen case is The soluion is 7.73) 7.74) τ Aτ) = as T ) + κθ Bu) du. 0 2ˆκbS T ) Bτ) = β 2 bs T ) eˆκτ 1) + 2ˆκeˆκτ, Aτ) = as T ) 2κθ ) Bτ)eˆκτ β 2 ln. bs T ) The fuures price on a coupon bond follows from 7.27). According o 7.29) he quoed Eurodollar fuures price is Ẽ T r, ) = 500 400e ÂT ) ˆBT )r, where he funcions  and ˆB in he CIR model can be compued o be 7.75) 7.76) 2ˆκb0.25) ˆBτ) = β 2 b0.25) eˆκτ 1) + 2ˆκeˆκτ, ) Âτ) = a0.25) 2κθ ˆBτ)eˆκτ β 2 ln. b0.25)

7.5 The Cox-Ingersoll-Ross model 192 7.5.5 Opion pricing To price European opions on zero-coupon bonds we can ry o compue C K,T,S r, ) = B T [ r, ) E QT r, max B S r T, T ) K, 0 )] using he disribuion of r T under he T -forward maringale measure Q T given r = r). However, his is much more complicaed in he CIR model han in he models of Meron and Vasicek. According o Cox, Ingersoll, and Ross he price is 7.77) C K,T,S r, ) = B S r, )χ 2 h 1 ; f, g 1 ) KB T r, )χ 2 h 2 ; f, g 2 ), where χ 2 ; f, g) is he cumulaive disribuion funcion for a non-cenrally χ 2 -disribued random variable wih f degrees of freedom and non-cenraliy parameer g. The formula has he same srucure as in he models of Meron and Vasicek, bu he relevan disribuion funcion is no longer he normal disribuion funcion. The wo probabiliies involved in he pricing formula are probabiliies of he opion finishing in he money under wo differen probabiliy measures cf. 5.24)). The opion will finish in he money if and only if he shor rae prevailing a he expiry dae of he opion is below a cerain criical level. Since he shor rae is non-cenrally χ 2 -disribued also under he forward maringale measures), he relevan probabiliies are given by he cumulaive disribuion funcion for he non-cenral χ 2 -disribuion. The parameers f, g i, and h i are given by f = 4κθ β 2, h 1 = 2 r ξ + ψ + bs T )), h 2 = 2 rξ + ψ), g 1 = 2ξ 2 γ[t ] re ξ + ψ + bs T ), and we have inroduced he auxiliary parameers g 2 = 2ξ2 re γ[t ], ξ + ψ ξ = 2γ β 2 [ e γ[t ] 1 ], ψ = ˆκ + γ as T ) + ln K β 2, r =. bs T ) Noe ha r is exacly he criical ineres rae, i.e. he value of he shor rae for which he opion finishes a he money, since B S r, T ) = K. To implemen he formula, he following approximaion of he χ 2 -disribuion funcion is useful: χ 2 h; f, g) Nd), where ) m h d = k l), f + g m = 1 2 f + g)f + 3g) 3 f + 2g) 2, k = 2m 2 p [1 p1 m)1 3m)] ) 1/2, l = 1 + mm 1)p 1 2 mm 1)2 m)1 3m)p2, p = f + 2g f + g) 2.

7.6 Non-affine models 193 This approximaion was originally suggesed by Sankaran 1963) and has subsequenly been applied in he CIR model by Longsaff 1993). Because of he complexiy of he formula i is difficul o evaluae how he call price depends on he parameers and variables involved. Of course, he call price is an increasing funcion of he ime o mauriy of he opion 9 and a decreasing funcion of he exercise price. An increase in he shor rae r has wo effecs on he call price: he presen value of he exercise price decreases, bu he value of he underlying bond also decreases. According o Cox, Ingersoll, and Ross 1985b), numerical compuaions indicae ha he laer effec dominaes he firs so ha he call price is a decreasing funcion of he ineres rae, as we saw i in Vasicek s model. For he pricing of European opions on coupon bonds we can again apply Jamshidian s rick of Theorem 7.3: 7.78) C K,T,cpn r, ) = Y i B Ti r, )χ 2 h 1i ; f, g 1i ) KB T r, )χ 2 h 2 ; f, g 2 ), where T i>t h 1i = 2r ξ + ψ + bt i T )), h 2 = 2r ξ + ψ), g 1i = 2ξ 2 γ[t ] re ξ + ψ + bt i T ), and f, g 2, ξ, and ψ are defined jus below 7.77). This resul was firs derived by Longsaff 1993). 7.6 Non-affine models The financial lieraure conains many oher one-facor models han hose ha fi ino he affine framework sudied in he previous secions. In his secion we will go hrough he non-affine models ha have araced mos aenion, which are models where he fuure values of he shor rae are lognormally disribued. An apparenly popular model among praciioners is he model suggesed by Black and Karasinski 1991) and, in paricular, he special case considered by Black, Derman, and Toy 1990), he so-called BDT model. The general ime homogeneous version of he Black-Karasinski model is 7.79) dln r ) = κ[θ ln r ] d + β dz Q, where κ, θ, and β are consans. Typically, praciioners replace he parameers κ, θ, and β by deerminisic funcions of ime which are chosen o ensure ha he model prices of bonds and caps are consisen wih curren marke prices. We will discuss his idea in Chaper 9 and sick o he model wih consan parameers in his secion. Relaive o he Vasicek model r is replaced by ln r in he sochasic differenial equaion. Since r T given r ) is normally disribued in he Vasicek model, i follows ha ln r T given r ) is normally disribued in he Black-Karasinski model, i.e. r T is lognormally disribued. A pleasan consequence of his is ha he ineres rae says posiive. Also his model exhibis a form of mean reversion. Assume ha κ > 0. If r < e θ, he drif rae of 9 There are no paymens on he underlying asse in he life of he opion, so a European call is equivalen o an American call, which clearly increases in value as ime o mauriy is increased.

7.6 Non-affine models 194 ln r is posiive so ha r is expeced o increase. Conversely if r > e θ. The parameer κ measures he speed wih which ln r is drawn owards θ. An applicaion of Iô s Lemma gives ha dr = [κθ + 12 ] ) β2 r κr ln r d + βr dz Q. There are no closed-form pricing expressions neiher for bonds nor forwards, fuures, and opions wihin his framework. Black and Karasinski implemen heir model in a binomial ree in which prices can be compued by he well-known backward ieraion procedure. In he Black-Karasinski model 7.79) he fuure shor rae is lognormally disribued. Anoher model wih his propery is he one where he shor rae follows a geomeric Brownian moion ] 7.80) dr = r [α d + β dz Q, where α and β are consans. Such a model was applied by Rendleman and Barer 1980). However, as he Black-Karasinski model, his lognormal model does no allow simple closed-form expressions for he prices we are ineresed in. 10 In addiion o he lack of nice pricing formulas, he lognormal models 7.79) and 7.80) have anoher very inappropriae propery. imply ha, for all, T, S wih < T < S, 7.81) E Q [ r, B S r T, T )) 1] =. As shown by Hogan and Weinraub 1993) hese models As noed by Sandmann and Sondermann 1997) his resul has wo inexpedien consequences, which we gaher in he following heorem. Theorem 7.6 In he lognormal one-facor models 7.79) and 7.80) he following holds: a) An invesmen in he bank accoun over any period of ime of sricly posiive lengh is expeced o give an infinie reurn, i.e. for T < S [ { }] S exp r u du =. E Q r, T b) The quoed Eurodollar fuures price is Ẽ T r, ) =. Proof: The firs par of he heorem follows by Jensen s inequaliy, which gives ha 11 [ { }] S B S r, T ) = E Q r,t exp r u du T { }) 1 [ { S }]) 1 S = E Q r,t exp r u du > exp r u du T E Q r,t T 10 Dohan 1978) and Hogan and Weinraub 1993) sae some very complicaed pricing formulas for zero-coupon bonds which involve complex numbers, Bessel funcions, and hyperbolic rigonomeric funcions! A seemingly fas and accurae recursive procedure for he compuaion of bond prices in he model 7.80) is described by Hansen and Jørgensen 2000). 11 Jensen s inequaliy says ha if X is a random variable and fx) is a convex funcion, hen E[fX)] > fe[x]).

7.6 Non-affine models 195 and hence [ { }] S B S r, T ) 1 < E Q r,t exp r u du. T Taking expecaions E Q r,[ ] on boh sides we ge 12 E Q r, [ { }] S exp r u du T > E Q [ r, B S r T, T )) 1] =. From 5.29) we have ha he quoed Eurodollar fuures price is Ẽ T r, T ) = 500 400 E Q [ r, B T +0.25 r T, T )) 1]. Insering 7.81) wih S = T +0.25 ino he expression above we ge he second par of he heorem. Since Eurodollar fuures is a highly liquid produc on he inernaional financial markes, i is very inappropriae o use a model which clearly misprices hese conracs. I can be shown ha he problemaic relaion 7.81) is avoided by assuming ha eiher he effecive annual ineres raes or he LIBOR ineres raes are lognormally disribued insead of he coninuously compounded ineres raes. Models wih lognormal LIBOR raes have become very popular in recen years, primarily because hey are consisen wih praciioners use of Black s pricing formula a leas o some exen). We will sudy such models closely in Chaper 11. In summary, he lognormal models 7.79) and 7.80) have he nice propery ha negaive raes are precluded, bu hey do no allow simple pricing formulas and hey clearly misprice an imporan class of asses. For hese reasons i is difficul o see why hey have gained such populariy. The CIR model, for example, also precludes negaive ineres raes, is analyically racable, and does no lead o obvious mispricing of any conracs. Furhermore, he model is consisen wih a general equilibrium of he economy. Of course, hese argumens do no imply ha he CIR model provides he bes descripion of he movemens of ineres raes over ime, cf. he discussion in he nex secion. Finally, le us menion some models ha are neiher affine nor lognormal. The model 7.82) dr = κ [θ r ] d + βr dz Q was suggesed by Brennan and Schwarz 1980) and Couradon 1982). Despie he relaively simple dynamics, no explici pricing formulas have been derived neiher for bonds nor derivaive asses. Longsaff 1989), Beaglehole and Tenney 1991, 1992), and Leippold and Wu 2002) consider so-called quadraic models where he shor rae is given as r = x 2, and x follows an Ornsein- Uhlenbeck process like he r-process in Vasicek s model). This specificaion ensures non-negaive ineres raes. By Iô s Lemma, he dynamics of r is of he form The price of a zero-coupon bond is of he form dr = α 1 + α 2 r + α 3 r ) d + β r dz Q. B T r, ) = e at ) bt )r ct )r2, ] 12 Here we apply he law of ieraed expecaions: E Q r, [E Q rt,t [Y ] = E Q r, [Y ] for any random variable Y.

7.7 Parameer esimaion and empirical ess 196 where he funcions a, b, and c solve ordinary differenial equaions. Relaive o he affine models, he erm ct )r 2 has been added. The quadraic models hus give a more flexible relaion beween zero-coupon bond prices and he shor rae. Leippold and Wu 2002) and Jamshidian 1996) obain some raher complex expressions for he prices of European bond opions and oher derivaives. 7.7 Parameer esimaion and empirical ess To implemen a model, one mus assume some values of he parameers. In pracice, he rue values of he parameers are unknown, bu values can be esimaed from observed ineres raes and prices. For concreeness we will ake he esimaion of he Vasicek model as an example, bu similar consideraions apply o oher models. The parameers of Vasicek s model are κ, θ, β, and λ. The esimaion mehods can be divided ino hree classes: Time series esimaion Wih his approach he parameers of he process for he shor rae are esimaed from a ime series of hisorical observaions of a shor erm ineres rae. The esimaion iself can be carried ou by means of differen saisical mehods, e.g. maximum likelihood [see Ogden 1987)] or various momen maching mehods [see Andersen and Lund 1997) and Chan, Karolyi, Longsaff, and Sanders 1992)]. An essenial, pracical problem is ha no ineres raes of zero mauriy are observable. Ineres raes of very shor mauriies are se a he money marke, bu due o he credi risk of he paries involved hese raes are no perfec subsiues for he ruly risk-free ineres rae of zero mauriy, which is represened by r in he models. Mos auhors use yields on governmen bonds wih shor mauriies, e.g. one or hree monhs, as an approximaion o he shor rae. However, sudies by Duffee 1996) and Honoré 1998) show ha his can have a significan effec on he parameer esimaes. Anoher problem in applying he ime series approach is ha no all parameers of he model can be idenified. In Vasicek s model only he parameers κ, θ, and β ha eners he real-world dynamics of he shor rae in 7.47) can be esimaed. The missing parameer λ only affecs he process for r under he risk adjused maringale measures, bu he ime series of ineres raes is of course observed in he real world, i.e. under he real-world probabiliy measure. A hird problem is ha a large number of observaions are required o give reasonably cerain parameer esimaes. However, he longer he observaion period is, he less likely i is ha he shor rae has followed he same process wih consan parameers) during he enire period. Furhermore, he ime series approach ignores he fac ha he ineres rae models no only describe he dynamics of he shor rae bu also describe he enire yield curve and is dynamics. Cross secion esimaion Alernaively, he parameers of he model can be esimaed as he values ha will lead o model prices of a cross secion of liquid bonds and possibly oher fixed income securiies) ha are as close as possible o he currenly observed prices of hese asses. Then he esimaed model can be applied o price less liquid asses in a way which is consisen wih he marke prices of he liquid asses. Typically, he parameer values are chosen o minimize he sum of squared deviaions of model prices from marke prices where he sum runs over all he asses in he chosen cross secion. Such a procedure is very simple o implemen.

7.7 Parameer esimaion and empirical ess 197 A cross secion esimaion canno idenify all parameers of he model eiher. The curren prices only depend on he parameers ha affec he shor rae dynamics under he risk adjused maringale measures. For Vasicek s model his is he case for ˆθ, β, and κ. The parameers θ and λ canno be esimaed separaely. However, if he only use of he model is o derive curren prices of oher asses, we only need he values of ˆθ, β, and κ. In view of he problems conneced wih observing he shor rae, he value of he shor rae is ofen esimaed in line wih he parameers of he model. A cross secion esimaion compleely ignores he ime series dimension of he daa. The esimaion procedure does no in any way ensure ha he parameer values esimaed a differen daes are of similar magniudes. 13 The model s resuls concerning he dynamics of ineres raes and asse prices are no used a all in he esimaion. Panel daa esimaion This esimaion approach combines he wo approaches described above by using boh he ime series and he cross secion dimension of he daa and he models. Typically, he daa used are ime series of seleced yields of differen mauriies. Wih a panel daa approach all he parameers can be esimaed. For example, Gibbons and Ramaswamy 1993) and Daves and Ehrhard 1993) apply his procedure o esimae he CIR model. If we wan o apply a model boh for pricing cerain asses and for assessing and managing he changes in ineres raes and prices over ime, we should also base our esimaion on boh cross secional and ime series informaion. Two relaively simple versions of he panel daa approach are obained by emphasizing eiher he ime series dimension or he cross secion dimension and only applying he oher dimension o ge all he parameers idenified. As discussed above he parameers κ, θ, and β of Vasicek s model can be esimaed from a ime series of observaions of approximaions of) he shor rae. The remaining parameer λ can hen be esimaed as he value ha leads o model prices using he already fixed esimaes of κ, θ, and β) ha are as close as possible o he curren prices on seleced, liquid asses. On he oher hand, one can esimae κ, ˆθ, and β from a cross secion and hen esimae θ from a ime series of ineres raes using he already fixed esimaes of κ and β). In his way an esimae of λ can be deermined such ha he relaion ˆθ = θ λβ/κ holds for he esimaed parameer values. In any esimaion he parameer values are chosen such ha he model fis he daa o he bes possible exen according o some specified crierion. Typically, an esimaion procedure will also generae informaion on how well he model fis he daa. Therefore, mos papers referred o above also conain a es of one or several models. Probably he mos frequenly cied reference on he esimaion, comparison, and es of onefacor diffusion models of he erm srucure is Chan, Karolyi, Longsaff, and Sanders 1992) [henceforh abbreviaed CKLS], who consider ime homogeneous models of he ype 7.83) dr = θ κr ) d + βr γ dz. By resricing he values of he parameers θ, κ, and γ, many of he models sudied earlier are 13 For example, Brown and Dybvig 1986) find ha he parameer esimaes of he CIR model flucuae considerably over ime.

7.7 Parameer esimaion and empirical ess 198 obained as special cases, for example he models of Meron κ = γ = 0), Vasicek γ = 0), CIR γ = 1/2), and he lognormal model 7.80) γ = 1, θ = 0). CKLS use he one-monh yield on governmen bonds as an approximaion o he shor-erm ineres rae and apply a ime series approach on U.S. daa over he period 1964 1989. They esimae eigh differen resriced models and he unresriced model and es how well hey perform in describing he evoluion in he shor rae over he given period. Their resuls indicae ha i is primarily he value ha he model assigns o he parameer γ which deermines wheher he model is rejeced or acceped. The unresriced esimae of γ is approximaely 1.5, and models having a lower γ-value are rejeced in heir es, including he Vasicek model and he CIR model. On he oher hand, he simple lognormal model 7.80) is acceped. Subsequenly he CKLS analysis has been criicized on several couns. Firsly, he one monh yield may be a poor approximaion o he zero-mauriy shor rae. In a one-facor model here is a one-o-one relaion beween he zero-coupon rae of any given mauriy and he rue shor rae. For he affine models his relaion is given by 7.13) where he funcions a and b are known in closed-form for some affine models Meron, Vasicek, and CIR), and for he oher affine models hey can be compued quickly and accuraely by solving he Ricai differenial equaions 7.8) and 7.9) numerically. For non-affine models he relaion can be found by numerically solving he PDE 7.3) for a zero-coupon bond wih he given ime o mauriy one monh in he CKLS case) and ransforming he price o a zero-coupon yield. In his way Honoré 1998) ransforms a ime series of zero-coupon raes wih a given mauriy o a ime series of implici zero-mauriy shor raes. Based on he ransformed ime series of shor raes he finds esimaes of he parameer γ in he inerval beween 0.8 and 1.0, which is much lower han he CKLS-esimae. Anoher criicism, advanced by Bliss and Smih 1997), is ha he daa se used by CKLS includes he period beween Ocober 1979 and Sepember 1982, when he Federal Reserve, i.e. he U.S. cenral bank, followed a highly unusual moneary policy he FED Experimen ) resuling in a non-represenaive dynamics in ineres raes, in paricular he shor raes. Hence, Bliss and Smih allow he parameers o have differen values in his sub-period han in he res of he period used by CKLS 1964 1989). Ouside he experimenal period he unresriced esimae of γ is 1.0, which is again considerably smaller han he CKLS-esimae. The only models ha are no rejeced on a 5% es level are he CIR model and he Brennan-Schwarz model 7.82). Finally, applying a differen esimaion mehod and a differen daa se weekly observaions of hree monh U.S. governmen bond yields over he period 1954-1995), Andersen and Lund 1997) esimae γ o 0.676, which is much lower han he esimae of CKLS. Chrisensen, Poulsen, and Sørensen 2001) discuss some general problems in esimaing a process like 7.83), and using a maximum likelihood esimaion procedure and 1982-1995 daa hey obain a γ-esimae of 0.78. The ess menioned above are based on a ime series of approximaions of) he shor rae. Similar ess of he CIR model using oher ime series are performed by Brown and Dybvig 1986) and Brown and Schaefer 1994). On he oher hand, Gibbons and Ramaswamy 1993) es he abiliy of he CIR model o simulaneously describe he evoluion in four zero-coupon raes, namely he 1, 3, 6, and 12 monh raes a panel daa es). Wih daa covering he same period as he CKLS sudy, hey accep he CIR model. By now i should be clear ha he exensive empirical lieraure canno give a clear answer o he quesion of which one-facor model fis he daa bes. The answer depends on he daa and

7.8 Concluding remarks 199 he esimaion echnique applied. In mos ess models wih consan ineres rae volailiy, such as he models of Meron and Vasicek, and ypically also all models wihou mean reversion are rejeced. The CIR model is acceped in mos ess, and since i boh has nice heoreical properies and allows relaively simple closed-form pricing formulas, i is widely used boh by academics and praciioners. 7.8 Concluding remarks In his chaper we have sudied ime homogeneous one-facor diffusion models of he erm srucure of ineres raes. They are all based on specific assumpions on he evoluion of he shor rae and on he marke price of ineres rae risk. The models of Vasicek and Cox, Ingersoll, and Ross are frequenly applied boh by praciioners for pricing and risk managemen and by academics for sudying he effecs of ineres rae uncerainy on various financial issues. Boh models are consisen wih a general economic equilibrium model, alhough his equilibrium is based on many simplifying and unrealisic assumpions on he economy and is agens. Boh models are analyically racable and generae relaively simple pricing formulas for many fixed income securiies. The CIR model has he economically mos appealing properies and perform beer han he Vasicek model in explaining he empirical bond marke daa. The assumpion of he models of his chaper ha he shor rae conains all relevan informaion abou he yield curve is very resricive and no empirically accepable. Several empirical sudies show ha a leas wo and possibly hree or four sae variables are needed o explain he observed variaions in yield curves. As we shall see in he nex chaper, many of he muli-facor models suggesed in he lieraure are generalizaions of he one-facor models of Vasicek and CIR. In all ime homogeneous one-facor models he curren yield curve is deermined by he curren shor rae and he relevan model parameers. No maer how he parameer values are chosen i is highly unlikely ha he yield curve derived from he model can be compleely aligned wih he yield curve observed in he marke. If he model is o be applied for he pricing of derivaives such as fuures and opions on bonds and caps, floors, and swapions, i is somewha disurbing ha he model canno price he underlying zero-coupon bonds correcly. As we will see in Chaper 9, a perfec model fi of he curren yield curve can be obained in a one-facor model by replacing one or more parameers by deerminisic funcions of ime. While hese ime inhomogeneous versions of he one-facor models may provide a beer basis for derivaive pricing, hey are no unproblemaic, however. Also noe ha ypically he curren yield curve is no direcly observable in he marke, bu has o be esimaed from prices of coupon bonds. For his purpose praciioners ofen use a cubic spline or a Nelson-Siegel parameerizaion as oulined in Chaper 1. If one insead applies he parameerizaion of he discoun funcion T B T r, ) ha comes ou of an economically beer founded model, such as he CIR model, he problems of ime inhomogeneous models can be avoided. 7.9 Exercises EXERCISE 7.1 Parallel shifs of he yield curve) The purpose of his exercise is o find ou under which assumpions he only possible shifs of he yield curve are parallel, i.e. such ha dȳ τ where ȳ τ = y +τ. is independen of τ

7.9 Exercises 200 a) Argue ha if he yield curve only changes in he form of parallel shifs, hen he zero-coupon yields a ime mus have he form y T = y T r, ) = r + ht ) for some funcion h wih h0) = 0 and ha he prices of zero-coupon bonds are hereby given as B T r, ) = e r[t ] ht )[T ]. b) Use he parial differenial equaion 7.3) on page 164 o show ha 1 *) 2 βr)2 T ) 2 ˆαr)T ) + h T )T ) + ht ) = 0 for all r, ) wih T, of course). c) Using *), show ha 1 2 βr)2 T ) 2 ˆαr)T ) mus be independen of r. Conclude ha boh ˆαr) and βr) have o be consans, so ha he model is indeed Meron s model. d) Describe he possible shapes of he yield curve in an arbirage-free model in which he yield curve only moves in erms of parallel shifs. Is i possible for he yield curve o be fla in such a model? EXERCISE 7.2 Call on zero-coupon bonds in Vasicek s model) Figure 7.9 on page 187 shows an example of he relaion beween he price of a European call on a zero-coupon bond and he curren shor rae r in he Vasicek model, cf. 7.61). The purpose of his exercise is o derive an explici expression for C/ r. a) Show ha B S r, )e 1 2 d 1r,) 2 = KB T r, )e 2 1 d 2r,) 2. b) Show ha B S r, )n d 1r, )) KB T r, )n d 2r, )) = 0, where ny) = exp y 2 /2)/ 2π is he probabiliy densiy funcion for a sandard normally disribued random variable. c) Show ha C K,T,S r r, ) = B S r, )bs )N d 1r, )) + KB T r, )bt )N d 2r, )). EXERCISE 7.3 Fuures on bonds) Show he las claim in Theorem 7.2. EXERCISE 7.4 CIR zero-coupon bond price) Show ha he funcions b and a given by 7.70) and 7.71) solve he ordinary differenial equaions 7.68) and 7.69). EXERCISE 7.5 Comparison of prices in he models of Vasicek and CIR) Compare he prices according o Vasicek s model 7.47) and he CIR-model 7.65) of he following securiies: a) 1 year and 10 year zero-coupon bonds; b) 3 monh European call opions on a 5 year zero-coupon bond wih exercise prices of 0.7, 0.75, and 0.8, respecively; c) a 10 year 8% bulle bond wih annual paymens; d) 3 monh European call opions on a 10 year 8% bulle bond wih annual paymens for hree differen exercise prices chosen o represen an in-he-money opion, a near-he-money opion, and an ou-ofhe-money opion. In he comparisons use κ = 0.3, θ = 0.05, and λ = 0 for boh models. The curren shor rae is r = 0.05, and he curren volailiy on he shor rae is 0.03 so ha β = 0.03 in Vasicek s model and β 0.05 = 0.03 in he CIR model.

Chaper 8 Muli-facor diffusion models 8.1 Wha is wrong wih one-facor models? The preceding chaper gave an overview over one-facor diffusion models of he erm srucure of ineres raes. All he models are based on an assumed dynamics in he coninuously compounded shor rae, r. In several of hese models we were able o derive relaively simple, explici pricing formulas for boh bonds and European opions on bonds and hence also for caps, floors, swaps, and European swapions, cf. Chaper 2. The models can generae yield curves of various realisic forms, and he parameers of he models can be esimaed quie easily from marke daa. Several of he empirical ess described in he lieraure have acceped seleced one-facor models. Furhermore, paricularly he CIR model is based on a dynamics of he shor rae ha has many realisic properies. However, all he one-facor models also have obviously unrealisic properies. Firs, hey are no able o generae all he yield curve shapes observed in pracice. For example, he Vasicek and CIR models can only produce an increasing curve, a decreasing curve, and a curve wih a small hump. While he zero-coupon yield curve ypically has one of hese shapes, i does occasionally have a differen shape, e.g. he yield curve is someimes decreasing for shor mauriies and hen increasing for longer mauriies. Second, he one-facor models are no able o generae all he ypes of yield curve changes ha have been observed. In he affine one-facor models he zero-coupon yield ȳ τ mauriy τ is of he form ȳ τ = aτ) τ + bτ) τ r, = y +τ for any cf. 7.13). If bτ) > 0, he change in he yield of any mauriy will have he same sign as he change in he shor rae. Therefore, hese models do no allow so-called wiss of he erm srucure of ineres raes, i.e. yield curve changes where shor-mauriy yields and long-mauriy yields move in opposie direcions. A hird criical poin, which is relaed o he second poin above, is ha he changes over infiniesimal ime periods of any wo ineres rae dependen variables will be perfecly correlaed. This is for example he case for any wo bond prices or any wo yields. This is due o he fac ha all unexpeced changes are proporional o he shock o he shor rae, dz. For example, he dynamics of he τ i -mauriy zero-coupon yield in any ime homogeneous one-facor model is of he 201

8.1 Wha is wrong wih one-facor models? 202 mauriy years) 0.25 0.5 1 2 5 10 20 30 0.25 1.00 0.85 0.80 0.72 0.61 0.52 0.46 0.46 0.50 0.85 1.00 0.90 0.85 0.76 0.68 0.63 0.62 1 0.80 0.90 1.00 0.94 0.87 0.79 0.73 0.73 2 0.72 0.85 0.94 1.00 0.95 0.88 0.82 0.82 5 0.61 0.76 0.87 0.95 1.00 0.96 0.92 0.91 10 0.52 0.68 0.79 0.88 0.96 1.00 0.97 0.96 20 0.46 0.63 0.73 0.82 0.92 0.97 1.00 0.97 30 0.46 0.62 0.73 0.82 0.91 0.96 0.97 1.00 Table 8.1: Esimaed correlaion marix of weekly changes in par yields on U.S. governmen bonds. The marix is exraced from Exhibi 1 in Canabarro 1995). form dȳ τi = µ y r, τ i ) d + σ y r, τ i ) dz, where he drif rae µ y and he volailiy σ y are model-specific funcions. change in he yield over an infiniesimal ime period is herefore The variance of he Var dȳ τi ) = σ y r, τ i ) 2 d. The covariance beween changes in wo differen zero-coupon yields is Cov dȳ τ1 Hence he correlaion beween he yield changes is Corr dȳ τ1, dȳ τ2 ) =, dȳ τ2 ) = σ y r, τ 1 )σ y r, τ 2 ) d. Cov dȳ τ1, dȳ τ2 Var dȳ τ1 ) ) Var dȳ τ2 ) = 1. This conflics wih empirical sudies which demonsrae ha he acual correlaion beween changes in zero-coupon yields of differen mauriies is far from one. Table 8.1 shows correlaions beween weekly changes in par yields on U.S. governmen bonds. The correlaions are esimaed by Canabarro 1995) from daa over he period from January 1986 o December 1991. A similar paern has been documened by oher auhors, e.g. Rebonao 1996, Ch. 2) who uses daa from he U.K. bond marke. Inuiively, muli-facor models are more flexible and should be able o generae addiional yield curve shapes and yield curve movemens relaive o he one-facor models. Furhermore, mulifacor models allow non-perfec correlaions beween differen ineres rae dependen variables, cf. he discussion in Secion 8.3.1. Several empirical sudies have invesigaed how many facors are necessary in order o obain a sufficienly precise descripion of he acual evoluion of he erm srucure of ineres raes and he correlaions beween yields of differen mauriies. Of course, o some exen he resul of such an invesigaion will depend on he chosen daa se, he observaion period, and he esimaion procedure. However, all sudies seem o indicae ha wo or hree facors are needed. One way o address his quesion is o perform a so-called principal componen analysis of he variance-covariance marix of changes in zero-coupon yields of seleced mauriies. Canabarro

8.2 Muli-facor diffusion models of he erm srucure 203 1995) finds ha a single facor can describe a mos 85.0% of he oal variaion in his daa from he period 1986 1991 on he U.S. bond marke. The second-mos imporan facor describes an addiional 10.3% of he variaion, while he hird-mos and he fourh-mos imporan facors provide addiional conribuions of 1.9% and 1.2%, respecively. Addiional facors conribue in oal wih less han 1.6%. Similar resuls are repored by Lierman and Scheinkman 1991), who also use U.S. bond marke daa, and by Rebonao 1996, Ch. 2), who applies U.K. daa over he period 1989 1992. A principal componen analysis does no provide a precise idenificaion of which facors bes describe he evoluion of he erm srucure, bu i can give some indicaion of he facors. The sudies menioned above give remarkably similar indicaions. They all find ha he mos imporan facor affecs yields of all mauriies similarly and hence can be inerpreed as a level facor. The second-mos imporan facor affecs shor-mauriy yields and long-mauriy yields in opposie direcions and can herefore be inerpreed as a slope facor. Finally, he hird-mos imporan facor affecs yields of very shor and long mauriies in he same direcion, bu yields of inermediae mauriies approx. 2-5 years) in he opposie direcion. We can inerpre his facor as a curvaure facor. Lierman and Scheinkman argue ha alernaively he hird facor can be inerpreed as a facor represening he erm srucure of yield volailiies, i.e. he volailiies of he zero-coupon yields of differen mauriies. Oher empirical papers have sudied how well specific muli-facor models can fi seleced bond marke daa. Empirical ess performed by Sambaugh 1988), Pearson and Sun 1991), Chen and Sco 1993), Brenner, Harjes, and Kroner 1996), Andersen and Lund 1997), Vezal 1997), Balduzzi, Das, and Foresi 1998), Dai and Singleon 2000), and Boudoukh, Richardson, Sanon, and Whielaw 1999) all conclude ha differen muli-facor models provide a much beer descripion of he shape and movemens of he erm srucure of ineres raes han he one-facor special cases of he models. 8.2 Muli-facor diffusion models of he erm srucure In his secion we review he noaion and he general resuls in muli-facor diffusion models, which were firs discussed in Chaper 5. In a general n-facor diffusion model of he erm srucure of ineres raes, he fundamenal assumpion is ha he sae of he economy can be represened by an n-dimensional vecor process x = x 1,..., x n ) of sae variables. In paricular, he process x follows a Markov diffusion process, 8.1) dx = αx, ) d + βx, ) dz, where z = z 1,..., z n ) is an n-dimensional sandard Brownian moion. Denoe by S R n he value space of he process, i.e. he se of possible saes. In he expression 8.1) above, α is a funcion from S R + ino R n, and β is a funcion from S R + ino he se of n n marices of real numbers, i.e. βx, ) is an n n marix. The funcions α and β mus saisfy cerain regulariy condiions o ensure ha he sochasic differenial equaion 8.1) has a unique soluion, cf. Øksendal 1998). We can wrie 8.1) componenwise as n dx i = α i x, ) d + β ij x, ) dz j = α i x, ) d + β i x, ) dz. j=1

8.2 Muli-facor diffusion models of he erm srucure 204 As discussed in Chaper 5, he absence of arbirage will imply he exisence of a vecor process λ = λ 1,..., λ n ) of marke prices of risk, so ha for any raded asse we have he relaion n µx, ) = rx, ) + σ j x, )λ j x, ), j=1 where µ denoes he expeced rae of reurn on he asse, and σ 1,..., σ n are he volailiy erms, i.e. he price process is dv = V µx, ) d + See for example 5.49) on page 126. n σ j x, ) dz j. We also know ha he n-dimensional process z Q = z Q 1,..., zq n ) defined by dz Q j = dz j + λ j x, ) d, j = 1,..., n, is a sandard Brownian moion under he spo maringale measure Q. Wih he noaion ˆαx, ) = αx, ) βx, )λx, ), i.e. n ˆα i x, ) = α i x, ) β ij x, )λ j x, ), j=1 j=1 we can wrie he dynamics of he sae variables under Q as dx = ˆαx, ) d + βx, ) dz Q or componenwise as n dx i = ˆα i x, ) d + β ij x, ) dz Q j. From he analysis in Chaper 5, we know ha he price V = V x, ) of a raded asse of he European ype can be found as he soluion o he parial differenial equaion j=1 8.2) V x, ) + n i=1 ˆα i x, ) V x i x, ) + 1 2 n n i=1 j=1 γ ij x, ) 2 V x i x j x, ) rx, )V x, ) = 0, x, ) S [0, T ), wih he appropriae erminal condiion V x, T ) = Hx), x S. Here, γ ij = n k=1 β ikβ jk is he i, j) h elemen of he variance-covariance marix β β. If ρ ij denoes he correlaion beween changes in he i h and he j h sae variables, we have ha γ ij = ρ ij β i β j. Alernaively, he price can be compued as an expecaion under he spo maringale measure, V x, ) = E Q x, [e ] T rxu,u) du Hx T ), or as an expecaion under he T -forward maringale measure 8.3) V x, ) = B T x, ) E QT x, [Hx T )],

8.3 Muli-facor affine diffusion models 205 or as an expecaion under anoher convenien maringale measure. Jus as in he analysis of one-facor models in Chaper 7, we focus on he ime homogeneous models in which he funcions ˆα and β, and also he shor rae r, do no depend on ime, bu only depend on he sae variables. In paricular, 8.4) dx = ˆαx ) d + βx ) dz Q. 8.3 Muli-facor affine diffusion models We focus firs on so-called affine models, which in a muli-facor seing were inroduced by Duffie and Kan 1996). In he affine muli-facor models he dynamics of he vecor of sae variables is of he form ν1 x ) 0... 0 8.5) dx = ˆϕ ) 0 ν2 x )... 0 ˆκx d + Γ..... dz Q.., 0 0... νn x ) where ν j x) = δ j0 + δ j x = δ j0 + n δ jk x k. Here, ˆϕ = ˆϕ 1,..., ˆϕ n ) and δ j = δ j1,..., δ jn ) for j = 1,..., n are all consan vecors, δ 10,..., δ n0 are consan scalars, and ˆκ and Γ are consan n n marices. A ime homogeneous muli-facor diffusion model is said o be affine if he dynamics of he sae variables under he spo maringale measure is of he form 8.5) and he shor rae r = rx ) is an affine funcion of x, i.e. a consan scalar ξ 0 and a consan n-dimensional vecor ξ = ξ 1,..., ξ n ) exis such ha 8.6) rx) = ξ 0 + ξ x = ξ 0 + k=1 n ξ i x i. The condiion on he shor rae is rivially saisfied in he one-facor models in Chaper 7 since hey all ake he shor rae iself as he sae variable. Similarly, he condiion is saisfied in he muli-facor models in which he shor rae is one of he sae variables. Noe ha if he vecor λx) of marke prices of risk is also an affine funcion of x, he drif of he sae variables will also be affine under he real-world probabiliy measure. We will firs consider wo-facor affine models, boh because he noaion and he saemen of resuls are simpler and because mos of he muli-facor models sudied in he lieraure have wo facors. Subsequenly, we will briefly exend he analysis o he general n-facor models. i=1 8.3.1 Two-facor affine diffusion models In a wo-facor affine model he dynamics of he vecor of sae variables is of he form 8.7) dx = ˆϕ ) ) ν1 x ) 0 ˆκx d + Γ dz Q, 0 ν2 x )

8.3 Muli-facor affine diffusion models 206 which can be wrien componenwise as 8.8) 8.9) dx 1 = ˆϕ 1 ˆκ 11 x 1 ˆκ 12 x 2 ) d + Γ 11 δ10 + δ 11 x 1 + δ 12 x 2 dz Q 1 + Γ 12 δ20 + δ 21 x 1 + δ 22 x 2 dz Q 2 dx 2 = ˆϕ 2 ˆκ 21 x 1 ˆκ 22 x 2 ) d + Γ 21 δ10 + δ 11 x 1 + δ 12 x 2 dz Q 1 + Γ 22 δ20 + δ 21 x 1 + δ 22 x 2 dz Q 2. The shor rae is given by 8.10) r = ξ 0 + ξ 1 x 1 + ξ 2 x 2. To ensure ha he square roos are well-defined, we require ha ν j x) = δ j0 + δ j1 x 1 + δ j2 x 2 0, j = 1, 2, for all he values ha x = x 1, x 2 ) can have, i.e. for all x S. Duffie and Kan sae wo condiions on he parameers ha imply boh ha he process x = x ) is well-defined by he sochasic differenial equaion above and ha ν j x ) > 0 for all wih probabiliy 1). These condiions are saisfied in he specific models we consider in he res of he chaper. Under he assumpion 8.5) he drif ˆαx) = ˆϕ ˆκx is clearly an affine funcion of x. Furhermore, he variance-covariance marix βx)βx) is also affine in x, in he sense ha any elemen of he marix is an affine funcion of x. To see his, we use 8.8) and 8.9) o derive ha he variance of he insananeous change in each of he sae variables is equal o Var Q dx i ) = γ i x 1, x 2 ) 2 d, where γ i x 1, x 2 ) 2 = Γ 2 i1 [δ 10 + δ 11 x 1 + δ 12 x 2 ] + Γ 2 i2 [δ 20 + δ 21 x 1 + δ 22 x 2 ] = δ 10 Γ 2 i1 + δ 20 Γ 2 i2) + δ11 Γ 2 i1 + δ 21 Γ 2 i2) x1 + δ 12 Γ 2 i1 + δ 22 Γ 2 i2) x2, which is an affine funcion of he sae variables. The covariance beween insananeous changes in he wo sae variables is Cov Q dx 1, dx 2 ) = γ 12 x 1, x 2 ), where γ 12 x 1, x 2 ) = Γ 11 Γ 21 [δ 10 + δ 11 x 1 + δ 12 x 2 ] + Γ 12 Γ 22 [δ 20 + δ 21 x 1 + δ 22 x 2 ] = Γ 11 Γ 21 δ 10 + Γ 12 Γ 21 δ 20 ) + Γ 11 Γ 21 δ 11 + Γ 12 Γ 21 δ 21 ) x 1 + Γ 11 Γ 21 δ 12 + Γ 12 Γ 21 δ 22 ) x 2, which is also an affine funcion of he sae variables. Hence, all he elemens of he variancecovariance marix are affine in he sae variables. In his way, he class of affine muli-facor models is a naural generalizaion of he class of affine one-facor models. Analogously o he one-facor analysis, we ge ha he price B T of a zero-coupon bond mauring a ime T in a muli-facor affine diffusion model can be wrien as an exponenial-affine funcion of he vecor of sae variables. The following heorem saes he precise resul: Theorem 8.1 In an affine wo-facor diffusion model where he shor rae is of he form 8.10) and he sae variables follow he process 8.7), he zero-coupon bond price B T = B T x 1, x 2, ) is given by he funcion 8.11) B T x 1, x 2, ) = exp { at ) b 1 T )x 1 b 2 T )x 2 },

8.3 Muli-facor affine diffusion models 207 where he funcions b 1, b 2, and a solve he following sysem of ordinary differenial equaions 8.12) 8.13) 8.14) b 1τ) = ˆκ 11 b 1 τ) ˆκ 21 b 2 τ) 1 2 δ 11 Γ 11 b 1 τ) + Γ 21 b 2 τ)) 2 1 2 δ 21 Γ 12 b 1 τ) + Γ 22 b 2 τ)) 2 + ξ 1, b 2τ) = ˆκ 12 b 1 τ) ˆκ 22 b 2 τ) 1 2 δ 12 Γ 11 b 1 τ) + Γ 21 b 2 τ)) 2 1 2 δ 22 Γ 12 b 1 τ) + Γ 22 b 2 τ)) 2 + ξ 2, a τ) = ˆϕ 1 b 1 τ) + ˆϕ 2 b 2 τ) 1 2 δ 10 Γ 11 b 1 τ) + Γ 21 b 2 τ)) 2 1 2 δ 20 Γ 12 b 1 τ) + Γ 22 b 2 τ)) 2 + ξ 0, wih he iniial condiions a0) = b 1 0) = b 2 0) = 0. The resul can be demonsraed by verifying ha he funcion B T x 1, x 2, ) given by 8.11) is a soluion o he parial differenial equaion 8.2) in he affine model if he funcions aτ), b 1 τ), and b 2 τ) solve he ordinary differenial equaions 8.12)-8.14). The erminal condiion on he price, B T x 1, x 2, T ) = 1 for all x 1, x 2 ) S, implies ha a0), b 1 0), and b 2 0) all have o be zero. Excep for he increased noaional complexiy, he proof is idenical o he one-facor case and is herefore omied. Noe ha o deermine a, b 1, and b 2, one mus firs solve he wo differenial equaions 8.12) and 8.13) simulaneously, and hen a follows from he b i s by an inegraion. In he following secions we will look a specific affine models in which he Ricai equaions 8.12)-8.14) have explici soluions. This is he case in Gaussian models and so-called muli-facor CIR models. For oher specificaions of he affine model, he Ricai equaions mus be solved numerically, see e.g. Duffie and Kan 1996). The Ricai equaions can be solved faser numerically han he parial differenial equaion. In an affine wo-facor model he zero-coupon yields ȳ τ = y +τ and he forward raes f τ = f +τ ȳ τ x 1, x 2 ) = aτ) τ + b 1τ) τ x 1 + b 2τ) x 2, τ = ln B +τ )/ T are of he form f τ x 1, x 2 ) = a τ) + b 1τ)x 1 + b 2τ)x 2. = ln B +τ )/τ ake he form Le us also look a he volailiy of he price of a zero-coupon bond wih a fixed mauriy dae T. Since B T / x i x 1, x 2, ) = b i T )B T x 1, x 2, ), Iô s Lemma for funcions of several variables see Theorem 3.5 on page 66) implies ha where db T B T = rx 1, x 2 ) d + σ T 1 x 1, x 2, ) dz Q 1 + σt 2 x 1, x 2, ) dz Q 2, σ1 T x 1, x 2, ) = b 1 T )Γ 11 + b 2 T )Γ 21 ) δ 10 + δ 11 x 1 + δ 12 x 2, σ2 T x 1, x 2, ) = b 1 T )Γ 12 + b 2 T )Γ 22 ) δ 20 + δ 21 x 1 + δ 22 x 2. The volailiy of he zero-coupon bond price is herefore σ T x 1, x 2, ), where σ T x 1, x 2, ) 2 = σ1 T x 1, x 2, ) 2 + σ2 T x 1, x 2, ) 2 = b 1 T )Γ 11 + b 2 T )Γ 21 ) 2 δ 10 + δ 11 x 1 + δ 12 x 2 ) + b 1 T )Γ 12 + b 2 T )Γ 22 ) 2 δ 20 + δ 21 x 1 + δ 22 x 2 ).

8.3 Muli-facor affine diffusion models 208 Similarly, he dynamics of he zero-coupon yield ȳ τ will be of he form where dȳ τ =... d + σ y1 x 1, x 2, τ) dz Q 1 + σ y2x 1, x 2, τ) dz Q 2, b1 τ) σ y1 x 1, x 2, τ) = τ b1 τ) σ y2 x 1, x 2, τ) = τ Γ 11 + b ) 2τ) δ10 Γ 21 + δ 11 x 1 + δ 12 x 2, τ Γ 12 + b ) 2τ) δ20 Γ 22 + δ 21 x 1 + δ 22 x 2, τ and we have omied he drif rae for clariy. We can see ha, in a wo-facor model, zero-coupon yields wih differen mauriies are no perfecly correlaed since he covariance is Cov Q dȳ τ1 and, hence, he correlaion is, dȳ τ2 ) = σ y1 x 1, x 2, τ 1 )σ y1 x 1, x 2, τ 2 ) + σ y2 x 1, x 2, τ 1 )σ y2 x 1, x 2, τ 2 )) d, Corr Q σ dȳ τ1, dȳ τ2 y1 x, τ 1 )σ y1 x, τ 2 ) + σ y2 x, τ 1 )σ y2 x, τ 2 ) ) = σy1 x, τ 1 ) 2 + σ y2 x, τ 1 ) 2 σ y1 x, τ 2 ) 2 + σ y2 x, τ 2 ), 2 which in general will be less han 1. The above analysis was based on wo unspecified sae variables x 1 and x 2. Due o he fac ha all prices, ineres raes, volailiies, ec., are funcions of x 1 and x 2, i is ypically possible o shif o anoher pair of sae variables x 1 and x 2 and express prices, raes, ec., in erms of he new variables. This is convenien if he new variables x 1 and x 2 are easier o observe han he original variables x 1 and x 2 since he price expressions are hen simpler o apply in pracice, and i will be easier o esimae he model parameers. Of course, we could have saed he model in erms of x 1 and x 2 from he beginning, bu i may be easier o develop he pricing formulas using x 1 and x 2. See Secion 8.5.2 for an imporan example. 8.3.2 n-facor affine diffusion models In he general n-facor affine model where he vecor of sae variables follows he process 8.5), and he shor rae is given as in 8.6), he zero-coupon bond price B T = B T x, ) is given by he funcion n 8.15) B T x, ) = exp { at ) bt ) x} = exp at ) b j T )x j, where he funcions aτ), b 1 τ),..., b n τ) solve he following sysem of ordinary differenial equaions: 8.16) 8.17) b iτ) = a τ) = n ˆκ ji b j τ) 1 2 j=1 n ˆϕ j b j τ) 1 2 j=1 n k=1 n k=1 δ k0 δ ki n Γ jk b j τ) j=1 n Γ jk b j τ) j=1 2 2 j=1 + ξ i, in = 1,..., n, + ξ 0, wih he iniial condiions a0) = b 1 0) = = b n 0) = 0. Conversely, under cerain regulariy condiions, he zero-coupon bond prices are only of he exponenial-affine form if ˆα, β β, and r are affine funcions of x, cf. Duffie and Kan 1996).

8.4 Muli-facor Gaussian diffusion models 209 In an affine n-facor model he zero-coupon yields ȳ τ = ln B +τ )/τ are 8.18) ȳ τ x) = aτ) τ + n j=1 b j τ) x j, τ and he forward raes f τ = f +τ are n f τ x) = a τ) + b τ)x j. j=1 The dynamics of he zero-coupon bond price B T is db T B T n = rx ) d + σj T x, ) dz Q j, j=1 where he sensiiviies σ T j are given by 8.19) σj T x, ) = n δ j0 + δ j1 x 1 +... δ jn x n Γ kj b k T ). 8.3.3 European opions on coupon bonds As demonsraed in Secion 7.2.3, he price of a European call opion on a coupon bond is in he one-facor affine models given as he price of a porfolio of European call opions on zero-coupon bonds, cf. 7.30) on page 170. This is he case whenever he price of any zero-coupon bond is a monoonic funcion of he shor rae. The same rick canno be applied in muli-facor models so ha he prices of opions on coupon bonds and consequenly also swapions, cf. Secion 2.9.2 on page 39) mus be compued using numerical mehods. However, i is possible o approximae very accuraely he price of a European opion on a coupon bond by he price of a single European opion on a carefully seleced zero-coupon bond. For deails on he approximaion, see Chaper 12 and Munk 1999). As we shall see below, several of he muli-facor models provide a closed-form expression for he price of a European opion on a zero-coupon bond so ha he approximae price of he coupon bond opion is easily obainable. k=1 8.4 Muli-facor Gaussian diffusion models 8.4.1 General analysis The simples affine muli-facor erm srucure models are he Gaussian models, which were firs sudied by Langeieg 1980). A Gaussian model is an affine model of he form 8.5) where he volailiy funcions ν j x) are consan so ha he vecors δ j are equal o zero. Since he diagonal marix is muliplied by he consan marix Γ, we can and will assume ha all he ν j x) s are equal o 1, i.e. δ j0 = 1. In he Gaussian models he dynamics of he sae variables is herefore 8.20) dx = ˆϕ ˆκx ) d + Γ dz Q. Analogously o he analysis for one-dimensional Ornsein-Uhlenbeck processes in Secion 3.8.2 on page 59, we ge ha fuure values of he vecor of sae variables are n-dimensionally normally disribued. In paricular, he individual sae variables are normally disribued. Since he shor

8.4 Muli-facor Gaussian diffusion models 210 rae is a linear combinaion of hese sae variables, i follows ha fuure values of he shor rae are normally disribued in hese models. The expressions for means, variances, and covariances of he sae variables and hence of he shor rae) will be simple only when he marix ˆκ is diagonal. 1 In a Gaussian model he sensiiviies of he zero-coupon bond prices σj T only on he ime o mauriy of he bond, σ T j ) = n Γ kj b k T ). k=1 defined in 8.19) depend Gaussian models are very racable and allow closed-form expressions for boh bond prices and prices of European opions on zero-coupon bonds. The bond prices follow from 8.15). According o 8.3), he price of a European call opion on a zero-coupon bond is C K,T,S x, ) = B T [ x, ) E QT x, max B S x T, T ) K, 0 )]. Since he vecor of sae variables x T given x = x is normally disribued, he zero-coupon bond price is lognormally disribued. As in he Gaussian one-facor models sudied in Chaper 7, we conclude ha C K,T,S x, ) = B S x, )N d 1 ) KB T x, )N d 2 ), where 8.21) 8.22) and 8.23) d 1 = 1 B S ) v, T, S) ln x, ) KB T + 1 v, T, S), x, ) 2 d 2 = d 1 v, T, S), v, T, S) 2 = Var QT n = = T [ j=1 T n j=1 ln F T,S T ] σ S j u) σ T j u) ) 2 du n 2 Γ kj [b k S u) b k T u)]) du. k=1 Le us focus on he case of wo facors. In a Gaussian wo-facor diffusion model he dynamics of he sae variables is of he form 8.24) dx 1 = ˆϕ 1 ˆκ 11 x 1 ˆκ 12 x 2 ) d + Γ 11 dz Q 1 + Γ 12 dz Q 2, 8.25) dx2 = ˆϕ2 ˆκ21x1 ˆκ22x2) d + Γ21 dz Q 1 + Γ 22 dz Q 2. The ordinary differenial equaions 8.12) and 8.13) reduce o 8.26) 8.27) b 1τ) = ˆκ 11 b 1 τ) ˆκ 21 b 2 τ) + ξ 1, b 1 0) = 0, b 2τ) = ˆκ 12 b 1 τ) ˆκ 22 b 2 τ) + ξ 2, b 2 0) = 0, 1 Generally he momens depend on he eigenvalues and he eigenvecors of he marix ˆκ, cf. he discussion in Langeieg 1980).

8.4 Muli-facor Gaussian diffusion models 211 while he equaion for he funcion a becomes 8.28) a τ) = ˆϕ 1 b 1 τ) + ˆϕ 2 b 2 τ) 1 2 Γ 11b 1 τ) + Γ 21 b 2 τ)) 2 1 2 Γ 12b 1 τ) + Γ 22 b 2 τ)) 2 + ξ 0, a0) = 0. According o 8.23), he variance erm in he opion price is given by 8.29) v, T, S) 2 = Γ 2 11 + Γ 2 ) T 12 [b 1 S u) b 1 T u)] 2 du + Γ 2 21 + Γ 2 ) T 22 [b 2 S u) b 2 T u)] 2 du + 2 Γ 11 Γ 21 + Γ 12 Γ 22 ) T 8.4.2 A specific example: he wo-facor Vasicek model [b 1 S u) b 1 T u)] [b 2 S u) b 2 T u)] du. Beaglehole and Tenney 1991) and Hull and Whie 1994a) have suggesed a Gaussian wofacor model, which is a relaively simple exension of he one-facor Vasicek model in which ] dr = κ [ˆθ r d + β dz Q = ˆϕ κr ) d + β dz Q, cf. Secion 7.4 on page 176. The exension is o le he long-erm level ˆθ follow a similar sochasic process. Hull and Whie formulae he generalized model as follows: 8.30) dr = ˆϕ + ε κ r r ) d + β r dz Q 1, 8.31) dε = κεε d + βερ dz Q 1 + β ε 1 ρ2 dz Q 2. The process ε = ε ) exhibis mean reversion around zero and represens he deviaion of he curren view on he long-erm level of he shor rae from he average view. Here, β ε is he volailiy of he ε-process, and ρ is he correlaion beween changes in he shor rae and changes in ε. All consan parameers are assumed o be posiive excep ρ, which can have any value in he inerval [ 1, 1]. This wo-facor model is he special case of he general Gaussian wo-facor model which is obained by leing ˆϕ 1 = ˆϕ, ˆκ 11 = κ r, ˆκ 12 = 1, Γ 11 = β r, Γ 12 = 0, ˆϕ 2 = 0, ˆκ 21 = 0, ˆκ 22 = κ ε, Γ 21 = β ε ρ, Γ 22 = β ε 1 ρ2. Since he shor rae is iself he firs sae variable, we mus in addiion pu ξ 1 = 1 and ξ 0 = ξ 2 = 0. Afer hese subsiuions he ordinary differenial equaions for b 1 and b 2 become 8.32) 8.33) b 1τ) = κ r b 1 τ) + 1, b 1 0) = 0, b 2τ) = b 1 τ) κ ε b 2 τ), b 2 0) = 0.

8.5 Muli-facor CIR models 212 The firs of hese is idenical o he differenial equaion solved in he original one-facor Vasicek model, cf. Secion 7.4.2 on page 178, so we know ha he soluion is b 1 τ) = 1 κ r 1 e κ rτ ). Nex, i can be verified ha he soluion o he equaion for b 2 is given by b 2 τ) = 1 1 κ r [κ r κ ε ] e κrτ κ ε [κ r κ ε ] e κετ + 1. κ r κ ε Finally, he equaion for he funcion a can be rewrien as from which i follows ha a τ) = ˆϕb 1 τ) 1 2 β2 r b 1 τ) 2 1 2 β2 εb 2 τ) 2 ρβ r β ε b 1 τ)b 2 τ), a0) = 0, aτ) = aτ) a0) = τ τ 0 a u) du τ = ˆϕ b 1 u) du 1 0 2 β2 r b 1 u) 2 du 1 τ 0 2 β2 ε b 2 u) 2 du ρβ r β ε b 1 u)b 2 u) du 0 0 = ˆϕ τ b 1 τ)) 1 1 κ r κ 2 r 2 β2 r ρβ rβ ε β 2 ) ε + τ κ r κ ε 2κ r κ ε ) 2 b 1 τ) 12 ) κ rb 1 τ) 2 β 2 ε 2τκε 4κ 3 3 + 4e κετ e 2κετ ) εκ r κ ε ) ρβ r β ε κ r κ ε κ r κ ε ) τ b 1 τ) 1 e κετ κ ε τ + 1 e κr+κε)τ κ r + κ ε The relevan variance v, T, S) 2 enering he price of an opion on a zero-coupon bond follows from 8.29): v, T, S) 2 = β 2 r T [b 1 S u) b 1 T u)] 2 du + β 2 ε T ). [b 2 S u) b 2 T u)] 2 du T + 2ρβ r β ε [b 1 S u) b 1 T u)] [b 2 S u) b 2 T u)] du, where he inegrals can be compued explicily. Hull and Whie 1994a) show furher how o obain a perfec fi of he model o an observed yield curve by replacing he consan ˆϕ by a suiable ime-dependen funcion. Oher Gaussian muli-facor models have been sudied by Langeieg 1980) and Beaglehole and Tenney 1991). 8.5 Muli-facor CIR models 8.5.1 General analysis A erm srucure model is said o be an n-facor CIR model if he shor rae equals he sum of he n sae variables, r = n j=1 x j, and he risk-neural dynamics of he n sae variables is of he form dx j = ˆϕ j ˆκ j x j ) d + β j xj dz Q j, j = 1,..., n, where ˆϕ j, ˆκ j, and β j are consans. Noe ha he sae variables are muually independen and ha each sae variable follows a square roo process, jus as he shor rae process in he onefacor CIR model, cf. Secion 7.5 on page 188. In paricular, he processes will have non-negaive

8.5 Muli-facor CIR models 213 values a all poins in ime. 2 Also noe ha a muli-facor CIR model is an affine model of he form 8.5), where i) he marix ˆκ is diagonal wih ˆκ 1,..., ˆκ n in he diagonal and zeros in all oher enries, ii) he marix Γ is he ideniy marix, i.e. he marix wih ones in he diagonal and zeros in all oher enries, and iii) ν j x ) = β 2 j x j. In he muli-facor CIR models he ordinary differenial equaions 8.16) and 8.17) can be solved explicily. The soluions can be compued from he known expressions for aτ) and bτ) in he one-facor CIR model. To see his, firs recall from Chaper 5 ha he zero-coupon bond price can be wrien as B T x, ) = E Q [exp { T r u du} x = x Using he relaion r u = x 1u + + x nu and he independence of he sae variables, we can rewrie he zero-coupon bond price as n B T x, ) = E Q exp j=1 { n = E Q exp = j=1 { n E [exp Q j=1 T T T ] x ju du x = x du} x ju x = x x ju du}. x j = x j Because each sae variable x j follows a process of he same ype as he shor rae in he one-facor CIR model, we ge where and γ j = E Q [exp { T x ju du} x j = x j 2e γjτ 1) b j τ) = γ j + ˆκ j )e γjτ, 1) + 2γ j a j τ) = 2 ˆϕ j β 2 j ] = exp { a j T ) b j T )x j }, ln2γ j ) + 1 2 ˆκ j + γ j )τ ln [γ j + ˆκ j )e γjτ 1) + 2γ j ] ˆκ 2 j + 2β2 j, cf. 7.67), 7.70), and 7.71). Consequenly, he zero-coupon bond price is B T x, ) = n n exp { a j T ) b j T )x j } = exp at ) b j T )x j, j=1 where aτ) = n j=1 a jτ). The dynamics of he zero-coupon bond price is ]. j=1 ), db T B T n = rx ) d + σj T x, ) dz Q j, j=1 where he sensiiviies σ T j are given by 8.34) σ T j x, ) = β j xj b j T ). 2 As in he one-facor CIR model, he process x j will be sricly posiive if 2 ˆϕ j β 2 j.

8.5 Muli-facor CIR models 214 8.5.2 A specific example: he Longsaff-Schwarz model Model descripion The prevalen muli-facor CIR model is he wo-facor model of Longsaff and Schwarz 1992a). As he one-facor CIR model, he Longsaff-Schwarz model is a special case of he general equilibrium model sudied by Cox, Ingersoll, and Ross 1985a). Wih some empirical suppor Longsaff and Schwarz assume ha he economy has one sae variable, x 1, ha affecs only expeced reurns on producive invesmens and anoher sae variable, x 2, ha affecs boh expeced reurns and he uncerainy abou he reurns on producive invesmens. The wo sae variables x 1 and x 2 are assumed o follow he independen processes dx 1 = ϕ 1 κ 1 x 1 ) d + β 1 x1 dz 1, dx 2 = ϕ 2 κ 2 x 2 ) d + β 2 x2 dz 2 under he real-world probabiliy measure. All he consans are posiive. Under cerain specificaions of preferences, endowmens, ec., of he agens in he economy and an appropriae scaling of he wo sae variables), he equilibrium shor rae is exacly he sum of he wo sae variables, 8.35) r = x 1 + x 2. Furhermore, he marke price of risk associaed wih x 1, i.e. λ 1 x, ), is equal o zero, while he marke price of risk associaed wih x 2 is of he form λ 2 x, ) = λ x 2 /β 2, where λ is a consan. Hence, he sandard Brownian moion under he spo maringale measure Q are given by 8.36) dz Q 1 = dz 1, dz Q 2 = dz 2 + λ x2 d. β 2 The dynamics of he sae variables under he spo maringale measure becomes dx 1 = ˆϕ 1 ˆκ 1 x 1 ) d + β 1 x1 dz Q 1, dx 2 = ˆϕ 2 ˆκ 2 x 2 ) d + β 2 x2 dz Q 2, where ˆϕ 1 = ϕ 1, ˆκ 1 = κ 1, ˆϕ 2 = ϕ 2, and ˆκ 2 = κ 2 + λ. The yield curve According o he analysis for general muli-facor CIR models, he zero-coupon bond price B T x 1, x 2, ) can be wrien as 8.37) B T x 1, x 2, ) = exp { at ) b 1 T )x 1 b 2 T )x 2 }, where aτ) = a 1 τ) + a 2 τ), and γ j = 2e γjτ 1) b j τ) = γ j + ˆκ j )e γjτ, j = 1, 2, 1) + 2γ j a j τ) = 2 ˆϕ j β 2 j ˆκ 2 j + 2β2 j. ln2γ j ) + 1 2 ˆκ j + γ j )τ ln [γ j + ˆκ j )e γjτ 1) + 2γ j ] ), j = 1, 2,

8.5 Muli-facor CIR models 215 The sae variables x 1 and x 2 are absrac variables ha are no direcly observable. Longsaff and Schwarz perform a change of variables o he shor rae, r, and he insananeous variance rae of he shor rae, v. Sricly speaking, hese variables canno be direcly observed eiher, bu hey can be esimaed from bond marke daa. In addiion, he new variables seem imporan for he pricing of bonds and ineres rae derivaives, and i is easier o relae o prices as funcions of r and v insead of funcions of x 1 and x 2. Since r is given by 8.35), we ge dr = dx 1 + dx 2, i.e. dr = ϕ 1 + ϕ 2 κ 1 x 1 κ 2 x 2 ) d + β 1 x1 dz 1 + β 2 x2 dz 2. The insananeous variance is Var dr ) = v d, where 8.38) v = β 2 1x 1 + β 2 2x 2, so ha he dynamics of v is dv = β 2 1ϕ 1 + β 2 2ϕ 2 β 2 1κ 1 x 1 β 2 2κ 2 x 2 ) d + β 3 1 x1 dz 1 + β 3 2 x2 dz 2. If β 1 β 2, he Equaions 8.35) and 8.38) imply ha 8.39) x 1 = β2 2r v β 2 2 β2 1 The dynamics of r and v can hen be rewrien as dr = ϕ 1 + ϕ 2 κ 1β2 2 κ 2 β1 2 β2 2 β2 1 8.40) 8.41), x 2 = v β1r 2 β2 2. β2 1 r κ ) 2 κ 1 β2 2 v d β2 1 β2 2 + β r v v β1 2 1 β2 2 dz 1 + β r 2 β2 1 β2 2 dz 2, β2 1 dv = β1ϕ 2 1 + β2ϕ 2 2 β1β 2 2 2 κ 1 κ 2 β2 2 r β2 2κ 2 β 2 ) 1κ 1 β2 1 β2 2 v d β2 1 + β1 3 β2 2r v β2 2 dz 1 + β 3 v β1 2r β2 2 1 β2 2 dz 2. β2 1 Since boh x 1 and x 2 say non-negaive, i follows from 8.39) ha v a any poin in ime will lie beween β 2 1r and β 2 2r. I can be shown ha 8.40) and 8.41) imply ha changes in r and v are posiively correlaed, which is in accordance wih empirical observaions of he relaion beween he level and he volailiy of ineres raes. Subsiuing 8.39) ino 8.37), we can wrie he zero-coupon bond price as a funcion of r and v: { 8.42) B T r, v, ) = exp at ) b 1 T )r b } 2 T )v, where b1 τ) = β2 2b 1 τ) β1b 2 2 τ) β2 2, β2 1 b2 τ) = b 2τ) b 1 τ) β2 2. β2 1 Noe ha he zero-coupon bond price involves six parameers, namely β 1, β 2, ˆκ 1, ˆκ 2, ˆϕ 1, and ˆϕ 2. The parial derivaives B T / r and B T / v can be eiher posiive or negaive so, in conras o

8.5 Muli-facor CIR models 216 he one-facor models in Chaper 7, he zero-coupon bond price is no a monoonically decreasing funcion of he shor rae. According o Longsaff and Schwarz, he derivaive B T / r is ypically negaive for shor-erm bonds, bu i can be posiive for long-erm bonds. The derivaive B T / v approaches zero for τ 0 so ha very shor-erm bonds are affeced primarily by he shor rae and only o a small exen by he variance of he shor rae. If he shor rae r a some poin in ime is zero in which case v is also zero), i will become sricly posiive immediaely aferwards and, hence, B T 0, 0, ) < 1. Finally, B T r, v, ) 0 for r in which case v ). The zero-coupon yield ȳ τ = y +τ is given by ȳ τ = ȳ τ r, v ), where ȳ τ r, v) = aτ) τ + b 1 τ) τ r + b 2 τ) v, τ which is an affine funcion of r and v. I can be shown ha ȳ τ r, v) r for τ 0 and ha he asympoic long rae is consan since ȳ τ r, v) ˆϕ 1 β 2 1 γ 1 ˆκ 1 ) + ˆϕ 2 β2 2 γ 2 ˆκ 2 ) for τ. According o Longsaff and Schwarz, he yield curve τ ȳ τ r, v) can have many differen shapes, for example i can be monoonically increasing or decreasing, humped i.e. firs increasing, hen decreasing), i can have a rough i.e. firs decreasing, hen increasing) or boh a hump and a rough. We can see mos of hese shapes in Figure 8.1. Noe ha for a given shor rae he shape of he yield curve may depend on he variance facor. Parial changes in r and v may imply a significan change of he shape of he yield curve, for example a wis so ha differen mauriy segmens of he yield curve move in opposie direcions. The Longsaff-Schwarz model is herefore much more flexible han he one-facor CIR model. The forward rae f τ = f +τ is given by f τ = f τ r, v ), where f τ r, v) = a τ) + b 1τ)r + b 2τ)v. All zero-coupon yields and forward raes are non-negaive in his model. The dynamics of he zero-coupon bond price is of he form db T B T = r d β 1 x1 b 1 T ) dz Q 1 β 2 x2 b 2 T ) dz Q 2 = r λx 2 b 2 T )) d β 1 x1 b 1 T ) dz 1 β 2 x2 b 2 T ) dz 2 ) λ = r + β2 2 b 2 T )β1r 2 v ) d β2 1 β2 2 β r v v β1 2 1 β2 2 b 1 T ) dz 1 β r 2 β2 1 β2 2 b 2 T ) dz 2, β2 1 where we have applied 8.34), 8.36), and 8.39). The so-called erm premium, i.e. he expeced rae of reurn on a zero-coupon bond in excess of he shor rae, is λb 2 T )β 2 1r v )/β 2 2 β 2 1), which is posiive if λ < 0. I is consisen wih empirical sudies ha he erm premium is affeced by wo sochasic facors r and v) and depends on he ineres rae volailiy. σ T r, v, ) of he zero-coupon bond price B T σ T r, v, ) 2 = β2 1β 2 2 β 2 2 β2 1 The volailiy is in he Longsaff-Schwarz model given by b1 T ) 2 b 2 T ) 2) r + β2 2b 2 T ) 2 β1b 2 1 T ) 2 β2 2 v. β2 1

8.5 Muli-facor CIR models 217 zero-coupon yield 5.0% 4.5% 4.0% 3.5% 3.0% 2.5% 0 5 10 15 20 years o mauriy a) Low curren shor rae zero-coupon yield 7.5% 7.0% 6.5% 6.0% 5.5% 5.0% 0 5 10 15 20 years o mauriy b) High curren shor rae 5.30% zero-coupon yield 5.20% 5.10% 5.00% 4.90% 0 5 10 15 20 years o mauriy c) Medium curren shor rae Figure 8.1: Zero-coupon yield curves in he Longsaff-Schwarz model. The parameer values are β 1 = 0.1, β 2 = 0.2, κ 1 = 0.3, κ 2 = 0.45, ϕ 1 = ϕ 2 = 0.01, and λ = 0. The asympoic long rae is 5.20%. The very hick lines are for a high value of v, namely 0.75β2 2 + 0.25β1)r; 2 he hin lines are for a medium value of v, namely 0.5β2 2 + 0.5β1)r; 2 and he medium hick lines are for a low value of v, namely 0.25β2 2 + 0.75β1)r. 2

8.5 Muli-facor CIR models 218 Since he funcion T σ T r, v, ) depends on boh of r and v, he wo-facor model is able o generae more flexible erm srucures of volailiies han he one-facor models. I can be shown ha he volailiy σ T r, v, ) is an increasing funcion of he ime o mauriy T. Opions and oher derivaives Longsaff and Schwarz sae a pricing formula for a European call opion on a zero-coupon bond, which in our noaion looks as follows: 8.43) where C K,T,S r, v, ) = B S r, v, )χ 2 θ 1, θ 2 ; 4ϕ 1 /β1, 2 4ϕ 2 /β2, 2 ω 1 [β2r 2 v], ω 2 [v β1r] 2 ) KB T r, v, )χ 2 ˆθ1, ˆθ ) 2 ; 4ϕ 1 /β1, 2 4ϕ 2 /β2, 2 ˆω 1 [β2r 2 v], ˆω 2 [v β1r] 2, θ i = 4γ2 i [as T ) + ln K] βi 2eγi[T ] 1) 2ˆb, i = 1, 2, i S ) ˆθ i = 4γi 2 [as T ) + ln K] βi 2eγi[T ] 1) 2ˆb, i = 1, 2, i T )ˆb i S T ) 4γ i e γi[t ]ˆb i S ) ω i = βi 2β2 2, i = 1, 2, β2 1 )eγi[t ] 1)ˆb i S T ) 4γ i e γi[t ]ˆb i T ) ˆω i = βi 2β2 2, i = 1, 2, β2 1 )eγi[t ] 1) ˆbi τ) = b iτ) e γiτ, i = 1, 2. 1 Here χ 2, ) is he cumulaive disribuion funcion for a wo-dimensional non-cenral χ 2 -disribuion. To be more precise, he value of he cumulaive disribuion funcion is [ θ1 ] θ2 uθ 2/θ 1 χ 2 θ 1, θ 2 ; c 1, c 2, d 1, d 2 ) = f χ2 c 1,d 1)u) f χ2 c 2,d 2)s) ds du, 0 where f χ2 c,d) is he probabiliy densiy funcion for a one-dimensional random variable which is non-cenrally χ 2 -disribued wih c degrees of freedom and non-cenraliy parameer d. Noe ha he inner inegral can be wrien as he cumulaive disribuion funcion for a one-dimensional non-cenral χ 2 -disribuion evaluaed in he poin θ 2 uθ 2 /θ 1. As discussed in he conex of he one-facor CIR model, see page 192, his one-dimensional cumulaive disribuion funcion can be approximaed by he cumulaive disribuion funcion of a sandard one-dimensional normal disribuion. The value of he wo-dimensional χ 2 -disribuion funcion can hen be obained by a numerical inegraion. Chen and Sco 1992) provide a deailed analysis of he compuaion of he wo-dimensional χ 2 -disribuion funcion. They conclude ha, despie he necessary numerical inegraion, he opion price can be compued much faser using 8.43) han using Mone Carlo simulaion or numerical soluion of he fundamenal parial differenial equaion. Longsaff and Schwarz sae ha he parial derivaives C/ r and C/ v can be eiher posiive or negaive, which is no surprising considering he fac ha he price of he underlying bond can also be eiher posiively or negaively relaed o r and v. In he Longsaff-Schwarz model he prices of many derivaive securiies can only be compued using numerical echniques. One approach is o numerically solve he fundamenal parial differenial equaion wih he appropriae erminal condiions, cf. Chaper 16. For ha purpose he 0

8.6 Oher muli-facor diffusion models 219 formulaion of he model in erms of he original sae variables, x 1 and x 2, is preferable. The PDE o be solved is V x 1, x 2, ) + ˆϕ 1 ˆκ 1 x 1 ) V x 1 x 1, x 2, ) + ˆϕ 2 ˆκ 2 x 2 ) V x 2 x 1, x 2, ) + 1 2 V 2 β2 1x 1 x 2 x 1, x 2, ) + 1 2 V 1 2 β2 2x 2 x 2 x 1, x 2, ) 2 x 1 + x 2 )V x 1, x 2, ) = 0, x 1, x 2, ) R + R + [0, T ). Noe ha since x 1 and x 2 are independen, here is no erm wih he mixed second-order derivaive 2 V/ x 1 x 2. This fac simplifies he numerical soluion. The PDE for he price funcion in erms of he variables r and v will involve a mixed second-order derivaive since r and v are no independen. Furhermore, he value space for he variables x 1 and x 2 is simpler han he value space for r and v since he possible values of v depend on he value of r. This will complicae he numerical soluion of he PDE involving r and v even furher. Addiional remarks To implemen he Longsaff-Schwarz model he curren values of he shor rae and he curren variance rae of he shor rae mus be deermined, and parameer values have o be esimaed. Longsaff and Schwarz discuss he esimaion procedure boh in he original aricle and in oher aricles, cf. Longsaff and Schwarz 1993a, 1994). Clewlow and Srickland 1994) and Rebonao 1996, Ch. 12) discuss several pracical problems in he parameer esimaion. Longsaff and Schwarz 1993a) explain how o obain a perfec fi of he model yield curve o he observed yield curve by replacing he parameer ˆκ 2 wih a suiable ime-dependen funcion. However, his exended version of he model exhibis ime inhomogeneous volailiies, which is problemaic as will be discussed in Secion 9.6. In Longsaff and Schwarz 1992b) he auhors consider he pricing of caps and swapions wihin heir wo-facor model, while in Longsaff and Schwarz 1993b) hey discuss he imporance of aking sochasic ineres rae volailiy ino accoun when measuring he ineres rae risk of bonds. 8.6 Oher muli-facor diffusion models 8.6.1 Models wih sochasic consumer prices Cox, Ingersoll, and Ross 1985b) inroduce several muli-facor versions of heir famous onefacor model. The shor rae in heir one-facor model is really he real shor rae, he bonds hey price are real bonds promising delivery of cerain prespecified consumpion unis, and he prices are also saed in consumpion unis. To derive prices in moneary unis e.g. dollars) of nominal securiies, i.e. securiies wih payoff specified in moneary unis, hey focus on including he consumer price index as an addiional sae variable. In heir exensions hey coninue o assume ha he real shor rae follows he process dr = κ[θ r ] d + β r dz 1, as in he one-facor model. geomeric Brownian moion The firs exension is o le he consumer price index p follow a dp = p [π d + β p dz 2 ],

8.6 Oher muli-facor diffusion models 220 where π denoes he expeced inflaion rae, and where he marke price associaed wih he consumer price uncerainy is zero. A well-known applicaion of Iô s Lemma implies ha dln p ) = π 1 ) 2 β2 p d + β p dz 2 so ha he exended model is affine in r and ln p. The price in moneary unis of a nominal zero-coupon bond mauring a ime T is { B T p, r, ) = p 1 exp [ at ) + π 1 ) ] } 2 β2 p T ) bt )r, where he funcions aτ) and bτ) are exacly as in he one-facor CIR model, cf. 7.70) and 7.71) on page 190. In heir second exension he expeced inflaion rae π is also assumed o be sochasic so ha dp = p [π d + β p π dz 2 ], dπ = κ π [θ π π ] d + ρβ π π dz 2 + 1 ρ 2 β π π dz 3. The resuling model is a hree-facor affine model wih he sae variables r, ln p, and π. For he precise expression of he price of a nominal zero-coupon bond we refer he reader o he aricle, Cox, Ingersoll, and Ross 1985b). 3 and Sco 1993). Oher affine models of his ype have been sudied by Chen 8.6.2 Models wih sochasic long-erm level and volailiy The Longsaff-Schwarz model described in Secion 8.5.2 is no he only model ha includes sochasic ineres rae volailiy. Vezal 1997) considers he model dr = ϕ 1 κ 1 r ) d + β r γ dz 1, dln β 2 ) = ϕ 2 κ 2 ln β 2 ) d + ρξ dz1 + 1 ρ 2 ξ dz 2, while Andersen and Lund 1997) sudy he special case where he wo processes are independen, i.e. ρ = 0. Boh aricles focus on he esimaion and esing of he models. None of he models are affine or able o produce closed-form expressions for prices on bonds or opions. Also Boudoukh, Richardson, Sanon, and Whielaw 1999) consider a model wih sochasic volailiy of he shor rae, bu hey use a non-parameric esimaion echnique in order o esimae he drif of he shor rae and boh he drif and he volailiy of he shor rae volailiy. Balduzzi, Das, Foresi, and Sundaram 1996) sugges a hree-facor affine model in which he hree sae variables are he shor rae r, he long-erm level θ of he shor rae, and he insananeous variance v of he shor rae. They assume ha he real-world dynamics is of he 3 In addiion, Cox, Ingersoll, and Ross 1985b) sae an explici, bu very complicaed, pricing formula for he nominal bond in he case where he expeced inflaion rae follows he process dπ = κ π[θ π π ] d + ρβ ππ 3/2 dz 2 + 1 ρ 2 β ππ 3/2 dz 3, and he dynamics of r and p are as before. This model does no belong o he affine class.

8.6 Oher muli-facor diffusion models 221 form dr = κ r [θ r ] d + v dz 1, dθ = κ θ [ θ θ ] d + β θ dz 2, dv = κ v [ v v ] d + ρβ v v dz 1 + 1 ρ 2 β v v dz 3. Here, ρ is he correlaion beween changes in he shor rae level and he variance of he shor rae. Furhermore, he marke prices of risk are assumed o have a form ha implies ha he dynamics under he spo maringale measure is dr = κ r [θ r ] λ r v ) d + v dz Q 1, dθ = ) κ θ [ θ θ ] λ θ β θ d + βθ dz Q 2, dv = κ v [ v v ] λ v v ) d + ρβ v v dz Q 1 + 1 ρ 2 β v v dz Q 3, where λ r, λ θ, and λ v are consans. The model is affine so ha he zero-coupon bond prices are B T r, θ, v, ) = exp { at ) b 1 T )r b 2 T )θ b 3 T )v}. The auhors find explici expressions for b 1 and b 2, bu a and b 3 mus o be found by numerical soluion of he appropriae ordinary differenial equaions, cf. 8.16) and 8.17). The model can produce a wide variey of ineresing yield curve shapes. The esimaion of he model is also discussed. Chen 1996) sudies a hree-facor model wih he same hree sae variables r, θ, and v. In he simples version of he model, he dynamics under he real-world probabiliy measure is given as dr = κ r [θ r ] d + v dz 1, dθ = κ θ [ θ θ ] d + β θ θ dz 2, dv = κ v [ v v ] d + β v v dz 3, and he marke prices of risk are such ha he dynamics of he sae variables have he same srucure under he spo maringale measure, bu wih differen consans κ r, κ θ, θ, κ v, and v. Since he model is affine, he zero-coupon bond prices are of he form B T r, θ, v, ) = exp { at ) b 1 T )r b 2 T )θ b 3 T )v}. Alhough he model is no a muli-facor CIR model, Chen is able o find explici, bu complicaed, expressions for he funcions a, b 1, b 2, and b 3. In addiion, Chen considers a more general hreefacor model, which is no included in he affine class of models. Dai and Singleon 2000) divide he class of affine models ino subclasses, and for each subclass hey find he mos general model ha saisfies he condiions ν j x) 0 j = 1,..., n) which ensure ha he process for he sae variables is well-defined. They focus on he hree-facor models and show how he hree-facor models suggesed in he lieraure e.g. he models of Chen and of Balduzzi, Das, Foresi, and Sundaram) fi ino his classificaion. They find ha he suggesed models can be exended and, based on an empirical es using he hisorical evoluion of zerocoupon yields of hree differen mauriies, hey conclude ha he exensions are imporan in order for he models o be realisic. However, explici expressions for he funcions a, b 1, b 2, and b 3 have no been found in he exended models.

8.6 Oher muli-facor diffusion models 222 8.6.3 A model wih a shor and a long rae One of he very firs wo-facor erm srucure models was suggesed by Brennan and Schwarz 1979). They ake he shor rae r and he long rae l o be he sae variables. The long rae is he yield on a consol an infinie mauriy bond) which yields a coninuous paymen a a consan rae c. This is in line wih empirical sudies since he shor rae can be seen as an indicaor for he level of he yield curve, while he difference beween he long and he shor rae is a measure of he slope of he yield curve. The specific long rae dynamics assumed by Brennan and Schwarz is unaccepable, however. The problem is ha he long rae is given by l = c/l, where L is he price of he consol, and we know ha his price follows from he shor rae process and he pricing formula L = E Q [ ] e s ru du c ds. The drif and he volailiy of L, and hence of l, are herefore closely relaed o he shor rae r. Brennan and Schwarz assume for example ha he volailiy of he long rae is proporional o he long rae and independen of he shor rae. For a more deailed discussion of his issue, see Hogan 1993) and Duffie, Ma, and Yong 1995). In addiion o he model formulaion problems, he Brennan-Schwarz model does no allow closed-form pricing formulas for bonds or derivaives. Alhough i should be possible o consruc a heoreically accepable model wih he shor and he long rae as he sae variables, no such model has apparenly been suggesed in he finance lieraure. 8.6.4 Key rae models In heir analysis of affine muli-facor models Duffie and Kan 1996) focus on models in which he sae variables are zero-coupon yields of seleced mauriies, e.g. he 1-year, he 5-year, he 10-year, and he 30-year zero-coupon yield. We will refer o he seleced ineres raes as key raes. A clear advanage of such a model is ha i is easy o observe or a leas esimae) he sae variables from marke daa, much easier han he shor rae volailiy for example. yield curve obained in hese models auomaically maches he marke yields for he seleced mauriies. Oher muli-facor models end o have difficulies in maching he long end of he yield curve, which is problemaic for he pricing and hedging of long-erm bonds and opions on long-erm bonds. Many praciioners measure he sensiiviy of differen securiies owards changes in differen mauriy segmens of he yield curve. For ha purpose i is clearly convenien o use a model ha gives a direc relaion beween securiy prices and represenaive yields for he differen mauriy segmens. As shown in 8.18), he zero-coupon yields ȳ τ = y +τ are given by ȳ τ = ȳ τ x ), where ȳ τ x) = aτ) τ + n j=1 The in a general n-facor affine diffusion model b j τ) x j. τ Here, he funcions a, b 1,..., b n solve he ordinary differenial equaions 8.16) and 8.17) wih he iniial condiions a0) = b j 0) = 0. If each sae variable x j is he zero-coupon yield for a given ime o mauriy τ j, i.e. ȳ τj x) = x j,

8.7 Final remarks 223 we mus have ha 8.44) b j τ j ) = τ j, aτ j ) = b i τ j ) = 0, in j. These condiions impose very complicaed resricions on he parameers in he drif and he volailiy erms in he dynamics of he sae variables, i.e. he key raes. Explici expressions for he funcions a and b 1,..., b n can only be found in he Gaussian models. In general, he Ricai equaions wih he exra condiions 8.44) have o be solved numerically. An alernaive procedure is o sar wih an affine model wih oher sae variables so ha he condiions 8.44) do no have o be imposed in he soluion of he Ricai equaions. Subsequenly, he variables can be changed o he desired key raes. Since he zero-coupon yields are affine funcions of he original sae variables, he model wih he ransformed sae variables i.e. he key raes) is also an affine model. 8.6.5 Quadraic models In Secion 7.6 we gave a shor inroducion o quadraic one-facor models, i.e. models in which he shor rae is he square of a sae variable which follows an Ornsein-Uhlenbeck process. There are also muli-facor quadraic erm srucure models. The vecor of sae variables x follows a muli-dimensional Ornsein-Uhlenbeck process dx = ˆϕ ˆκx ) d + Γ dz Q, and he shor rae is a quadraic funcion of he sae variables, i.e. r = ξ + ψ x + x Θ x = ξ + The zero-coupon bond prices are hen of he form n ψ i x i + i=1 n i=1 j=1 n Θ ij x i x j. B T x, ) = exp { at ) bt ) x x ct )x } n = exp at ) n n b i T )x i c ij T )x i x j, i=1 i=1 j=1 where he funcions a, b i, and c ij can be found by solving a sysem of ordinary differenial equaions. These equaions have explici soluions only in very simple cases, bu efficien numerical soluion echniques exis. Special cases of his model class have been sudied by Beaglehole and Tenney 1992) and Jamshidian 1996), whereas Leippold and Wu 2002) provide a general characerizaion of he quadraic models. 8.7 Final remarks To give a precise descripion of he evoluion of he erm srucure of ineres raes over ime, i seems o be necessary o use models wih more han one sae variable. However, i is more complicaed o esimae and apply muli-facor models han one-facor models. Is he addiional effor worhwhile? Do muli-facor models generae prices and hedge raios ha are significanly differen from hose generaed by one-facor models? Of course, he answer will depend on he precise resuls we wan from he model.

8.7 Final remarks 224 Buser, Hendersho, and Sanders 1990) compare he prices on seleced opions on long-erm bonds compued wih differen ime homogeneous models. They conclude ha when he model parameers are chosen so ha he curren shor rae, he slope of he yield curve, and some ineres rae volailiy measure are he same in all he models, he model prices are very close, excep when he ineres rae volailiy is large. However, hey only consider specific derivaives and do no compare hedge sraegies, only prices. For a comparison of derivaive prices in differen models o be fair, he models should produce idenical prices of he underlying asses, which in he case of ineres rae derivaives are he zero-coupon bonds of all mauriies. As we will discuss in more deail in Chaper 9, he models sudied so far can be generalized in such a way ha he erm srucure of ineres raes produced by he model and he observed erm srucure mach exacly. The model is said o be calibraed o he observed erm srucure. Basically, one of he model parameers has o be replaced by a carefully chosen ime-dependen funcion, which resuls in a ime inhomogeneous version of he model. The models can also be calibraed o mach prices of derivaive securiies. Several auhors assume a presumably reasonable wo-facor model and calibrae a simpler one-facor model o he yield curve of he wo-facor model using he exension echnique described above. They compare he prices on various derivaives and he efficiency of hedging sraegies for he wo-facor and he calibraed one-facor model. We will ake a closer a look a hese sudies in Secion 9.9. The overall conclusion is ha he calibraed one-facor models should be used only for he pricing of securiies ha resemble he securiies o which he model is calibraed. For he pricing of oher securiies and, in paricular, for he consrucion of hedging sraegies i is imporan o apply muli-facor models ha provide a good descripion of he acual evoluion of he erm srucure of ineres raes. Anoher conclusion of Chaper 9 is ha he calibraed facor models should be used wih cauion. They have some unrealisic properies ha may affec he prices of derivaive securiies. In Chapers 10 and 11 we will consider models ha from he ouse are developed o mach he observed yield curve. Jus as a he end of Chaper 7 we will come o he defense of he ime homogeneous diffusion models. In pracice, he zero-coupon yield curve is no direcly observable, bu has o be esimaed, ypically by using observed prices on coupon bonds. Frequenly, he esimaion procedure is based on a relaively simple parameerizaion of he discoun funcion as e.g. a cubic spline or a Nelson- Siegel parameerizaion described in Chaper 1. Probably, an equally good fi o he observed marke prices and an economically more appropriae yield curve esimae can be obained by applying he parameerizaion of he discoun funcion T B T ha comes from an economically founded model such as hose discussed in his chaper. Hence, i may be beer o use he ime homogeneous version of he model han o calibrae a ime inhomogeneous version perfecly o an esimae of he curren yield curve.

Chaper 9 Calibraion of diffusion models 9.1 Inroducion In Chapers 7 and 8 we have sudied diffusion models in which he drif raes, he variance raes, and he covariance raes of he sae variables do no explicily depend on ime, bu only on he curren value of he sae variables. Such diffusion processes are called ime homogeneous. The drif raes, variances, and covariances are simple funcions of he sae variables and a small se of parameers. The derived prices and ineres raes are also funcions of he sae variables and hese few parameers. Consequenly, he resuling erm srucure of ineres raes will ypically no fi he currenly observable erm srucure perfecly. I is generally impossible o find values of a small number of parameers so ha he model can perfecly mach he infiniely many values ha a erm srucure consiss of. This propery appears o be inappropriae when he models are o be applied o he pricing of derivaive securiies. If he model is no able o price he underlying securiies i.e. he zero-coupon bonds) correcly, why rus he model prices for derivaive securiies? In order o be able o fi he observable erm srucure we need more parameers. This can be obained by replacing one of he model parameers by a carefully chosen ime-dependen funcion. The model is said o be calibraed o he marke erm srucure. The resuling model is ime inhomogeneous. The calibraed model is consisen wih he observed erm srucure and is herefore a relaive pricing model or pure no-arbirage model), cf. he classificaion inroduced in Secion 5.8. The model can also be calibraed o oher marke informaion such as he erm srucure of ineres rae volailiies. This requires ha an addiional parameer is allowed o depend on ime. For ime homogeneous diffusion models he curren prices and yields and he disribuion of fuure prices and yields do no depend direcly on he calendar dae, only on he ime o mauriy. For example, he zero-coupon yield y +τ does no depend direcly on, bu is deermined by he mauriy τ and value of he sae variables x. Consequenly, if he sae variables have he same values a wo differen poins in ime, he yield curve will also be he same. Time homogeneiy seems o be a reasonable propery of a erm srucure model. When ineres raes and prices change over ime, i is due o changes in he economic environmen he sae variables) raher han he simple passing of ime. In conras, he ime inhomogeneous models discussed in his chaper involve a direc dependence on calendar ime. We have o be careful no o inroduce unrealisic ime dependencies ha are likely o affec he prices of he derivaive securiies we are ineresed in. In his chaper we consider he calibraion of he one-facor models discussed in Chaper 7. 225

9.2 Time inhomogeneous affine models 226 Similar echniques can be applied o he muli-facor models of Chaper 8, bu in order o focus on he ideas and keep he noaion simple we will consider only one-facor models. The approach aken in his chaper is basically o srech an equilibrium model by inroducing some paricular ime-dependen funcions in he dynamics of he sae variable. A more naural approach for obaining a model ha fis he erm srucure is aken in Chapers 10 and 11, where he dynamics of he enire yield curve is modeled in an arbirage-free way assuming ha he iniial yield curve is he one currenly observed in he marke. 9.2 Time inhomogeneous affine models Replacing he consans in he ime homogeneous affine model 7.6) on page 164 by deerminisic funcions, we ge he shor rae dynamics 9.1) dr = ˆϕ) ˆκ)r ) d + δ 1 ) + δ 2 )r dz Q under he spo maringale measure, i.e. in a hypoheical, risk-neural world. In his exended version of he model, he disribuion of he shor rae r +τ prevailing τ years from now will depend boh on he ime horizon τ and he curren calendar ime. In he ime homogeneous models he disribuion of r +τ is independen of. Despie he exension we obain more or less he same pricing resuls as for he ime homogeneous affine models. Analogously o Theorem 7.1, we have he following characerizaion of bond prices: Theorem 9.1 In he model 9.1) he ime price of a zero-coupon bond mauring a T is given as B T = B T r, ), where 9.2) B T a,t ) b,t )r r, ) = e and he funcions a, T ) and b, T ) saisfy he following sysem of differenial equaions: 9.3) 1 2 δ 2)b, T ) 2 + ˆκ)b, T ) b, T ) 1 = 0, a 9.4), T ) + ˆϕ)b, T ) 1 2 δ 1)b, T ) 2 = 0 wih he condiions at, T ) = bt, T ) = 0. The only difference relaive o he resul for ime homogeneous models is ha he funcions a and b and hence he bond price) now depend separaely on and T, no jus on he difference T. The proof is almos idenical o he proof of Theorem 7.1 and is herefore omied. The funcions a, T ) and b, T ) can be deermined from he Equaions 9.3) and 9.4) by firs solving 9.3) for b, T ) and hen subsiuing ha soluion ino 9.4), which can hen be solved for a, T ). I follows immediaely from he above heorem ha he zero-coupon yields and he forward raes are given by 9.5) y T r, ) = a, T ) T + b, T ) T r and 9.6) f T r, ) = a b, T ) +, T )r. T T

9.2 Time inhomogeneous affine models 227 Boh expressions are affine in r. Nex, le us look a he erm srucures of volailiies in hese models, i.e. he volailiies on zero-coupon bond prices P τ = B +τ, zero-coupon yields ȳ τ = y +τ, and forward raes f τ = f +τ as funcions of he ime o mauriy τ. These volailiies involve he volailiy of he shor rae βr, ) = δ 1 ) + δ 2 )r and he funcion b, T ). The dynamics of he zero-coupon bond prices is dp +τ = P +τ [r λr, )βr, )b, + τ)) d b, + τ)βr, ) dz ], while he dynamics of zero-coupon yields and forward raes is given by 9.7) dy +τ =... d + b, + τ) βr, ) dz τ and 9.8) df +τ =... d + b, T ) T βr, ) dz, T =+τ respecively. Focusing on he volailiies, we have omied he raher complicaed drif erms. From 9.3) we see ha if he funcions δ 2 ) and ˆκ) are consan, hen we can wrie b, T ) as bt ) where he funcion bτ) solves he same differenial equaion as in he ime homogeneous affine models, i.e. he ordinary differenial equaion 7.8) on page 165. If δ 1 ) is also consan, he shor rae volailiy βr, ) = δ 1 ) + δ 2 )r will be ime homogeneous. Consequenly, when ˆκ), δ 1 ), and δ 2 ) bu no necessarily ˆϕ) are consans, he erm srucures of volailiies of he model are ime homogeneous in he sense ha he volailiies of B +τ, y +τ, and f +τ depend only on τ and he curren shor rae, no on. Due o he ime inhomogeneiy, he fuure volailiy srucure can be very differen from he curren volailiy srucure, even for a similar yield curve. This propery is inappropriae and no realisic. Furhermore, he prices of many derivaive securiies are highly dependen on he evoluion of volailiies, see e.g. Carverhill 1995) and Hull and Whie 1995). A model wih unreasonable volailiy srucures will probably produce unreasonable prices and hedge sraegies. For hese reasons i is ypically only he parameer ˆϕ ha is allowed o depend on ime. Below we will discuss such exensions of he models of Meron, of Vasicek, and of Cox, Ingersoll, and Ross. For a paricular choice of he funcion ˆϕ) hese exended models are able o mach he observed yield curve exacly, i.e. he models are calibraed o he marke yield curve. Noe ha if only ˆϕ depends on ime, he funcion bτ) is jus as in he original ime homogeneous version of he model, whereas he a funcion will be differen. Since and at, T ) = 0, Eq. 9.4) implies ha at, T ) a, T ) = T a u, T ) du u In paricular, a, T ) = T ˆϕu)bT u) du δ 1 2 T bt u) 2 du. 9.9) a0, T ) = T 0 ˆϕ)bT ) d δ 1 2 T 0 bt ) 2 d.

9.3 The Ho-Lee model exended Meron) 228 We wan o pick he funcion ˆϕ) so ha he curren ime 0) model prices on zero-coupon bonds, B T r 0, 0), are idenical o he observed prices, P T ), i.e. 9.10) a0, T ) = bt )r 0 ln P T ). We can hen deermine ˆϕ) by comparing 9.9) and 9.10). In he exensions of he models of Meron and Vasicek we are able o find an explici expression for ˆϕ), while numerical mehods mus be applied in he exension of he CIR model. 9.3 The Ho-Lee model exended Meron) Ho and Lee 1986) developed a recombining binomial model for he evoluion of he enire yield curve aking he currenly observed yield curve as given. Subsequenly, Dybvig 1988) has demonsraed ha he coninuous ime limi of heir binomial model is a model wih 9.11) dr = ˆϕ) d + β dz Q, which exends Meron s model described in Secion 7.3. The prices of zero-coupon bonds are of he form where bτ) = τ jus as in Meron s model, and 9.12) a, T ) = T B T r, ) = e a,t ) bt )r, ˆϕu)T u) du 1 T 2 β2 T u) 2 du, cf. he discussion in he preceding secion. The following heorem shows how o choose he funcion ˆϕ) in order o mach any given iniial yield curve. Theorem 9.2 Le f) be he curren erm srucure of forward raes and assume ha his funcion is differeniable. Then he erm srucure of ineres raes in he Ho-Lee model 9.11) wih 9.13) ˆϕ) = f ) + β 2 will be idenical o he curren erm srucure. In his case we have 9.14) a, T ) = ln P T ) P ) T ) f) + 1 2 β2 T ) 2. Proof: Subsiuing bt ) = T and δ 1 = β 2 ino 9.9), we obain a0, T ) = T 0 ˆϕ)T ) d 1 T 2 β2 T ) 2 d = T 0 0 ˆϕ)T ) d 1 6 β2 T 3. Compuing he derivaive wih respec o T, using Leibniz rule, 1 we obain a T T 0, T ) = 0 ˆϕ) d 1 2 β2 T 2, 1 If h, T ) is a deerminisic funcion, which is differeniable in T, hen T ) T h h, T ) d = ht, T ) +, T ) d. T 0 0 T

9.3 The Ho-Lee model exended Meron) 229 and anoher differeniaion gives 9.15) We wish o saisfy he relaion 9.10), i.e. 2 a T 2 0, T ) = ˆϕT ) β2 T. a0, T ) = T r 0 ln P T ). Recall from 1.14) on page 9 he following relaion beween he discoun funcion P and he erm srucure of forward raes f: Hence, a0, T ) mus saisfy ha and herefore 9.16) ln P T ) T = P T ) P T ) = ft ). a T 0, T ) = r 0 + ft ) 2 a T 2 0, T ) = f T ), where we have assumed ha he erm srucure of forward raes is differeniable. Comparing 9.15) and 9.16), we obain he saed resul. Subsiuing 9.13) ino 9.12), we ge a, T ) = T Parial inegraion gives T T f u)t u) du + β 2 ut u) du 1 T 2 β2 T u) 2 du. T f u)t u) du = T ) f) + fu) du = T ) f) ln ) P T ), P ) where we have used he relaion beween forward raes and zero-coupon bond prices o conclude ha 9.17) T fu) du = T Furhermore, edious calculaions yield ha 0 ) P fu) du fu) du = ln P T ) + ln P T ) ) = ln. 0 P ) T β 2 ut u) du 1 T 2 β2 T u) 2 du = 1 2 β2 T ) 2. Now, a, T ) can be wrien as saed in he Theorem. In he Ho-Lee model he shor rae follows a generalized Brownian moion wih a imedependen drif). From he analysis in Chaper 3 we ge ha he fuure shor rae is normally disribued, i.e. he Ho-Lee model is a Gaussian model. The pricing of European opions is similar o he original Meron model. The price of a call opion on a zero-coupon bond is given by 7.43) on page 175 wih he same expression for he variance v, T, S). As usual, he price of a call opion on a coupon bond follows from Jamshidian s rick.

9.4 The Hull-Whie model exended Vasicek) 230 9.4 The Hull-Whie model exended Vasicek) When we replace he parameer ˆθ in Vasicek s model 7.50) by a ime-dependen funcion ˆθ), we ge he following shor rae dynamics under he spo maringale measure: ] 9.18) dr = κ [ˆθ) r d + β dz Q. This model was inroduced by Hull and Whie 1990a) and is called he Hull-Whie model or he exended Vasicek model. As in he original Vasicek model, he process has a consan volailiy β > 0 and exhibis mean reversion wih a consan speed of adjusmen κ > 0, bu in he exended version he long-erm level is ime-dependen. The risk-adjused process 9.18) may be he resul of a real-world dynamics of dr = κ [θ) r ] d + β dz, and an assumpion ha he marke price of risk depends a mos on ime, λ). In ha case we will have ˆθ) = θ) β κ λ). Despie he sligh exension, i follows from he discussion in Secion 3.8.2 on page 59 ha he model remains Gaussian. To be more precise, he fuure shor rae r T is normally disribued wih he same variance as in he original Vasicek model, bu a differen mean, namely Var r, [r T ] = Var Q r,[r T ] = β 2 T under he spo maringale measure and under he real-world probabiliy measure. e 2κ[T u] du = β2 1 e 2κ[T ]), 2κ T E Q r,[r T ] = e κ[t ] r + κ e κ[t u] ˆθu) du T E r, [r T ] = e κ[t ] r + κ e κ[t u] θu) du According o Theorem 9.1 and he subsequen discussion, he zero-coupon bond prices in he Hull-Whie model are given by 9.19) B T r, ) = e a,t ) bt )r, where 9.20) bτ) = 1 κ 1 e κτ ), 9.21) a, T ) = κ T ˆθu)bT u) du + β2 4κ bt )2 + β2 bt ) T )). 2κ2 This expression holds for any given funcion ˆθ. Now assume ha a ime 0 we observe he curren shor rae r 0 and he enire discoun funcion T P T ) or, equivalenly, he erm srucure of forward raes T ft ). The following resul shows how o choose he funcion ˆθ so ha he model discoun funcion maches he observed discoun funcion exacly.

9.4 The Hull-Whie model exended Vasicek) 231 Theorem 9.3 Le f) be he curren ime 0) erm srucure of forward raes and assume ha his funcion is differeniable. model 9.18) wih Then he erm srucure of ineres raes in he Hull-Whie 9.22) ˆθ) = f) + 1 κ f ) + β2 2κ 2 1 e 2κ ) will be idenical o he curren erm srucure of ineres raes. In his case we have 9.23) a, T ) = ln P T ) P ) bt ) f) + β2 4κ bt )2 1 e 2κ). Proof: From Eq. 9.21) i follows ha 9.24) a0, ) = κ ˆθu)b u) du + β2 0 4κ b)2 + β2 b) ). 2κ2 Repeaed differeniaions yield 9.25) and a0, ) 2 a0, ) 2 = κˆθ) κ 2 = κ ˆθu)e κ[ u] du 1 0 2 β2 b) 2 = κˆθ) κ 0 a0, ) ˆθu)e κ[ u] du β 2 b)be κ β2 1 e 2κ ), 2κ where we have applied Leibniz rule see foonoe 1, page 228). Consequenly, 9.26) ˆθ) = β 2 2κ 2 1 e 2κ ) + 1 κ 2 a0, ) 2 + a0, ). Differeniaion of he expression 9.10), which we wan o be saisfied, yields 9.27) a0, ) = P ) P ) r 0e κ = f) r 0 e κ and 2 a0, ) 2 = f ) + κr 0 e κ. Subsiuing hese expressions ino 9.26), we obain 9.22). Subsiuing 9.26) ino 9.21), we ge a, T ) = Noe ha T + β2 2κ 2 fu) 1 e κ[t u]) du + 1 T κ T Parial inegraion yields 1 T κ f u) 1 e κ[t u]) du 1 e 2κu ) 1 e κ[t u]) du + β2 4κ bt )2 + β2 bt ) T )). 2κ2 T f u) du = ft ) f). f u)e κ[t u] du = 1 κ ft ) 1 T κ f)e κ[t ] fu)e κ[t u] du.

9.5 The exended CIR model 232 From 9.17) we ge ha a, T ) = ln + β2 ) P T ) T P ) f)bt ) + β2 1 e 2κu ) 2κ 2 1 e κ[t u]) du 4κ bt )2 + β2 bt ) T )). 2κ2 Afer some sraighforward, bu edious, manipulaions we arrive a he desired relaion 9.23). Due o he fac ha he Hull-Whie model is Gaussian, he prices of European call opions on zero-coupon bonds can be derived jus as in he Vasicek model. Since he b funcion and he variance of he fuure shor rae are he same in he Hull-Whie model as in Vasicek s model, we obain exacly he same opion pricing formula, i.e. 9.28) C K,T,S r, ) = B S r, )N d 1 ) KB T r, )N d 2 ), where d 1 = v, T, S) = 1 B S ) v, T, S) ln r, ) KB T + 1 v, T, S), r, ) 2 d 2 = d 1 v, T, S), β 2κ 3 1 e κ[s T ]) 1 e 2κ[T ]) 1/2. Since he zero-coupon bond price is a decreasing funcion of he shor rae, we can apply Jamshidian s rick saed in Theorem 7.3 for he pricing of European opions on coupon bonds in erms of a porfolio of European opions on zero-coupon bonds. 9.5 The exended CIR model Exending he CIR model analyzed in Secion 7.5 in he same way as we exended he models of Meron and Vasicek, he shor rae dynamics becomes 2 9.29) dr = κθ) ˆκr ) d + β r dz Q. For he process o be well-defined θ) has o be non-negaive. This will ensure a non-negaive drif when he shor rae is zero so ha he shor rae says non-negaive and he square roo erm makes sense. To ensure sricly posiive ineres raes we mus furher require ha 2κθ) β 2 for all. For an arbirary non-negaive funcion θ) he zero-coupon bond prices are B T r, ) = e a,t ) bt )r, where bτ) is exacly as in he original CIR model, cf. 7.70) on page 190, while he funcion a is now given by T a, T ) = κ θu)bt u) du. 2 This exension was suggesed already in he original aricle by Cox, Ingersoll, and Ross 1985b).

9.6 Calibraion o oher marke daa 233 Suppose ha he curren discoun funcion is P T ) wih he associaed erm srucure of forward raes given by ft ) = P T )/ P T ). To obain P T ) = B T r 0, 0) for all T, we have o choose θ) so ha T a0, T ) = ln P T ) bt )r 0 = κ θu)bt u) du, T > 0. 0 Differeniaing wih respec o T, we ge ft ) = b T )r 0 + κ T 0 θu)b T u) du, T > 0. According o Heah, Jarrow, and Moron 1992, p. 96) i can be shown ha his equaion has a unique soluion θ), bu i canno be wrien in an explici form so a numerical procedure mus be applied. We canno be sure ha he soluion complies wih he condiions ha guaranee a well-defined shor rae process. Clearly, a necessary condiion for θ) o be non-negaive for all is ha 9.30) ft ) r0 b T ), T > 0. No all forward rae curves saisfy his condiion, cf. Exercise 9.1. Consequenly, in conras o he Meron and he Vasicek models, he CIR model canno be calibraed o any given erm srucure. No explici opion pricing formulas have been found in he exended CIR model. Opion prices can be compued by numerically solving he parial differenial equaion associaed wih he model, e.g. using he echniques oulined in Chaper 16. 9.6 Calibraion o oher marke daa daa. Many praciioners wan a model o be consisen wih basically all reliable curren marke The objecive may be o calibrae a model o he prices of liquid bonds and derivaive securiies, e.g. caps, floors, and swapions, and hen apply he model for he pricing of less liquid securiies. In his manner he less liquid securiies are priced in a way which is consisen wih he indispuable observed prices. Above we discussed how an equilibrium model can be calibraed o he curren yield curve i.e. curren bond prices) by replacing he consan in he drif erm wih a ime-dependen funcion. If we replace oher consan parameers by carefully chosen deerminisic funcions, we can calibrae he model o furher marke informaion. Le us ake he Vasicek model as an example. If we allow κ o depend on ime, he shor rae dynamics becomes ] dr = κ) [ˆθ) r d + β dz Q = [ ˆϕ) κ)r ] d + β dz Q. The price of a zero-coupon bond is sill given by Theorem 9.1 as B T r, ) = exp{ a, T ) b, T )r}. According o Eqs. 15) and 16) in Hull and Whie 1990a), he funcions κ) and ˆϕ) are κ) = 2 b0, )/ 2 b0, )/, ) 2 a0, ) ˆϕ) = κ) + 2 a0, ) b0, ) ) 2 β 2 du, b0, u)/ u 0

9.7 Iniial and fuure erm srucures in calibraed models 234 and can hence be deermined from he funcions a0, ) and b0, ) and heir derivaives. From 9.7) we ge ha he model volailiy of he zero-coupon yield y +τ σy +τ ) = β b, + τ). τ = y +τ r, ) is In paricular, he ime 0 volailiy is σ τ y 0) = βb0, τ)/τ. If he curren erm srucure of zerocoupon yield volailiies is represened by he funcion σ y ), we can obain a perfec mach of hese volailiies by choosing b0, ) = τ β σ y). The funcion a0, ) can hen be deermined from b0, ) and he curren discoun funcion P ) as described in he previous secions. Noe ha he erm srucure of volailiies can be esimaed eiher from hisorical flucuaions of he yield curve or as implied volailiies derived from curren prices of derivaive securiies. Typically he laer approach is based on observed prices of caps. Finally, we can also le he shor rae volailiy be a deerminisic funcion β) so ha we ge he fully exended Vasicek model 9.31) dr = κ)[ˆθ) r ] d + β) dz Q. Choosing β) in a specific way, we can calibrae he model o furher marke daa. Despie all hese exensions, he model remains Gaussian so ha he opion pricing formula 9.28) sill applies. However, he relevan volailiy is now v, T, S), where v, T, S) 2 = T T [ ] 2 βu) 2 [bu, S) bu, T )] 2 du = [b0, S) b0, T )] 2 βu) du, b0, u)/ u cf. Hull and Whie 1990a). Jamshidian s resul 7.30) for European opions on coupon bonds is sill valid if he esimaed b, T ) funcion is posiive. If eiher κ or β or boh) are ime-dependen, he volailiy srucure in he model becomes ime inhomogeneous, i.e. dependen on he calendar ime, cf. he discussion in Secion 9.2. Since he volailiy srucure in he marke seems o be prey sable when ineres raes are sable), his dependence on calendar ime is inappropriae. Broadly speaking, o le κ or β depend on ime is o srech he model oo much. I should no come as a surprise ha i is hard o find a reasonable and very simple model which is consisen wih boh yield curves and volailiy curves. If only he parameer θ is allowed o depend on ime, he volailiy srucure of he model is ime homogeneous. The drif raes of he shor rae, he zero-coupon yields, and he forward raes are sill ime inhomogeneous, which is cerainly also unrealisic. The drif raes may change over ime, bu only because key economic variables change, no jus because of he passage of ime. However, Hull and Whie and oher auhors argue ha ime inhomogeneous drif raes are less criical for opion prices han ime inhomogeneous volailiy srucures. See also he discussion in Secion 9.9 below. 9.7 Iniial and fuure erm srucures in calibraed models In he preceding secion we have implicily assumed ha he curren erm srucure of ineres raes is direcly observable. In pracice, he erm srucure of ineres raes is ofen esimaed from

9.7 Iniial and fuure erm srucures in calibraed models 235 he prices of a finie number of liquid bonds. As discussed in Secion 1.6, his is ypically done by expressing he discoun funcion or he forward rae curve as some given funcion wih relaively few parameers. The values of hese parameers are chosen o mach he observed prices as closely as possible. A cubic spline esimaion of he discoun funcion will frequenly produce unrealisic esimaes for he forward rae curve and, in paricular, for he slope of he forward rae curve. This is problemaic since he calibraion of he equilibrium models depends on he forward rae curve and is slope as can be seen from he earlier secions of his chaper. In conras, he Nelson-Siegel parameerizaion 9.32) f) = c1 + c 2 e k + c 3 e k, cf. 1.30), ensures a nice and smooh forward rae curve and will presumably be more suiable in he calibraion procedure. No maer which of hese parameerizaions is used, i will no be possible o mach all he observed bond prices perfecly. Hence, i is no sricly correc o say ha he calibraion procedure provides a perfec mach beween model prices and marke prices of he bonds. See also Exercise 9.2. Recall ha he cubic spline and he Nelson-Siegel parameerizaions are no based on any economic argumens, bu are simply curve fiing echniques. The heoreically beer founded dynamic equilibrium models of Chapers 7 and 8 also resul in a parameerizaion of he discoun funcion, e.g. 7.67) and he associaed expressions for a and b in he Cox-Ingersoll-Ross model. Why no use such a parameerizaion insead of he cubic spline or he Nelson-Siegel parameerizaion? And if he parameerizaion generaed by an equilibrium model is used, why no use ha equilibrium model for he pricing of fixed income securiies raher han calibraing a differen model o he chosen parameerized form? In conclusion, he objecive mus be o use an equilibrium model ha produces yield curve shapes and yield curve movemens ha resemble hose observed in he marke. If such a model is oo complex, one can calibrae a simpler model o he yield curve esimae semming from he complex model and hope ha he calibraed simpler model provides prices and hedge raios which are reasonably close o hose in he complex model. A relaed quesion is wha shapes he fuure yield curve may have, given he chosen parameerizaion of he curren yield curve and he model dynamics of ineres raes. For example, if we use a Nelson-Siegel parameerizaion 9.32) of he curren yield curve and le his yield curve evolve according o a dynamic model, e.g. he Hull-Whie model, will he fuure yield curves also be of he form 9.32)? Inuiively, i seems reasonable o use a parameerizaion which is consisen wih he model dynamics, in he sense ha he possible fuure yield curves can be wrien on he same parameerized from, alhough possibly wih oher parameer values. Which parameerizaions are consisen wih a given dynamic model? This quesion was sudied by Björk and Chrisensen 1999) using advanced mahemaics, so le us jus lis some of heir conclusions: The simple affine parameerizaion f) = c 1 +c 2 is consisen wih he Ho-Lee model 9.11), i.e. if he iniial forward rae curve is a sraigh line, hen he fuure forward rae curves in he model are also sraigh lines.

9.8 Calibraed non-affine models 236 The simples parameerizaion of he forward rae curve, which is consisen wih he Hull- Whie model 9.18), is f) = c 1 e k + c 2 e 2k. The Nelson-Siegel parameerizaion 9.32) is consisen neiher wih he Ho-Lee model nor he Hull-Whie model. However, he exended Nelson-Siegel parameerizaion f) = c 1 + c 2 e k + c 3 e k + c 4 e 2k is consisen wih he Hull-Whie model. Furhermore, i can be shown ha he Nelson-Siegel parameerizaion is no consisen wih any non-rivial one-facor diffusion model, cf. Filipović 1999). 9.8 Calibraed non-affine models In Secion 7.6 on page 193 we looked a some non-affine one-facor models wih consan parameers. These models can also be calibraed o marke daa by replacing he consan parameers by ime-dependen funcions. The Black-Karasinksi model 7.79) can hus be exended o 9.33) dln r ) = κ)θ) ln r ) d + β) dz Q, where κ, θ, and β are deerminisic funcions of ime. Despie he generalizaion, he fuure values of he shor rae remain lognormally disribued. Black and Karasinksi implemen heir model in a binomial ree and choose he funcions κ, θ, and β so ha he yields and he yield volailiies compued wih he ree exacly mach hose observed in he marke. There are no explici pricing formulas, and he consrucion of he calibraed binomial ree is quie complicaed. The BDT model inroduced by Black, Derman, and Toy 1990) is he special case of he Black- Karasinski model where β) is a differeniable funcion and κ) = β ) β). Sill, no explici pricing formulas have been found, and also his model is ypically implemened in a binomial ree. 3 To avoid he difficulies arising from ime-dependen volailiies, β) has o be consan. In ha case, κ) = 0 in he BDT model, and he model is reduced o he simple model dr = 1 2 β2 r d + βr dz Q, which is a special case of he Rendleman-Barer model 7.80) and canno be calibraed o he observed yield curve. Theorem 7.6 showed ha he ime homogeneous lognormal models produce compleely wrong Eurodollar-fuures prices. The ime inhomogeneous versions of he lognormal models exhibi he same unpleasan propery. 3 I is no clear from Black, Derman, and Toy 1990) how a calibraed ree can be consruced, bu Jamshidian 1991) fills his gap in heir presenaion.

9.9 Is a calibraed one-facor model jus as good as a muli-facor model? 237 9.9 Is a calibraed one-facor model jus as good as a muli-facor model? In he opening secion of Chaper 8 we argued ha more han one facor is needed in order o give a reasonable descripion of he evoluion of he erm srucure of ineres raes. However, muli-facor models are harder o esimae and apply han one-facor models. If here are no significan differences in he prices and hedge raios obained in a muli-facor and a one-facor model, i will be compuaionally convenien o use he one-facor model. Bu will a simple onefacor model provide he same prices and hedge raios as a more realisic muli-facor model? We iniiaed he discussion of his issue in Secion 8.7, where we focused on he ime homogeneous models. Inuiively, he prices of derivaive securiies in wo differen models should be closer when he wo models produce idenical prices o he underlying asses. Several auhors compare a ime homogeneous wo-facor model o a ime inhomogeneous one-facor model which has been perfecly calibraed o he yield curve generaed by he wo-facor model. Hull and Whie 1990a) compare prices of seleced derivaive securiies in differen models ha have been calibraed o he same iniial yield curve. They firs assume ha he ime homogeneous CIR model wih cerain parameer values) provides a correc descripion of he erm srucure, and hey compue prices of European call opions on a 5-year bulle bond and of various caps, boh wih he original CIR model and he exended Vasicek model calibraed o he CIR yield curve. They find ha he prices in he wo models are generally very close, bu ha he percenage deviaion for ou-of-he-money opions and caps can be considerable. Nex, hey compare prices of European call opions on a 5-year zero-coupon bond in he exended Vasicek model o prices compued using wo differen wo-facor models, namely a wo-facor Gaussian model and a wofacor CIR model. In each of he comparisons he wo-facor model is assumed o provide he rue yield curve, and he exended Vasicek-model is calibraed o he yield curve of he wo-facor model. The price differences are very small. Hence, Hull and Whie conclude ha alhough he rue dynamics of he yield curve is consisen wih a complex one-facor CIR) or a wo-facor model, one migh as well use he simple exended Vasicek model calibraed o he rue yield curve. Hull and Whie consider only a few differen derivaive securiies and only wo relaively simple muli-facor models, and hey compare only prices, no hedge sraegies. Canabarro 1995) performs a more adequae comparison. Firs, he argues ha he wo-facor models used in he comparison of Hull and Whie are degenerae and describe he acual evoluion of he yield curve very badly. For example, he shows ha, in he wo-facor CIR model hey use, one of he facors wih he parameer values used by Hull and Whie) will explain more han 99% of he oal variaion in he yield curve and hence he second facor explains less han 1%. As discussed in Secion 8.1 on page 201, he finds empirically ha he mos imporan facor can explain only 85% and he second-mos imporan facor more han 10% of he variaion in he yield curve. Moreover, Hull and Whie s wo-facor CIR model gives unrealisically high correlaions beween zero-coupon yields of differen mauriies. For example, he correlaion beween he 3-monh and he 30-year par yields is as high as 0.96 in ha model, which is far from he empirical esimae of 0.46, cf. Table 8.1. Therefore, a comparison o his wo-facor model will provide very lile informaion on wheher i is reasonable or no o use a simple calibraed one-facor model o represen he complex real-world dynamics. In his comparisons Canabarro also uses a differen wo-facor model, namely he model of Brennan and Schwarz 1979), which was briefly described a he end of

9.10 Final remarks 238 Secion 8.6. Despie he heoreical deficiencies of he Brennan-Schwarz model, Canabarro shows ha he model provides reasonable values for he correlaions and he explanaory power of each of he facors. Each of he wo wo-facor models is compared o wo calibraed one-facor models. The firs is he exended one-facor CIR model and he second is he BDT model dr = ˆϕ) ˆκ)r ) d + β r dz Q, dln r ) = β ) β) θ) ln r ) d + β) dz Q. The ime-dependen funcions in hese models are chosen so ha he models produce he same iniial yield curve and he same prices of caps wih a given cap-rae, bu differen mauriies, as he wo-facor model which is assumed o be he rue model. Noe ha boh hese calibraed one-facor models exhibi ime-dependen volailiy srucures, which in general should be avoided. For boh of he wo benchmark wo-facor models, Canabarro finds ha using a calibraed onefacor model insead of he correc wo-facor model resuls in price errors ha are very small for relaively simple securiies such as caps and European opions on bonds. For so-called yield curve opions ha have payoffs given by he difference beween a shor-erm and a long-erm zero-coupon yield, he errors are much larger and non-negligible. These findings are no surprising since he onefacor models do no allow for wiss in he yield curve, i.e. yield curve movemens where he shor end and he long end move in opposie direcions. I is exacly hose movemens ha make yield curve opions valuable. Regarding he efficiency of hedging sraegies, he calibraed one-facor models perform very badly. This is rue even for he hedging of simple securiies ha resemble he securiies used in he calibraion of he models. Boh pricing errors and hedging errors are ypically larger for he BDT model han for he calibraed one-facor CIR model. In general, he errors are larger when he one-facor models have been calibraed o he more realisic Brennan-Schwarz model han o he raher degenerae wo-facor CIR model used in he comparison of Hull and Whie. The conclusion o be drawn from hese sudies is ha he calibraed one-facor models should be used only for pricing securiies ha are closely relaed o he securiies used in he calibraion of he model. For he pricing of oher securiies and, in paricular, for he design of hedging sraegies i is imporan o apply muli-facor models ha give a good descripion of he acual movemens of he yield curve. 9.10 Final remarks This chaper has shown how one-facor equilibrium models can be perfecly fied o he observed yield curve by replacing consan parameers by cerain ime-dependen funcions. However, we have also argued ha his calibraion approach has inappropriae consequences and should be used only wih grea cauion. Similar procedures apply o muli-facor models. Since muli-facor models ypically involve more parameers han one-facor models, hey can give a closer fi o any given yield curve wihou inroducing ime-dependen funcions. In his sense he gain from a perfec calibraion is less for

9.11 Exercises 239 muli-facor models. In he following wo chapers we will look a a more direc way o consruc models ha are consisen wih he observable yield curve. 9.11 Exercises EXERCISE 9.1 Calibraion of he CIR model) Compue b τ) in he CIR model by differeniaion of 7.70) on page 190. Find ou which ypes of iniial forward rae curves he CIR model can be calibraed o, by compuing using a spreadshee for example) he righ-hand side of 9.30) for reasonable values of he parameers and he iniial shor rae. Vary he parameers and he shor rae and discuss he effecs. EXERCISE 9.2 The Hull-Whie model calibraed o he Vasicek yield curve) Suppose he observable bond prices are fied o a discoun funcion of he form *) P ) = e a) b)r 0, where b) = 1 κ 1 e κ ), a) = y [ b)] + β2 4κ b)2, where y, κ, and β are consans. This is he discoun funcion of he Vasicek model, cf. 7.51) 7.53) on page 178. a) Express he iniial forward raes f) and he derivaives f ) in erms of he funcions a and b. b) Show by subsiuion ino 9.22) ha he funcion ˆθ) in he Hull-Whie model will be given by he consan ˆθ) = y + β2 2κ, 2 when he iniial observable discoun funcion is of he form *), i.e. as in he Vasicek model.

Chaper 10 Heah-Jarrow-Moron models 10.1 Inroducion In Chaper 7 and Chaper 8 we discussed various models of he erm srucure of ineres raes which assume ha he enire erm srucure is governed by a low-dimensional Markov vecor diffusion process of sae variables. Among oher hings, we concluded from hose chapers ha a ime-homogeneous diffusion model generally canno produce a erm srucure consisen wih all observed bond prices, bu as discussed in Chaper 9 a simple exension o a ime-inhomogeneous model allows for a perfec fi o any or almos any) given erm srucure of ineres raes. A more naural way o achieve consisency wih observed prices is o sar from he observed erm srucure and hen model he evoluion of he enire erm srucure of ineres raes in a manner ha precludes arbirage. This is he approach inroduced by Heah, Jarrow, and Moron 1992), henceforh abbreviaed HJM. 1 The HJM models are relaive pricing models and focus on he pricing of derivaive securiies. This chaper gives an overview of he HJM class of erm srucure models. We will discuss he main characerisics, advanages, and drawbacks of he general HJM framework and consider several model specificaions in more deail. In paricular, we shall sudy he relaionship beween HJM models and he diffusion models discussed in previous chapers. We ake an applied perspecive and, alhough he exposiion is quie mahemaical, we shall no go oo deep ino all echnicaliies, bu refer he ineresed reader o he original HJM paper and he oher references given below for deails. 10.2 Basic assumpions As before, we le f T be he coninuously compounded) insananeous forward rae prevailing a ime for a loan agreemen over an infiniesimal ime inerval saring a ime T. We shall refer o f T as he T -mauriy forward rae a ime. Suppose ha we know he erm srucure of ineres raes a ime 0 represened by he forward rae funcion T f0 T. Assume ha, for any fixed T, he T -mauriy forward rae evolves according o n 10.1) df T = α, T, f s ) s ) d + β i, T, f s ) s ) dz i, 0 T, i=1 1 The binomial model of Ho and Lee 1986) can be seen as a forerunner of he more complee and horough HJM-analysis. 240

10.2 Basic assumpions 241 where z 1,..., z n are n independen sandard Brownian moions under he real-world probabiliy measure. The f s ) s erms indicae ha boh he forward rae drif α and he forward rae volailiy erms β i a ime may depend on he enire forward rae curve presen a ime. We call 10.1) an n-facor HJM model of he erm srucure of ineres raes. Noe ha n is he number of random shocks Brownian moions), and ha all he forward raes are affeced by he same n shocks. The diffusion models discussed in Chaper 7 and Chaper 8 are based on he evoluion of a low-dimensional vecor diffusion process of sae variables. The general HJM model does no fi ino ha framework. As discussed in Chaper 5 i is no possible o use he parial differenial equaion approach for pricing in such general models, bu we can sill price by compuing relevan expecaions under he appropriae maringale measures. However, we can hink of he general model 10.1) as an infinie-dimensional diffusion model, since he infiniely many forward raes can affec he dynamics of any forward rae. 2 In Secion 10.6 we shall discuss when an HJM model can be represened by a low-dimensional diffusion model. The basic idea of HJM is o direcly model he enire erm srucure of ineres raes. Recall from Chaper 1 ha he erm srucure a some ime is equally well represened by he discoun funcion T B T or he yield curve T y T as by he forward rae funcion T f T, due o he following relaions 10.2) B T = e T f s ds = e yt T ), 10.3) 10.4) f T = ln BT T y T = 1 T T = y T + T ) yt T, f s ds = 1 T ln BT. We could herefore have specified he dynamics of he zero-coupon bond prices or he yield curve insead of he forward raes. However, here are a leas) hree reasons for choosing he forward raes as he modeling objec. Firsly, he forward raes are he mos basic elemens of he erm srucure. Boh he zero-coupon bond prices and yields involve sums/inegrals of forward raes. Secondly, we know from our analysis in Chaper 5 ha one way of pricing derivaives is o find he expeced discouned payoff under he spo maringale measure, where he discouning is in erms of he shor-erm ineres rae r. The shor rae is relaed o he forward raes, he yield curve, and he discoun funcion as r = f = lim T y T = lim T ln B T T Obviously, he relaion of he forward raes o he shor rae is much simpler han ha of boh he yield curve and he discoun funcion, so his moivaes he HJM choice of modeling basis. Thirdly, he volailiy srucure of zero coupon bond prices is more complicaed han ha of ineres raes. For example, he volailiy of he bond price mus approach zero as he bond approaches mauriy, and, o avoid negaive ineres raes, he volailiy of a zero coupon bond price mus approach zero as he price approaches one. Such resricions need no be imposed on he volailiies of forward raes. 2 In fac, he resuls of Theorem 10.1, 10.2, and 10.4 are valid in he more general seing, where he drif and volailiies of he forward raes also depend he forward rae curves a previous daes. Since no models wih ha feaure have been sudied in he lieraure, we focus on he case where only he curren forward rae curve affecs he dynamics of he curve over he nex infiniesimal period of ime..

10.3 Bond price dynamics and he drif resricion 242 10.3 Bond price dynamics and he drif resricion In his secion we will discuss how we can change he probabiliy measure in he HJM framework o he spo maringale measure Q. As a firs sep, he following heorem gives he dynamics under he real-world probabiliy measure of he zero coupon bond prices B T assumpion 10.1). Theorem 10.1 Under he assumed forward rae dynamics 10.1), he price B T bond mauring a ime T evolves as [ 10.5) db T = B T where 10.6) 10.7) µ T, f s ) s ) d + ] n σi T, f s ) s ) dz i, i=1 T µ T, f s ) s ) = r α, u, f s ) s ) du + 1 2 T σi T, f s ) s ) = β i, u, f s ) s ) du, Proof: For simpliciy we only proof he claim for he case n = 1, where under he HJM of a zero coupon n 2 T β i, u, f s ) s ) du), i=1 df T = α, T, f s ) s ) d + β, T, f s ) s ) dz, 0 T, for any T. Inroduce he auxiliary sochasic process Y = T f u du. Then we have from 10.2) ha he zero coupon bond price is given by B T = e Y. If we can find he dynamics of Y, we can herefore apply Iô s Lemma o derive he dynamics of he zero-coupon bond price B T. Since Y is a funcion of infiniely many forward raes f u wih dynamics given by 10.1), i is however quie complicaed o derive he dynamics of Y. Due o he fac ha appears boh in he lower inegraion bound and in he inegrand iself, we mus apply Leibniz rule for sochasic inegrals saed in Theorem 3.3 on page 55, which in his case yields ) T ) T dy = r + α, u, f s ) s ) du d + β, u, f s ) s ) du dz, where we have applied ha r = f, ). Since B T = gy ), where gy ) = e Y wih g Y ) = e Y and g Y ) = e Y, Iô s Lemma see Theorem 3.4 on page 56) implies ha he dynamics of he zero coupon bond prices is ) T ) 2 db T = e Y r + α, u, f s ) s ) du + 1 T β, u, f s 2 e Y ) s ) du d ) T e Y β, u, f s ) s ) du dz [ T 2 = B T r α, u, f s ) s ) du + 1 T β, u, f s ) s ) du) d 2 ) T β, u, f s ) s ) du dz ],

10.3 Bond price dynamics and he drif resricion 243 which gives he one-facor version of 10.5). Now we urn o he behavior under he spo maringale measure Q. The forward rae will have he same volailiy erms β i, T, f s ) s ) as under he real-world probabiliy measure, bu a differen drif. More precisely, we have from Chaper 5 ha he n-dimensional process z Q = z Q 1,..., zq n ) defined by dz Q i = dz i + λ i d is a sandard Brownian moion under he spo maringale measure Q, where he λ i processes are he marke prices of risk. Subsiuing his ino 10.1), we ge where df T = ˆα, T, f s ) s ) d + n β i, T, f s ) s ) dz Q i, i=1 ˆα, T, f s ) s ) = α, T, f s ) s ) n β i, T, f s ) s )λ i. As in Theorem 10.1 we ge ha he drif rae of he zero coupon bond price becomes T r ˆα, u, f s ) s ) du + 1 2 i=1 n T β i, u, f s ) s ) du under he spo maringale measure Q. Bu we also know ha his drif rae has o be equal o r. This can only be rue if i=1 ) 2 T ˆα, u, f s ) s ) du = 1 2 n 2 T β i, u, f s ) s ) du). i=1 Differeniaing wih respec o T, we ge he following key resul: Theorem 10.2 The forward rae drif under he spo maringale measure Q saisfies 10.8) ˆα, T, f s ) s ) = n i=1 T β i, T, f s ) s ) β i, u, f s ) s ) du. The relaion 10.8) is called he HJM drif resricion. The drif resricion has imporan consequences: Firsly, he forward rae behavior under he spo maringale measure Q is fully characerized by he iniial forward rae curve, he number of facors n, and he forward rae volailiy erms β i, T, f s ) s ). The forward rae drif is no o be specified exogenously. This is in conras o he diffusion models considered in he previous chapers, where boh he drif and he volailiy of he sae variables were o be specified. Secondly, since derivaive prices depend on he evoluion of he erm srucure under he spo maringale measure and oher maringale measures, i follows ha derivaive prices depend only on he iniial forward rae curve and he forward rae volailiy funcions β i, T, f s ) s ). paricular, derivaives prices do no depend on he marke prices of risk. We do no have o make any assumpions or equilibrium derivaions of he marke prices of risk o price derivaives in an HJM model. In his sense, HJM models are pure no-arbirage models. Again, his is in conras wih he diffusion models of Chapers 7 and 8. In he one-facor diffusion models, for example, he enire erm srucure is assumed o be generaed by he movemens of he very shor end and In

10.4 Three well-known special cases 244 he resuling erm srucure depends on he marke price of shor rae risk. In he HJM models we use he informaion conained in he curren erm srucure and avoid o separaely specify he marke prices of risk. 10.4 Three well-known special cases Since he general HJM framework is quie absrac, we will in his secion look a hree specificaions ha resul in well-known models. 10.4.1 The Ho-Lee exended Meron) model Le us consider he simples possible HJM-model: a one-facor model wih β, T, f s ) s ) = β > 0, i.e. he forward rae volailiies are idenical for all mauriies independen of T ) and consan over ime independen of ). From he HJM drif resricion 10.8), he forward rae drif under Q is T ˆα, T, f s ) s ) = β β du = β 2 [T ]. Wih his specificaion he fuure value of he T -mauriy forward rae is given by f T = f T 0 + 0 β 2 [T u] du + 0 β dz Q u, which is normally disribued wih mean f0 T + β 2 [T /2] and variance 0 β2 du = β 2. In paricular, he fuure value of he shor rae is By Iô s Lemma, r = f = f 0 + 1 2 β2 2 + 10.9) dr = ˆϕ) d + β dz Q, where ˆϕ) = f 0/ + β 2. 0 β dz Q u. From 10.9), we see ha his specificaion of he HJM model is equivalen o he Ho-Lee exension of he Meron model, which was sudied in Secion 9.3 on page 228. relaion where I follows ha zero coupon bond prices are given in erms of he shor rae by he a, T ) = B T = e a,t ) T )r, T ˆϕu)T u) du β2 6 T )3. Furhermore, he price C K,T,S of a European call opion mauring a ime T wih exercise price K wrien on he zero coupon bond mauring a S is 10.10) C K,T,S = B S N d 1 ) KB T N d 2 ), where 10.11) 10.12) 10.13) d 1 = 1 B S v, T, S) ln KB T d 2 = d 1 v, T, S), v, T, S) = β[s T ] T. ) + 1 v, T, S), 2

10.4 Three well-known special cases 245 In addiion, Jamshidian s rick for he pricing of European opions on coupon bonds see Theorem 7.3 on page 170) can be applied since B S T is a monoonic funcion of r T. 10.4.2 The Hull-Whie exended Vasicek) model Nex, le us consider he one-facor model wih he forward rae volailiy funcion 10.14) β, T, f s κ[t ] ) s ) = βe for some posiive consans β and κ. Here he forward rae volailiy is an exponenially decaying funcion of he ime o mauriy. By he drif resricion, he forward rae drif under Q is ˆα, T, f s ) s ) = βe κ[t ] T so ha he fuure value of he T -mauriy forward rae is f T = f0 T β 2 + 0 κ e κ[t u] 1 e κ[t u]) du + In paricular, he fuure shor rae is where he deerminisic funcion g is defined by βe κ[u ] du = β2 κ e κ[t ] κ[t 1 e ]) r = f = g) + βe κ e κu dzu Q, g) = f0 β 2 + 0 κ e κ[ u] 1 e κ[ u]) du = f 0 + β2 2κ 2 1 e κ ) 2. 0 0 βe κ[t u] dz Q u. Again, he fuure values of he forward raes and he shor rae are normally disribued. Le us find he dynamics of he shor rae. Wriing R = 0 eκu dz Q u, we have r = G, R ), where G, R) = g) + βe κ R. We can now apply Iô s Lemma wih G/ = g ) κβe κ R, G/ R = βe κ, and 2 G/ R 2 = 0. Since dr = e κ dz Q we ge and g ) = f 0 + β2 κ e κ 1 e κ), dr = [ g ) κβe κ ] R d + βe κ e κ dz Q [ f = 0 + β2 κ e κ 1 e κ) ] κβe κ R d + β dz Q. Insering he relaion r g) = βe κ R, we can rewrie he above expression as [ f dr = 0 + β2 κ e κ 1 e κ) ] κ[r g)] d + β dz Q = κ[ˆθ) r ] d + β dz Q, where ˆθ) = f 0 + 1 κ f 0 + β2 2κ 2 1 e 2κ ).

10.4 Three well-known special cases 246 A comparison wih Secion 9.4 on page 230 reveals ha he HJM one-facor model wih forward rae volailiies given by 10.14) is equivalen o he Hull-Whie or exended-vasicek) model. Therefore, we know ha he zero coupon bond prices are given by where bτ) = 1 κ a, T ) = κ 1 e κτ ), T B T = e a,t ) bt )r, ˆθu)bT u) du + β2 4κ bt )2 + β2 bt ) T )). 2κ2 The price of a European call on a zero coupon bond is again given by 10.10), bu where 10.15) v, T, S) = β 1 e κ[s T ]) 1 e 2κ[T ]) 1/2. 2κ 3 Again, Jamshidian s rick can be used for European opions on coupon bonds. 10.4.3 The exended CIR model We will now discuss he relaion beween he HJM models and he Cox-Ingersoll-Ross CIR) model sudied in Secion 7.5 wih is exension examined in Secion 9.5. In he exended CIR model he shor rae is assumed o follow he process dr = κθ) ˆκr ) d + β r dz Q under he spo maringale measure. The zero-coupon bond prices are of he form B T r, ) = exp{ a, T ) bt )r }, where bτ) = 2e γτ 1) γ + ˆκ)e γτ 1) + 2γ wih γ = ˆκ 2 + 2β 2, and he funcion a is no imporan for wha follows. Therefore, he volailiy of he zero-coupon bond price is σ T r, ) = bt )β r. On he oher hand, in a one-facor HJM se-up he zero-coupon bond price volailiy is given in erms of he forward rae volailiy funcion β, T, f s ) s ) by 10.7). To be consisen wih he CIR model, he forward rae volailiy mus hence saisfy he relaion T Differeniaing wih respec o T, we ge β, u, f s ) s ) du = bt )β r. β, T, f s ) s ) = b T )β r. A sraighforward compuaion of b τ) allows his condiion o be rewrien as 10.16) β, T, f s ) s ) = 4γ 2 γ[t ] e γ + ˆκ)e γ[t ] 1) + 2γ ) 2 β r. As discussed in Secion 9.5, such a model does no make sense for all ypes of iniial forward rae curves.

10.5 Gaussian HJM models 247 10.5 Gaussian HJM models In he firs wo models sudied in he previous secion, he fuure values of he forward raes are normally disribued. Models wih his propery are called Gaussian. Clearly, Gaussian models have he unpleasan and unrealisic feaure of yielding negaive ineres raes wih a sricly posiive probabiliy, cf. he discussion in Chaper 7. racable. On he oher hand, Gaussian models are highly An HJM model is Gaussian if he forward rae volailiies β i are deerminisic funcions of ime and mauriy, i.e. β i, T, f s ) s ) = β i, T ), i = 1, 2,..., n. To see his, firs noe ha from he drif resricion 10.8) i follows ha he forward rae drif under he spo maringale measure Q is also a deerminisic funcion of ime and mauriy: n T ˆα, T ) = β i, T ) β i, u) du. I follows ha, for any T, he T -mauriy forward raes evolves according o n f T = f0 T + ˆαu, T ) du + β i u, T ) dz Q iu. 0 i=1 Because β i u, T ) a mos depends on ime, he sochasic inegrals are normally disribued, cf. Theorem 3.2 on page 54. The fuure forward raes are herefore normally disribued under Q. The shor-erm ineres rae is r = f, i.e. 10.17) r = f 0 + 0 ˆαu, ) du + n i=1 i=1 0 0 β i u, ) dz Q iu, 0, which is also normally disribued under Q. In paricular, here is a posiive probabiliy of negaive ineres raes. 3 To demonsrae he high degree of racabiliy of he general Gaussian HJM framework, he following heorem provides a closed-form expression for he price C K,T,S of a European call on he zero-coupon bond mauring a S. Theorem 10.3 In he Gaussian n-facor HJM model in which he forward rae volailiy coefficiens β i, T, f s ) s ) only depend on ime and mauriy T, he price of a European call opion mauring a T wrien wih exercise price K on a zero-coupon bond mauring a S is given by 10.18) C K,T,S = B S N d 1 ) KB T N d 2 ), where 10.19) 10.20) 10.21) d 1 = 1 B S v, T, S) ln KB T ) + 1 v, T, S), 2 d 2 = d 1 v, T, S), n [ T 2 S v, T, S) = β i u, y) dy] du i=1 T 1/2. 3 Of course, his does no imply ha ineres raes are necessarily normally disribued under he rue, real-world probabiliy measure P, bu since he probabiliy measures P and Q are equivalen, a posiive probabiliy of negaive raes under Q implies a posiive probabiliy of negaive raes under P.

10.6 Diffusion represenaions of HJM models 248 Proof: We will apply he same procedure as we did in he diffusion models of Chaper 7, see e.g. he derivaion of he opion price in he Vasicek model in Secion 7.4.5. The opion price is given by 10.22) C K,T,S = B T [ E QT max B S T K, 0 )] = B T E QT [ )] max F T,S T K, 0, where Q T denoes he T -forward maringale measure inroduced in Secion 5.3.1 on page 113. We will find he disribuion of he underlying bond price BT S a expiraion of he opion, which is idenical o he forward price of he bond wih immediae delivery, F T,S T. The forward price for delivery a T is given a ime as F T,S = B S /B T. We know ha he forward price is a Q T - maringale, and by Iô s Lemma we can express he volailiy erms of he forward price by he volailiy erms of he bond prices, which according o 10.7) is given by σi S) = S β i, y) dy and σi T ) = T β i, y) dy. Therefore, we ge ha df T,S = n i=1 σ S i ) σ T i ) ) F T,S ) S dzi T = β i, y) dy T } {{ } h i) F T,S dz T i. I follows see Chaper 3) ha ln F T,S T = ln F T,S 1 2 n i=1 T h i u) 2 du + n i=1 T h i u) dz T iu. From Theorem 3.2 we ge ha ln BT S T,S = ln FT is normally disribued wih variance n T n T 2 S v, T, S) 2 = h i u) 2 du = β i u, y) dy) du. i=1 The resul now follows from an applicaion of Theorem A.4 in Appendix A. i=1 T Consider, for example, a wo-facor Gaussian HJM model wih forward rae volailiies β 1, T ) = β 1 and β 2, T ) = β 2 e κ[t ], where β 1, β 2, and κ are posiive consans. This is a combinaion of wo one-facor examples of Secion 10.4. In his model we have [ T 2 S [ T 2 S v, T, S) 2 = β 1 dy] du + β 2 e dy] κ[y u] du T T = β1[s 2 T ] 2 [T ] + β2 2 2κ 3 1 e κ[s T ]) 2 1 e 2κ[T ]), cf. 10.13) and 10.15). I is generally no possible o express he fuure zero coupon bond price BT S as a monoonic funcion of r T, no even when we resric ourselves o a Gaussian model. Therefore, we can generally no use Jamshidian s rick o price European opions on coupon bonds. 10.6 Diffusion represenaions of HJM models As discussed immediaely below he basic assumpion 10.1) on page 240, he HJM models are generally non-markov in he sense ha he ineres rae processes and derived processes do

10.6 Diffusion represenaions of HJM models 249 generally no have he Markov propery. For compuaional purposes here is a grea advanage in applying a low-dimensional Markov diffusion model as we will argue below. As discussed earlier in his chaper, we can hink of he enire forward rae curve as following an infinie-dimensional Markov diffusion process. On he oher hand, we have already seen some specificaions of he HJM model framework which imply ha he shor-erm ineres rae follows a Markov diffusion process. In his secion, we will discuss when such a low-dimensional diffusion represenaion of an HJM model is possible. 10.6.1 On he use of numerical echniques for diffusion and non-diffusion models For he purpose of using numerical echniques for derivaive pricing, i is crucial wheher or no he shor rae or more generally some low-dimensional vecor of sae variables has he Markov propery. A Markov process can be approximaed by a recombining ree, whereas a nonrecombining ree mus be used for non-markov processes since he fuure evoluion can depend on he pah followed hus far. The number of nodes in a non-recombining ree explodes. A onevariable binomial ree wih n ime seps has n + 1 endnodes if i is recombining, bu 2 n endnodes if i is non-recombining. This makes i pracically impossible o use rees o compue prices of long-erm derivaives in non-diffusion erm srucure models. In a Markov diffusion model we can use parial differenial equaions PDEs) for pricing, cf. he analysis in Chaper 5. Such PDEs can be efficienly solved by numerical mehods for boh European- and American-ype derivaives as long as he dimension of he sae variable vecor does no exceed hree or maybe four. If i is impossible o express he model in some low-dimensional vecor of sae variables, he PDE approach does no work. The hird frequenly used numerical pricing echnique is he Mone Carlo simulaion approach. The Mone Carlo approach can be applied even for non-diffusion models. The basic idea is o simulae, from now and o he mauriy dae of he coningen claim, he underlying Brownian moions and, hence, he relevan underlying ineres raes, bond prices, ec., under an appropriaely chosen maringale measure. Then he payoff from he coningen claim can be compued for his paricular simulaed pah of he underlying variables. Doing his a large number of imes, he average of he compued payoffs leads o a good approximaion o he heoreical value of he claim. In is original formulaion, Mone Carlo simulaion can only be applied o European-syle derivaives. The wish o price American-ype derivaives in non-diffusion HJM models has recenly induced some suggesions on he use of Mone Carlo mehods for American-syle asses, see, e.g., Boyle, Broadie, and Glasserman 1997), Broadie and Glasserman 1997b), Carr and Yang 1997), Andersen 2000), and Longsaff and Schwarz 2001). Generally, Mone Carlo pricing of even European-syle asses in non-diffusion HJM models is compuaionally inensive since he enire erm srucure has o be simulaed, no jus one or wo variables. 10.6.2 In which HJM models does he shor rae follow a diffusion process? We seek o find condiions under which he shor-erm ineres rae in an HJM model follows a Markov diffusion process. Firs, we will find he dynamics of he shor rae in he general HJM framework 10.1). For he pricing of derivaives i is he dynamics under he spo maringale measure or relaed maringale measures which is relevan. The following heorem gives he shor

10.6 Diffusion represenaions of HJM models 250 rae dynamics under he spo maringale measure Q. Theorem 10.4 In he general HJM framework 10.1) he dynamics of he shor rae r under he spo maringale measure is given by 10.23) dr = Proof: measure Q is { f 0 + n i=1 0 β i u,, fu) s [ s u ) u n + i=1 0 ] β i u, x, fu) s s u ) dx du+ β i u,, f s u) s u ) dz Q iu } d + n i=1 0 β i u,, f s u) s u ) 2 du n β i,, f s ) s ) dz Q i. For each T, he dynamics of he T -mauriy forward rae under he spo maringale df T = ˆα, T, f s ) s ) d + i=1 n β i, T, f s ) s ) dz Q i, where ˆα is given by he drif resricion 10.8). This implies ha i=1 f T = f0 T + ˆαu, T, fu) s s u ) du + 0 n i=1 0 β i u, T, f s u) s u ) dz Q iu. Since he shor rae is simply he zero-mauriy forward rae, r = f, i follows ha 10.24) r = f 0 + = f 0 + 0 n i=1 ˆαu,, f s u) s u ) du + 0 n i=1 0 β i u,, f s u) s u ) dz Q iu [ ] β i u,, fu) s s u ) β i u, x, fu) s s u ) dx du + u n i=1 0 β i u,, f s u) s u ) dz Q iu. To find he dynamics of r, we proceed as in he simple examples of Secion 10.4. Le R i = 0 β iu,, fu) s s u ) dz Q iu for i = 1, 2,..., n. Then [ dr i = β i,, f s ) s ) dz Q i + β i u,, fu) s ] s u ) dz Q iu d by Leibniz rule for sochasic inegrals see Theorem 3.3 on page 55). Define he funcion G i ) = 0 β iu,, fu) s s u )H i u, ) du, where H i u, ) = u β iu, x, fu) s s u ) dx. By Leibniz rule for ordinary inegrals, G i) = β i,, f s ) s )H i, ) + 0 [β iu,, fu) s s u )H i u, )] du [ βi u,, f = u) s s u ) H i u, ) + β i u,, f s 0 u) s u ) H ] iu, ) du [ βi u,, f s = u) s u ) ] β i u, x, f s u) s u ) dx + β i u,, fu) s s u ) 2 0 where we have used he chain rule and he fac ha H i, ) = 0. Noe ha u r = f 0 + 0 n G i ) + i=1 n R i, i=1 du,

10.6 Diffusion represenaions of HJM models 251 where he G i s are deerminisic funcions and R i ) are sochasic processes. By Iô s Lemma, we ge [ f n ] 0 dr = + G i) d + i=1 n dr i. Subsiuing in he expressions for G i ) and dr i, we arrive a he expression 10.23). i=1 From 10.23) we see ha he drif erm of he shor rae generally depends on pas values of he forward rae curve and pas values of he Brownian moion. Therefore, he shor rae process is generally no a diffusion process in an HJM model. However, if we know ha he iniial forward rae curve belongs o a cerain family, he shor rae may be Markovian. If, for example, he iniial forward rae curve is on he form generaed by he original one-facor CIR diffusion model, hen he shor rae in he one-facor HJM model wih forward rae volailiy given by 10.16) will, of course, be Markovian since he wo models are hen indisinguishable. Under wha condiions on he forward rae volailiy funcions β i, T, f s ) s ) will he shor rae follow a diffusion process for any iniial forward rae curve? Hull and Whie 1993) and Carverhill 1994) answer his quesion. Their conclusion is summarized in he following heorem. Theorem 10.5 Consider an n-facor HJM model. Suppose ha deerminisic funcions g i and h exis such ha β i, T, f s ) s ) = g i )ht ), i = 1, 2,..., n, and h is coninuously differeniable, non-zero, and never changing sign. 4 dynamics 10.25) dr = [ f 0 + h)2 n i=1 0 ] g i u) 2 du + h ) h) r f0) d + Then he shor rae has n g i )h) dz Q i, so ha he shor rae follows a diffusion process for any given iniial forward rae curve. i=1 Proof: We will only consider he case n = 1 and show ha r indeed is a Markov diffusion process when 10.26) β, T, f s ) s ) = g)ht ), where g and h are deerminisic funcions and h is coninuously differeniable, non-zero, and never changing sign. Firs noe ha 10.24) and 10.26) imply ha 10.27) r = f 0 + h) 0 gu) 2 [ u ] hx) dx du + h) 0 gu) dz Q u, and, hus, 10.28) 0 gu) dz Q u = 1 h) r f 0) 0 gu) 2 [ u ] hx) dx du. 4 Carverhill claims ha he h funcion can be differen for each facor, i.e., β i, T, f s ) s ) = g i )h i T ), bu his is incorrec.

10.6 Diffusion represenaions of HJM models 252 The dynamics of r in Equaion 10.23) specializes o dr = [ f 0 + h ) 0 gu) 2 [ u ] hx) dx du + h) 2 gu) 2 du 0 + h ) 0 gu) dz Q u ] d + g)h) dz Q, which by applying 10.28) can be wrien as he one-facor version of 10.25). Noe ha he Ho-Lee model and he Hull-Whie model sudied in Secion 10.4 boh saisfy he condiion 10.26). Obviously, he HJM models where he shor rae is Markovian are members of he Gaussian class of models discussed in Secion 10.5. In paricular, he price of a European call on a zero-coupon bond is given by 10.18). I can be shown ha wih a volailiy specificaion of he form 10.26), he fuure price B T of a zero-coupon bond can be expressed as a monoonic funcion of ime and he shor rae r a ime. I follows ha Jamshidian s rick inroduced in Secion 7.2.3 on page 169 can be used for pricing European opions on coupon bonds in his special seing. The Markov propery is one aracive feaure of a erm srucure model. We also wan a model o exhibi ime homogeneous volailiy srucures in he sense ha he volailiies of, e.g., forward raes, zero-coupon bond yields, and zero-coupon bond prices do no depend on calendar ime in iself, cf. he discussion in Chaper 9. For he forward rae volailiies in an HJM model o be ime homogeneous, β i, T, f s ) s ) mus be of he form β i T, f s ) s ). I hen follows from 10.7) ha he zero coupon bond prices B T will also have ime homogeneous volailiies. Similarly for he zero-coupon yields y T. Hull and Whie 1993) have shown ha here are only wo models of he HJM-class ha have boh a Markovian shor rae and ime homogeneous volailiies, namely he Ho-Lee model and he Hull-Whie model of Secion 10.4. As discussed above, he HJM models wih a Markovian shor rae are Gaussian models. While Gaussian models have a high degree of compuaional racabiliy, hey also allow negaive raes, which cerainly is an unrealisic feaure of a model. Furhermore, he volailiy of he shor rae and oher ineres raes empirically seems o depend on he shor rae iself. Therefore, we seek o find HJM models wih non-deerminisic forward rae volailiies ha are sill compuaionally racable. 10.6.3 A wo-facor diffusion represenaion of a one-facor HJM model Richken and Sankarasubramanian 1995) show ha in a one-facor HJM model wih a forward rae volailiy of he form 10.29) β, T, f s ) s ) = β,, f s ) s )e T κx) dx for some deerminisic funcion κ, i is possible o capure he pah dependence of he shor rae by a single variable, and ha his is only possible, when 10.29) holds. The evoluion of he erm srucure will depend only on he curren value of he shor rae and he curren value of his addiional variable. The addiional variable needed is ϕ = 0 βu,, f s u) s u ) 2 du = 0 βu, u, f s u) s u ) 2 e 2 u κx) dx du,

10.7 HJM-models wih forward-rae dependen volailiies 253 which is he accumulaed forward rae variance. The fuure zero coupon bond price B T can be expressed as a funcion of r and ϕ in he following way: B T = e a,t ) b1,t )r b2,t )ϕ, where B T a, T ) = ln 0 b 1, T ) = T B 0 b 2, T ) = 1 2 b 1, T ) 2. ) b 1, T )f 0, e u κx) dx du, The dynamics of r and ϕ under he spo maringale measure Q is given by ) f dr = 0 + ϕ κ)[r f0] d + β,, f s ) s ) dz Q, dϕ = β,, f s ) s ) 2 2κ)ϕ ) d. The wo-dimensional process r, ϕ) will be Markov if he shor rae volailiy depends on, a mos, he curren values of r and ϕ, i.e. if here is a funcion β r such ha β,, f s ) s ) = β r r, ϕ, ). In ha case, we can price derivaives by wo-dimensional recombining rees or by numerical soluions of wo-dimensional PDEs no closed-form soluions have been repored). 5 One allowable specificaion is β r r, ϕ, ) = βr γ for some non-negaive consans β and γ, which, e.g., includes a CIR-ype volailiy srucure for γ = 1 2 ). The volailiies of he forward raes are relaed o he shor rae volailiy hrough he deerminisic funcion κ, which mus be specified. If κ is consan, he forward rae volailiy is an exponenially decaying funcion of he ime o mauriy. Empirically, he forward rae volailiy seems o be a humped firs increasing, hen decreasing) funcion of mauriy. This can be achieved by leing he κx) funcion be negaive for small values of x and posiive for large values of x. Also noe ha he volailiy of some T -mauriy forward rae f T is no allowed o depend on he forward rae f T iself, bu only he shor rae r and ime. For furher discussion of he circumsances under which an HJM model can be represened as a diffusion model, he reader is referred o Jeffrey 1995), Cheyee 1996), Bhar and Chiarella 1997), Inui and Kijima 1998), Bhar, Chiarella, El-Hassan, and Zheng 2000), and Björk and Landén 2002). 10.7 HJM-models wih forward-rae dependen volailiies In he models considered unil now, he forward rae volailiies are eiher deerminisic funcions of ime he Gaussian models) or a funcion of ime and he curren shor rae he exended CIR model and he Richken-Sankarasubramanian model). The mos naural way o inroduce 5 Li, Richken, and Sankarasubramanian 1995) show how o build a ree for his model, in which boh Europeanand American-ype erm srucure derivaives can be efficienly priced.

10.8 Concluding remarks 254 non-deerminisic forward rae volailiies is o le hem be a funcion of ime and he curren value of he forward rae iself, i.e. of he form 10.30) β i, T, f s ) s ) = β i, T, f T ). A model of his ype, inspired by he Black-Scholes sock opion pricing model, is obained by leing 10.31) β i, T, f T ) = γ i, T )f T, where γ i, T ) is a posiive, deerminisic funcion of ime. The forward rae drif will hen be n ˆα, T, f s ) s ) = γ i, T )f T i=1 T γ i, u)f u du. The specificaion 10.31) will ensure non-negaive forward raes saring wih a erm srucure of posiive forward raes) since boh he drif and volailiies are zero for a zero forward rae. Such models have a serious drawback, however. A process wih he drif and volailiies given above will explode wih a sricly posiive probabiliy in he sense ha he value of he process becomes infinie. 6 Wih a sricly posiive probabiliy of infinie ineres raes, bond prices mus equal zero, and his, obviously, implies arbirage opporuniies. Heah, Jarrow, and Moron 1992) discuss he simple one-facor model wih a capped forward rae volailiy, β, T, f T ) = β minf T, ξ), where β and ξ are posiive consans, i.e. he volailiy is proporional for small forward raes and consan for large forward raes. They showed ha wih his specificaion he forward raes do no explode, and, furhermore, hey say non-negaive. The assumed forward rae volailiy is raher far-feched, however, and seems unrealisic. Milersen 1994) provides a se of sufficien condiions for HJM-models of he ype 10.30) o yield non-negaive and non-exploding ineres raes. One of he condiions is ha he forward rae volailiy is bounded from above. This is, obviously, no saisfied for proporional volailiy models, i.e. models where 10.31) holds. 10.8 Concluding remarks Empirical sudies of various specificaions of he HJM model framework have been performed on a variey of daa ses by, e.g., Amin and Moron 1994), Flesaker 1993), Heah, Jarrow, and Moron 1990), Milersen 1998), and Pearson and Zhou 1999). However, hese papers do no give a clear picure of how he forward rae volailiies should be specified. To implemen an HJM-model one mus specify boh he forward rae volailiy funcions β i, T, f s ) s ) and an iniial forward rae curve u f0 u given as a parameerized funcion of mauriy. In he ime homogeneous Markov diffusion models sudied in he Chapers 7 and 8, he forward rae curve in a given model can a all poins in ime be described by he same parameerizaion alhough possibly wih differen parameers a differen poins in ime due o changes in he sae variables). For example in he Vasicek one-facor model, we know from 7.57) on page 184 6 This was shown by Moron 1988).

10.8 Concluding remarks 255 ha he forward raes a ime are given by f T = 1 e κ[t ]) ) y + β2 ] e κ[t + e κ[t ] r 2κ2 ) β 2 = y + 2κ 2 + r y e κ[t ] β2 2κ 2 e 2κ[T ], which is always he same kind of funcion of ime o mauriy T, alhough he muliplier of e κ[t ] is non-consan over ime due o changes in he shor rae. As discussed in Secion 9.7 ime inhomogeneous diffusion models do generally no have his nice propery, and neiher do he HJMmodels sudied in his chaper. If we use a given parameerizaion of he iniial forward curve, hen we canno be sure ha he fuure forward curves can be described by he same parameerizaion even if we allow he parameers o be differen. We will no discuss his issue furher bu simply refer he ineresed reader o Björk and Chrisensen 1999), who sudy when he iniial forward rae curve and he forward rae volailiy are consisen in he sense ha fuure forward rae curves have he same form as he iniial curve. If he iniial forward rae curve is aken o be of he form given by a ime homogeneous diffusion model and he forward rae volailiies are specified in accordance wih ha model, hen he HJMmodel will be indisinguishable from ha diffusion model. For example, he ime 0 forward rae curve in he one-facor CIR model is of he form ] = r 0 + ˆκ [ˆθ r bt ) 1 2 β2 rbt ) 2, f T 0 cf. 7.72) on page 191, where he funcion bt ) is given by 7.70). Wih such an iniial forward rae curve, he one-facor HJM model wih forward rae volailiy funcion given by 10.16) is indisinguishable from he original ime homogeneous one-facor CIR model.

Chaper 11 Marke models 11.1 Inroducion The erm srucure models sudied in he previous chapers have involved assumpions abou he evoluion in one or more coninuously compounded ineres raes, eiher he shor rae r or he insananeous forward raes f T. However, many securiies raded in he money markes, e.g. caps, floors, swaps, and swapions, depend on periodically compounded ineres raes such T,T +δ as spo LIBOR raes l +δ, forward LIBOR raes L, spo swap raes l δ, and forward swap raes. For he pricing of hese securiies i seems appropriae o apply models ha are based L T,δ on assumpions on he LIBOR raes or he swap raes. Also noe ha hese ineres raes are direcly observable in he marke, whereas he shor rae and he insananeous forward raes are heoreical consrucs and no direcly observable. We will use he erm marke models for models based on assumpions on periodically compounded ineres raes. All he models sudied in his chaper ake he currenly observed erm srucure of ineres raes as given and are herefore o be classified as relaive pricing or pure no-arbirage models. Consequenly, hey offer no insighs ino he deerminaion of he curren ineres raes. We will disinguish beween LIBOR marke models ha are based on assumpions T,T +δ on he evoluion of he forward LIBOR raes L and swap marke models ha are based on assumpions on he evoluion of he forward swap raes. By consrucion, he marke models are no suiable for he pricing of fuures and opions on governmen bonds and similar conracs ha do no depend on he money marke ineres raes. In he recen lieraure several marke models have been suggesed, bu mos aenion has been given o he so-called lognormal LIBOR marke models. In such a model he volailiies of T,T +δ a relevan selecion of he forward LIBOR raes L are assumed o be proporional o he level of he forward rae so ha he disribuion of he fuure forward LIBOR raes is lognormal under an appropriae forward maringale measure. As discussed in Secion 7.6 on page 193, lognormally disribued coninuously compounded ineres raes have unpleasan consequences, bu Sandmann and Sondermann 1997) show ha models wih lognormally disribued periodically compounded raes are no subjec o he same problems. Below, we will demonsrae ha a lognormal assumpion on he disribuion of forward LIBOR raes implies pricing formulas for caps and floors ha are idenical o Black s pricing formulas saed in Chaper 5. Similarly, lognormal swap marke models imply European swapion prices consisen wih he Black formula for swapions. Hence, he lognormal marke models provide some suppor for he widespread use 256

11.2 General LIBOR marke models 257 of Black s formula for fixed income securiies. However, he assumpions of he lognormal marke models are no necessarily descripive of he empirical evoluion of LIBOR raes, and herefore we will also briefly discuss alernaive marke models. 11.2 General LIBOR marke models In his secion we will inroduce a general LIBOR marke model, describe some of he model s basic properies, and discuss how derivaive securiies can be priced wihin he framework of he model. The presenaion is inspired by Jamshidian 1997) and Musiela and Rukowski 1997, Chs. 14 and 16). 11.2.1 Model descripion As described in Secion 2.8, a cap is a conrac ha proecs a floaing rae borrower agains paying an ineres rae higher han some given rae K, he so-called cap rae. We le T 1,..., T n denoe he paymen daes and assume ha T i T i 1 = δ for all i. In addiion we define T 0 = T 1 δ. A each ime T i i = 1,..., n) he cap gives a payoff of ) ) = Hδ max l Ti T K, 0 i δ = Hδ max L Ti δ,ti T i δ K, 0, C i T i where H is he face value of he cap. A cap can be considered as a porfolio of caples, namely one caple for each paymen dae. The definiion of he forward maringale measure in Chaper 5 implies ha he value of he above payoff can be found as he produc of he expeced payoff compued under he T i -forward maringale measure and he curren discoun facor for ime T i paymens, i.e. 11.1) C i = HδB Ti E QT i The price of a cap can herefore be deermined as 11.2) C = Hδ n i=1 B Ti [ )] max L Ti δ,ti T i δ K, 0, < T i δ. E QT i [ )] max L Ti δ,ti T i δ K, 0, < T 0. For T 0 he firs-coming paymen of he cap is known so ha is presen value is obained by muliplicaion by he riskless discoun facor, while he remaining payoffs are valued as above. For more deails see Secion 2.8. The price of he corresponding floor is 11.3) F = Hδ n i=1 B Ti E QT i [ )] max K L Ti δ,ti T i δ, 0, < T 0. In order o compue he cap price from 11.2), we need knowledge of he disribuion of L Ti δ,ti T i δ under he T i -forward maringale measure Q Ti for each i = 1,..., n. For his purpose i is naural o model he evoluion of L Ti δ,ti probabiliy measure he drif rae of L Ti δ,ti from Eq. 1.9) on page 8 ha 11.4) L Ti δ,ti under Q Ti. The following argumen shows ha under he Q Ti = 1 δ is zero, i.e. L Ti δ,ti ) B Ti δ 1. B Ti is a Q Ti -maringale. Remember

11.2 General LIBOR marke models 258 Under he T i -forward maringale measure Q Ti he raio beween he price of any asse and he zerocoupon bond price B Ti is a maringale. In paricular, he raio B Ti δ /B Ti is a Q Ti -maringale so ha he expeced change of he raio over any ime inerval is equal o zero under he Q Ti measure. From he formula above i follows ha also he expeced change over any ime inerval) in he periodically compounded forward rae L Ti δ,ti following heorem: is zero under Q Ti. We summarize he resul in he Theorem 11.1 The forward rae L Ti δ,ti is a Q Ti -maringale. Consequenly, a LIBOR marke model is fully specified by he number of facors i.e. he number of sandard Brownian moions) ha influence he forward raes and he forward rae volailiy funcions. For simpliciy, we focus on he one-facor models ) 11.5) dl Ti δ,ti = β, T i δ, T i, L Tj,δ ) Tj dz Ti, i = 1,..., n, where z Ti is a one-dimensional sandard Brownian moion under he T i -forward maringale measure Q Ti. The symbol L Tj,δ ) Tj indicaes as in Chaper 10) ha he ime value of he volailiy funcion β can depend on he curren values of all he modeled forward raes. 1 In he lognormal LIBOR marke models we will sudy in Secion 11.3, we have ) β, T i δ, T i, L Tj,δ ) Tj = γ, T i δ, T i )L Ti δ,ti for some deerminisic funcion γ. However, unil hen we coninue o discuss he more general specificaion 11.5). We see from he general cap pricing formula 11.2) ha he cap price also depends on he curren discoun facors B T1, B T2,..., B Tn. According o 11.4) hese discoun facors can be deermined by B T0 and he curren values of he modeled forward raes, i.e. L T0,T1, L T1,T2,..., L Tn 1,Tn. Similarly o he HJM models in Chaper 10, he LIBOR marke models ake he currenly observable values of hese raes as given. 11.2.2 The dynamics of all forward raes under he same probabiliy measure The basic assumpion 11.5) for he LIBOR marke model involves n differen forward maringale measures. In order o beer undersand he model and o simplify he compuaion of some securiy prices we will describe he evoluion of he relevan forward raes under he same common probabiliy measure. As discussed in he nex subsecion, Mone Carlo simulaion is ofen used o compue prices of cerain securiies in LIBOR marke models. I is much simpler o simulae he evoluion of he forward raes under a common probabiliy measure han o simulae he evoluion of each forward rae under he maringale measure associaed wih he forward rae. One possibiliy is o choose one of he n differen forward maringale measures used in he assumpion of he model. Noe ha he T i -forward maringale measure only makes sense up o ime T i. Therefore, i is appropriae o use he forward maringale measure associaed wih he las paymen dae, i.e. he T n -forward maringale measure Q Tn, since his measure applies o he enire relevan ime period. In his conex Q Tn is someimes referred o as he erminal measure. Anoher obvious 1 As for he HJM models in Chaper 10, he general resuls for he marke models hold even when earlier values of he forward raes affec he curren dynamics of he forward raes, bu such a generalizaion seems worhless.

11.2 General LIBOR marke models 259 candidae for he common probabiliy measure is he spo maringale measure. Le us look a hese wo alernaives in more deail. The erminal measure We wish o describe he evoluion in all he modeled forward raes under he T n -forward maringale measure. For ha purpose we shall apply he following heorem which oulines how o shif beween he differen forward maringale measures of he LIBOR marke model. Theorem 11.2 Assume ha he evoluion in he LIBOR forward raes L Ti δ,ti for i = 1,..., n, where T i = T i 1 + δ, is given by 11.5). Then he processes z Ti δ and z Ti are relaed as follows: ) 11.6) dz Ti δβ = dz Ti δ +, T i δ, T i, L Tj,δ 1 + δl Ti δ,ti ) Tj Proof: From Secion 5.3.1 we have ha he T i -forward maringale measure Q Ti by he fac ha he process z Ti is a sandard Brownian moion under Q Ti, where ) dz Ti = dz + λ σ Ti d. Here, σ Ti sochasic. Similarly, d. is characerized denoes he volailiy of he zero-coupon bond mauring a ime T i, which may iself be A simple compuaion gives ha 11.7) dz Ti ) dz Ti δ = dz + λ σ Ti δ d. [ = dz Ti δ + σ Ti δ ] σ Ti d. As shown in Theorem 11.1, L Ti δ,ti is a Q Ti -maringale and, hence, has an expeced change of zero under his probabiliy measure. According o 11.4) he forward rae L Ti δ,ti is a funcion of he zero-coupon bond prices B Ti δ oal, he dynamics is dl Ti δ,ti and B Ti = BTi δ δb Ti σ Ti δ = 1 1 + δlti δ,ti ) δ Comparing wih 11.5), we can conclude ha δβ 11.8) σ Ti δ σ Ti = so ha he volailiy follows from Iô s Lemma. In ) σ Ti dz Ti σ Ti δ, T i δ, T i, L Tj,δ 1 + δl Ti δ,ti ) σ Ti dz Ti. ) Tj ). Subsiuing his relaion ino 11.7), we obain he saed relaion beween he processes z Ti z Ti δ. and Using 11.6) repeaedly, we ge ha dz Tn n 1 = dz Ti + j=i δβ, T j, T j+1, L Tj,δ 1 + δf s, T j, T j+1 ) ) Tj ) d.

11.2 General LIBOR marke models 260 Consequenly, for each i = 1,..., n, we can wrie he dynamics of L Ti δ,ti as 11.9) dl Ti δ,ti = β, T i δ, T i, L T k,δ = β, T i δ, T i, L T k,δ n 1 = j=i δβ ) Tk ) Tk ), T i δ, T i, L T k,δ + β, T i δ, T i, L T k,δ dz Ti ) dz Tn ) Tk n 1 ) β j=i 1 + δl Tj,Tj+1 ) dz Tn. ) Tk δβ under he Q Tn -measure, T j, T j+1, L T k,δ 1 + δl Tj,Tj+1, T j, T j+1, L T k,δ ) Tk ) ) Tk Noe ha he drif may involve some or all of he oher modeled forward raes. Therefore, he vecor of all he forward raes L T0,T1,..., L Tn 1,Tn ) will follow an n-dimensional diffusion process so ha a LIBOR marke model can be represened as an n-facor diffusion model. Securiy prices are hence soluions o a parial differenial equaion PDE), bu in ypical applicaions he dimension n, i.e. he number of forward raes, is so big ha neiher explici nor numerical soluion of he PDE is feasible. 2 d ) d For example, o price caps, floors, and swapions ha depend on 3-monh ineres raes and have mauriies of up o 10 years, one mus model 40 forward raes so ha he model is a 40-facor diffusion model! Nex, le us consider an asse wih a single payoff a some poin in ime T [T 0, T n ]. The payoff H T may in general depend on he value of all he modeled forward raes a and before ime T. Le V denoe he ime value of his asse measured in moneary unis, e.g. dollars). From he definiion of he T n -forward maringale measure Q Tn i follows ha [ ] and hence V B Tn V = B Tn = E QTn E QTn In paricular, if T is one of he ime poins of he enor srucure, say T = T k, we ge [ ] From 11.4) we have ha 1 B Tn T k = B = = T k T k B T k+1 T k T B k+1 T k B T k+2 T k V = B Tn E QTn... B T n 1 T k B Tn T k H T B Tn T [ H T B Tn T H Tk B Tn T k [ ] [ ] 1 + δl T k,t k+1 T k 1 + δl T k+1,t k+2 T k... n 1 j=k [ ] 1 + δl Tj,Tj+1 T k, ].. [ ] 1 + δl Tn 1,Tn T k 2 However, Andersen and Andreasen 2000) inroduce a rick ha may reduce he compuaional complexiy considerably.

11.2 General LIBOR marke models 261 so ha he price can be rewrien as 11.10) V = B Tn E QTn H Tk n 1 j=k [ ] 1 + δl Tj,Tj+1 T k. The righ-hand side may be approximaed using Mone Carlo simulaions in which he evoluion of he forward raes under Q Tn is used, as oulined in 11.9). If he securiy maures a ime T n, he price expression is even simpler: 11.11) V = B Tn E QTn [H Tn ]. In ha case i suffices o simulae he evoluion of he forward raes ha deermine he payoff of he securiy. The spo LIBOR maringale measure The spo maringale measure Q, which we defined and discussed in Chaper 5, is associaed wih he use of a bank accoun earning he coninuously compounded shor rae as he numeraire, cf. he discussion in Secion 5.3. However, he LIBOR marke model does no a all involve he shor rae so he radiional spo maringale measure does no make sense in his conex. The LIBOR marke counerpar is a roll over sraegy in he shores zero-coupon bonds. To be more precise, he sraegy is iniiaed a ime T 0 by an invesmen of one dollar in he zero-coupon bond mauring a ime T 1, which allows for he purchase of 1/B T1 T 0 unis of he bond. A ime T 1 he payoff of 1/B T1 T 0 dollars is invesed in he zero-coupon bond mauring a ime T 2, ec. Le us define I) = min {i {1, 2,..., n} : T i } so ha T I) denoes he nex paymen dae afer ime. In paricular, IT i ) = i so ha T ITi) = T i. A any ime T 0 he sraegy consiss of holding N = 1 B T1 1 T 0 B T2 T 1... 1 B T I) T I) 1 unis of he zero-coupon bond mauring a ime T I). The value of his posiion is 11.12) A = B T I) N = B T I) I) 1 j=0 1 B Tj+1 T j = B TI) I) 1 j=0 [ ] 1 + δl Tj,Tj+1 T j, where he las equaliy follows from he relaion 11.4). Since A is posiive, i is a valid numeraire. The corresponding maringale measure is called he spo LIBOR maringale measure and is denoed by Q. Le us look a a securiy wih a single paymen a a ime T [T 0, T n ]. The payoff H T depend on he values of all he modeled forward raes a and before ime T. Le us by V denoe he dollar value of his asse a ime. From he definiion of he spo LIBOR maringale measure Q i follows ha and hence V A = E Q V = E Q [ A [ HT A T A T ], H T ]. may

11.2 General LIBOR marke models 262 From he calculaion A A T = B T I) B T IT ) T = BT I) B T IT ) T we ge ha he price can be rewrien as 11.13) V = B T I) E Q [ I) 1 j=0 IT ) 1 j=0 IT ) 1 j=i) H T B T IT ) T 1 + δl Tj,Tj+1 [ ] 1 + δl Tj,Tj+1 T j T j ] [ 1 + δl Tj,Tj+1 T j ] 1, IT ) 1 j=i) [ 1 + δl Tj,Tj+1 T j ] 1. In paricular, if T is one of he daes in he enor srucure, say T = T k, we ge 11.14) V = B T k 1 [ ] 1 I) E Q H Tk 1 + δl Tj,Tj+1 T j j=i) since IT k ) = k and B T IT k ) T k = B T k T k = 1. In order o compue ypically by simulaion) he expeced value on he righ-hand side, we need o know he evoluion of he forward raes L Tj,Tj+1 under he spo LIBOR maringale measure Q. I can be shown ha he process z defined by dz = dz Ti [ σ T I) ] σ Ti d is a sandard Brownian moion under he probabiliy measure Q. As usual, σ T denoes he volailiy of he zero-coupon bond mauring a ime T. Repeaed use of 11.8) yields ) σ T i 1 δβ, T j, T j+1, L T k,δ ) Tk I) σ Ti = 1 + δl Tj,Tj+1 so ha 11.15) dz = dz Ti j=i) i 1 j=i) δβ, T j, T j+1, L T k,δ 1 + δl Tj,Tj+1 ) Tk Subsiuing his relaion ino 11.5), we can rewrie he dynamics of he forward raes under he spo LIBOR maringale measure as = β 11.16) dl Ti δ,ti, T i δ, T i, L T k,δ ) Tk ) dz Ti = β, T i δ, T i, L T k,δ ) Tk ) dz + = i 1 j=i) δβ, T i δ, T i, L T k,δ + β, T i δ, T i, L T k,δ ) Tk ) β i 1 j=i) 1 + δl Tj,Tj+1 ) dz. ) Tk ) d. δβ, T j, T j+1, L T k,δ, T j, T j+1, L T k,δ 1 + δl Tj,Tj+1 ) Tk ) ) Tk Noe ha he drif in he forward raes under he spo LIBOR maringale measure follows from he specificaion of he volailiy funcion β and he curren forward raes. The relaion beween he drif and he volailiy is he marke model counerpar o he drif resricion of he HJM models, cf. 10.8) on page 243. d ) d

11.3 The lognormal LIBOR marke model 263 11.2.3 Consisen pricing As indicaed above, he model can be used for he pricing of all securiies ha only have paymen daes in he se {T 1, T 2,..., T n }, and where he size of he paymen only depends on he modeled forward raes and no oher random variables. This is rue for caps and floors on δ-period ineres raes of differen mauriies where he price can be compued from 11.2) and 11.3). The model can also be used for he pricing of swapions ha expire on one of he daes T 0, T 1,..., T n 1, and where he underlying swap has paymen daes in he se {T 1,..., T n } and is based on he δ-period ineres rae. For European swapions he price can be wrien as 11.14). For Bermuda swapions ha can be exercised a a subse of he swap paymen daes {T 1,..., T n }, one mus maximize he righ-hand side of 11.14) over all feasible exercise sraegies. See Andersen 2000) for deails and a descripion of a relaively simple Mone Carlo based mehod for he approximaion of Bermuda swapion prices. The LIBOR marke model 11.5) is buil on assumpions abou he forward raes over he ime inervals [T 0, T 1 ], [T 1, T 2 ],..., [T n 1, T n ]. However, hese forward raes deermine he forward raes for periods ha are obained by connecing succeeding inervals. For example, we have from Eq. 1.9) on page 8 ha he forward rae over he period [T 0, T 2 ] is uniquely deermined by he forward raes for he periods [T 0, T 1 ] and [T 1, T 2 ] since ) 1 B L T0,T2 T0 = 1 T 2 T 0 B T2 11.17) B T0 B T1 B T1 B T2 1 = T 2 T 0 = 1 [ 2δ 1 + δl T0,T1 ] [ 1 ) 1 + δl T1,T2 ] ) 1, where δ = T 1 T 0 = T 2 T 1 as usual. Therefore, he disribuions of he forward raes L T0,T1 and L T1,T2, implied by he LIBOR marke model 11.18), deermine he disribuion of he forward rae L T0,T2. A LIBOR marke model based on hree-monh ineres raes can hence also be used for he pricing of conracs ha depend on six-monh ineres raes, as long as he paymen daes for hese conracs are in he se {T 0, T 1,..., T n }. More generally, in he consrucion of a model, one is only allowed o make exogenous assumpions abou he evoluion of forward raes for non-overlapping periods. 11.3 The lognormal LIBOR marke model 11.3.1 Model descripion The marke sandard for he pricing of caps is Black s formula, i.e. formula 5.72) on page 132. As discussed in Chaper 5, he radiional derivaion of Black s formula is based on inappropriae assumpions. The lognormal LIBOR marke model provides a more reasonable framework in which he Black cap formula is valid. The model was originally developed by Milersen, Sandmann, and Sondermann 1997), while Brace, Gaarek, and Musiela 1997) sor ou some echnical deails and inroduce an explici, bu approximaive, expression for he prices of European swapions in he lognormal LIBOR marke model. Whereas Milersen, Sandmann, and Sondermann derive he cap price formula using PDEs, we will follow Brace, Gaarek, and Musiela and use he forward

11.3 The lognormal LIBOR marke model 264 maringale measure echnique discussed in Chaper 5 since his simplifies he analysis considerably. In he developmen of Black s cap price formula in Chaper 5, we assumed among oher hings ha he forward rae L Ti δ,ti was a maringale under he spo maringale measure Q and ha he fuure value L Ti δ,ti T i δ was lognormally disribued under Q. However, as shown in Theorem 11.1 his forward rae is a maringale under he T i -forward maringale measure and will herefore no be a maringale under he Q-measure. Remember: a change of measure corresponds o changing he drif rae.) Looking a he general cap pricing formula 11.2), i is clear ha we can obain a pricing formula of he same form as Black s formula by assuming ha L Ti δ,ti T i δ is lognormally disribued under he T i -forward maringale measure Q Ti. This is exacly he assumpion of he lognormal LIBOR marke model: 11.18) dl Ti δ,ti = L Ti δ,ti γ, T i δ, T i ) dz Ti, i = 1, 2,..., n, where γ, T i δ, T i ) is a bounded, deerminisic funcion. Here we assume ha he relevan forward raes are only affeced by one Brownian moion, bu below we shall briefly consider muli-facor lognormal LIBOR marke models. A familiar applicaion of Iô s Lemma implies ha from which we see ha ln L Ti δ,ti T i δ dln L Ti δ,ti ) = 1 2 γ, T i δ, T i ) 2 d + γ, T i δ, T i ) dz Ti, = ln L Ti δ,ti 1 2 Ti δ γu, T i δ, T i ) 2 du + Ti δ γu, T i δ, T i ) dz Ti u. Because γ is a deerminisic funcion, i follows from Theorem 3.2 on page 54 ha ) Ti δ γu, T i δ, T i ) dz Ti u N under he T i -forward maringale measure. Hence, ln L Ti δ,ti T i δ N ln L Ti δ,ti 1 2 Ti δ 0, Ti δ γu, T i δ, T i ) 2 du, γu, T i δ, T i ) 2 du Ti δ γu, T i δ, T i ) 2 du ) so ha L Ti δ,ti T i δ no surprise: is lognormally disribued under Q Ti. The following resul should now come as Theorem 11.3 Under he assumpion 11.18) he price of he caple wih paymen dae T i a any ime < T i δ is given by 11.19) C i = HδB Ti [ ] L Ti δ,ti Nd 1i ) KNd 2i ), where 11.20) 11.21) 11.22) d 1i = ) ln L Ti δ,ti /K v L, T i δ, T i ) + 1 2 v L, T i δ, T i ), d 2i = d 1i v L, T i δ, T i ), 1/2 Ti δ v L, T i δ, T i ) = γu, T i δ, T i ) du) 2.

11.3 The lognormal LIBOR marke model 265 Proof: I follows from Theorem A.4 in Appendix A ha [ )] [ E QT i max L Ti δ,ti T i δ K, 0 = E QT i L Ti δ,ti T i δ ] Nd 1i ) KNd 2i ) = L Ti δ,ti Nd 1i ) KNd 2i ), where he las equaliy is due o he fac ha L Ti δ,ti from 11.1). is a Q Ti -maringale. The claim now follows Noe ha v L, T i δ, T i ) 2 is he variance of ln L Ti δ,ti T i δ under he T i -forward maringale measure given he informaion available a ime. The expression 11.19) is idenical o Black s formula 5.68) if we inser σ i = v L, T i δ, T i )/ T i δ. An immediae consequence of he heorem above is he following cap pricing formula in he lognormal one-facor LIBOR marke model: Theorem 11.4 Under he assumpion 11.18) he price of a cap a any ime < T 0 is given as 11.23) C = Hδ n i=1 B Ti where d 1i and d 2i are as in 11.20) and 11.21). [ ] L Ti δ,ti Nd 1i ) KNd 2i ), For T 0 he firs-coming paymen of he cap is known and is herefore o be discouned wih he riskless discoun facor, while he remaining paymens are o be valued as above. For deails, see Secion 2.8. Analogously, he price of a floor under he assumpion 11.18) is 11.24) F = Hδ n i=1 B Ti [ ] KN d 2i ) L Ti δ,ti N d 1i ), T 0. The deerminisic funcion γ, T i δ, T i ) remains o be specified. We will discuss his maer in Secion 11.6. If he erm srucure is affeced by n exogenous sandard Brownian moions, he assumpion 11.18) is replaced by 11.25) dl Ti δ,ti = L Ti δ,ti n j=1 γ j, T i δ, T i ) dz Ti j, where all γ j, T i δ, T i ) are bounded and deerminisic funcions. Again, he cap price is given by 11.23) wih he small change ha v L, T i δ, T i ) is o be compued as n 11.26) v L, T i δ, T i ) = j=1 Ti δ γ j u, T i δ, T i ) 2 du 1/2. 11.3.2 The pricing of oher securiies No exac, explici soluion for European swapions has been found in he lognormal LIBOR marke seing. In paricular, Black s formula for swapions is no correc under he assumpion 11.18). The reason is ha when he forward LIBOR raes have volailiies proporional o heir level, he volailiy of he forward swap rae will no be proporional o he level of he forward swap rae. As described in Secion 11.2, he swapion price can be approximaed by a Mone

11.3 The lognormal LIBOR marke model 266 Carlo simulaion, which is ofen quie ime-consuming. Brace, Gaarek, and Musiela 1997) derive he following Black-ype approximaion o he price of a European payer swapion wih expiraion dae T 0 and exercise rae K under he lognormal LIBOR marke model assumpions: 11.27) P = Hδ n i=1 B Ti [ ] L Ti δ,ti Nd 1i) KNd 2i), < T 0, where d 1i and d 2i are quie complicaed expressions involving he variances and covariances of he ime T 0 values of he forward raes involved. These variances and covariances are deermined by he γ-funcion of he assumpion 11.18). This approximaion delivers he price much faser han a Mone Carlo simulaion. Brace, Gaarek, and Musiela provide numerical examples in which he price compued using he approximaion 11.27) is very close o he correc price compued using Mone Carlo simulaions). Of course, a similar approximaion applies o he European receiver swapion. Under he assumpions of he lognormal LIBOR marke model Milersen, Sandmann, and Sondermann 1997) derive an explici pricing formula for European opions on zero-coupon bonds, bu only for opions expiring a one of he ime poins T 0, T 1,..., T n 1, and where he underlying zero-coupon bond maures a he following dae in his sequence. In oher words, he ime disance beween he mauriy of he opion and he mauriy of he underlying zero-coupon bond mus be equal o δ. Represening he exercise price by K, he pricing formula for a European call opion is: 11.28) C K,Ti δ,ti = 1 K)B Ti Ne 1i ) K[B Ti δ B Ti ]Ne 2i ) where e 1i = 1 v L, T i δ, T i ) ln e 2i = e 1i v L, T i δ, T i ), 1 K)B Ti K[B Ti δ B Ti ] ) + 1 2 v L, T i δ, T i ), and, as above, v L, T i δ, T i ) is given by 11.22) in he one-facor seing and by 11.26) in he muli-facor seing. The price of he corresponding European pu opion follows from he pu-call pariy 2.9) on page 29: 11.29) π K,Ti δ,ti = K[B Ti δ B Ti ]N e 2i ) 1 K)B Ti N e 1i ). These formulas can be derived from he caple formula 11.19) and he relaions beween caples, floorles, and European bond opions known from Chaper 2, cf. Exercise 11.1. As argued in Secion 11.2, in any LIBOR marke model based on he δ-period ineres raes one can also price securiies ha depend on ineres raes over periods of lengh 2δ, 3δ, ec., as long as he paymen daes of hese securiies are in he se {T 0, T 1,..., T n }. Of course, his is also rue for he lognormal LIBOR marke model. For example, le us consider conracs ha depend on ineres raes covering periods of lengh 2δ. From 11.17) we have ha L T0,T2 = 1 2δ [ 1 + δl T0,T1 ] [ 1 + δl T1,T2 ] ) 1. According o he assumpion 11.18) of he lognormal δ-period LIBOR marke model, each of he forward raes on he righ-hand side has a volailiy proporional o he level of he forward rae.

11.4 Alernaive LIBOR marke models 267 An applicaion of Iô s Lemma o he above relaion shows ha he same proporionaliy does no hold for he 2δ-period forward rae L T0,T2. Consequenly, Black s cap formula canno be correc boh for caps on he 3-monh rae and caps on he 6-monh rae. To price caps on he 6-monh rae consisenly wih he assumpions of he lognormal LIBOR marke model for he 3-monh rae one mus resor o numerical mehods, e.g. Mone Carlo simulaion. I follows from he above consideraions ha he model canno jusify praciioners frequen use of Black s formula for boh caps and swapions and for conracs wih differen frequencies δ. Of course, he differences beween he prices generaed by Black s formula and he correc prices according o some reasonable model may be so small ha his inconsisency can be ignored, bu so far his issue has no been saisfacorily invesigaed in he lieraure. 11.4 Alernaive LIBOR marke models The lognormal LIBOR marke model specifies he forward rae volailiy in he general LIBOR marke model 11.5) as β, T i δ, T i, L Tj,δ ) Tj ) = L Ti δ,ti γ, T i δ, T i ), where γ is a deerminisic funcion. As we have seen, his specificaion has he advanage ha he prices of some) caps and floors are given by Black s formula. However, oher volailiy specificaions may be more realisic see Secion 11.6), which makes i worhwhile o look a alernaive specificaions. marke model. Below we will consider a racable and empirically relevan alernaive LIBOR European sock opion prices are ofen ransformed ino implici volailiies using he Black- Scholes-Meron formula. Similarly, for each caple we can deermine an implici volailiy for he corresponding forward rae as he value of he parameer σ i ha makes he caple price compued using Black s formula 5.68) idenical o he observed marke price. Suppose ha several caples are raded on he same forward rae and wih he same paymen dae, bu wih differen cap raes i.e. exercise raes) K. Then we ge a relaion σ i K) beween he implici volailiies and he cap rae. If he forward rae has a proporional volailiy, Black s model will be correc for all hese caples. In ha case all he implici volailiies will be equal so ha σ i K) corresponds o a fla line. However, according o Andersen and Andreasen 2000) σ i K) is ypically decreasing in K, which is referred o as a volailiy skew. Such a skew is inconsisen wih he volailiy assumpion of he LIBOR marke model 11.18). 3 Andersen and Andreasen consider a so-called CEV LIBOR marke model where he forward rae volailiy is given as β, T i δ, T i, L Tj,δ ) Tj ) = ) αγ, L Ti δ,ti Ti δ, T i ), i = 1,..., n, where γ is a bounded, deerminisic funcion, and α is a posiive consan. 4 For α = 1, he model is idenical o he lognormal LIBOR marke model. Andersen and Andreasen derive a closed form expression for he price of caps. The pricing formula is very similar o Black s formula, bu he relevan probabiliies are given by he disribuion funcion for a non-cenral χ 2 -disribuion. 3 Hull 2003, Ch. 15) has a deailed discussion of he similar phenomenon for sock and currency opions. 4 CEV is shor for Consan Elasiciy of Variance. This erm arises from he fac ha he elasiciy of he

11.5 Swap marke models 268 They show ha a CEV model wih α < 1 can generae he volailiy skew observed in pracice. In addiion, hey give an explici approximaion o he price of a European swapion in heir CEV LIBOR marke model. Also his pricing formula is of he same form as Black s formula, bu involves he disribuion funcion for he non-cenral χ 2 -disribuion insead of he normal disribuion. 11.5 Swap marke models Jamshidian 1997) inroduced he so-called swap marke models ha are based on assumpions abou he evoluion of cerain forward swap raes. Under he assumpion of a proporional volailiy of hese forward swap raes, he models will imply ha Black s formula for European swapions, i.e. 5.74) on page 132, is correc, a leas for some swapions. Given ime poins T 0, T 1,..., T n, where T i = T i 1 + δ for all i = 1,..., n. We will refer o a payer swap wih sar dae T k and final paymen dae T n i.e. paymen daes T k+1,..., T n ) as a k, n)-payer swap. Here we mus have 1 k < n. Le us by L T k,δ denoe he forward swap rae prevailing a ime T k for a k, n)-swap. Analogous o 2.30) on page 38, we have ha 11.30) LT k,δ where we have inroduced he noaion 11.31) G k,n = = BT k B Tn δg k,n n i=k+1 B Ti, which is he value of an annuiy bond paying 1 dollar a each dae T k+1,..., T n. A European payer k, n)-swapion gives he righ a ime T k o ener ino a k, n)-payer swap where he fixed rae K is idenical o he exercise rae of he swapion. From 2.33) on page 39 we know ha he value of his swapion a he expiraion dae T k is given by ) 11.32) P k,n T k = G k,n T k Hδ max k,δ LT T k K, 0. As discussed in Secion 5.3.2 on page 114, i is compuaionally convenien o use he annuiy as he numeraire. We refer o he corresponding maringale measure Q k,n as he k, n)-swap maringale measure. Since G k,k+1 = B T k+1, we have in paricular ha he k, k + 1)-swap maringale measure Q k,k+1 is idenical o he T k+1 -forward maringale measure Q T k+1. By he definiion of Q k,n, he ime price V of a securiy paying H Tk a ime T k is given by [ ] V H G k,n = E Qk,n Tk G k,n, T k volailiy wih respec o he forward rae level is equal o he consan α since β, T i δ, T i, L T ) j,δ ) Tj /β, T i δ, T i, L T ) j,δ ) Tj L T i δ,t i /L T i δ,t i β, T i δ, T i, L T ) j,δ ) Tj = L T i δ,t i = α L T i δ,t i ) α 1 γ, Ti δ, T i ), β β L T i δ,t i, T i δ, T i, L T j,δ L T i δ,t i, T i δ, T i, L T j,δ Cox and Ross 1976) sudy a similar varian of he Black-Scholes-Meron model for sock opions. ) Tj ) ) Tj ) = α.

11.5 Swap marke models 269 and hence 11.33) V = G k,n E Qk,n [ ] H Tk G k,n. T k The pricing formula 11.33) is paricularly convenien for he k, n)-swapion. Insering he payoff from 11.32), we obain a price of 11.34) P k,n [ )] = G k,n Hδ E Qk,n max k,δ LT T k K, 0. To price he swapion i suffices o know he disribuion of he swap rae L T k,δ T k swap maringale measure Q k,n. Here he following resul comes in handy: under he k, n)- Theorem 11.5 The forward swap rae L T k,δ is a Q k,n -maringale. Proof: According o 11.30), he forward swap rae is given as ) L T k,δ = BT k B Tn δg k,n = 1 B T k δ G k,n BTn G k,n. By definiion of he k, n)-swap maringale measure he price of any securiy relaive o he annuiy is a maringale under his probabiliy measure. In paricular, boh B T k /G k,n and B Tn /G k,n are Q k,n -maringales. Therefore, he expeced change in hese raios is zero under Q k,n. I follows from he above formula ha he expeced change in he forward swap rae L T k,δ is also zero under Q k,n so ha L T k,δ is a Q k,n -maringale. Consequenly, he evoluion in he forward swap rae L T k,δ is fully specified by i) he number of Brownian moions affecing his and oher modeled forward swap raes and ii) he volailiy funcions ha show he sensiiviy of he forward swap raes o changes in each of he Brownian moions. Le us again focus on a one-facor model. A swap marke model is based on he assumpion d L T k,δ ) = β k,n, L Tj,δ ) Tj dz k,n, where z k,n is a Brownian moion under he k, n)-swap maringale measure Q k,n, and he volailiy funcion β k,n Tj,δ hrough he erm L ) Tj can depend on he curren values of all he modeled forward swap raes. Under he assumpion ha β k,n is proporional o he level of he forward swap rae, i.e. 11.35) d L T k,δ = L T k,δ γ k,n ) dz k,n where γ k,n ) is a bounded, deerminisic funcion, we ge ha he fuure value of he forward swap rae is lognormally disribued. This model is herefore referred o as he lognormal swap marke model. In such a model he swapion price in formula 11.34) can be compued explicily: Theorem 11.6 Under he assumpion 11.35) he price of a European k, n)-payer swapion is given by 11.36) P k,n = n i=k+1 B Ti ) ] Hδ k,δ [ LT Nd 1 ) KNd 2 ), < T k,

11.6 Furher remarks 270 where d 1 = ) ln k,δ LT /K v k,n ) d 2 = d 1 v k,n ), + 1 2 v k,n), 1/2 Tk v k,n ) = γ k,n u) du) 2. The proof of his resul is analogous o he proof of Theorem 11.3 and is herefore omied. The pricing formula is idenical o Black s formula 5.74) wih σ given by σ 2 [T k ] = v k,n ) 2. Hence, he lognormal swap marke model provides some heoreical suppor of he Black swapion pricing formula. In a previous secion we concluded ha in a LIBOR marke model i is no jusifiable o exogenously specify he processes for all forward raes, only he processes for non-overlapping periods. In a swap marke model Musiela and Rukowski 1997, Secion 14.4) demonsrae ha he T1,δ T2,δ Tn 1,δ processes for he forward swap raes L, L,..., L can be modeled independenly. These are forward swap raes for swaps wih he same final paymen dae T n, bu wih differen sar daes T 1,..., T n 1 and hence differen mauriies. In paricular, he lognormal assumpion 11.35) can hold for all hese forward swap raes, which implies ha all he swapion prices P 1,n,..., P n 1,n are given by Black s swapion pricing formula. However, under such an assumpion neiher he forward LIBOR raes L Ti 1,Ti nor he forward swap raes for swaps wih oher final paymen daes can have proporional volailiies. Consequenly, Black s formula canno be correc neiher for caps, floors nor swapions wih oher mauriy daes. The correc prices of hese securiies mus be compued using numerical mehods, e.g. Mone Carlo simulaion. Also in his case i is no clear by how much he Black pricing formulas miss he heoreically correc prices. In he conex of he LIBOR marke models we have derived relaions beween he differen forward maringale measures. For he swap marke models we can derive similar relaions beween he differen swap maringale measures and hence describe he dynamics of all he forward T1,δ T2,δ Tn 1,δ swap raes L, L,..., L under he same probabiliy measure. Then all he relevan processes can be simulaed under he same probabiliy measure. For deails he reader is referred o Jamshidian 1997) and Musiela and Rukowski 1997, Secion 14.4). 11.6 Furher remarks De Jong, Driessen, and Pelsser 2001) invesigae he exen o which differen lognormal LIBOR and swap marke models can explain empirical daa consising of forward LIBOR ineres raes, forward swap raes, and prices of caples and European swapions. The observaions are from he U.S. marke in 1995 and 1996. For he lognormal one-facor LIBOR marke model 11.18) hey find ha i is empirically more appropriae o use a γ-funcion which is exponenially decreasing in he ime-o-mauriy T i δ of he forward raes, γ, T i δ, T i ) = γe κ[ti δ ], i = 1,..., n, han o use a consan, γ, T i δ, T i ) = γ. This is relaed o he well-documened mean reversion of ineres raes ha makes long ineres raes relaively less volaile han shor ineres raes.

11.7 Exercises 271 They also calibrae wo similar model specificaions perfecly o observed caple prices, bu find ha in general he prices of swapions in hese models are furher from he marke prices han are he prices in he ime homogeneous models above. In all cases he swapion prices compued using one of hese lognormal LIBOR marke models exceed he marke prices, i.e. he lognormal LIBOR marke models overesimae he swapion prices. All heir specificaions of he lognormal one-facor LIBOR marke model give a relaively inaccurae descripion of marke daa and are rejeced by saisical ess. De Jong, Driessen, and Pelsser also show ha wo-facor lognormal LIBOR marke models are no significanly beer han he one-facor models and conclude ha he lognormaliy assumpion is probably inappropriae. Finally, hey presen similar resuls for lognormal swap marke models and find ha hese models are even worse han he lognormal LIBOR marke models when i comes o fiing he daa. 11.7 Exercises EXERCISE 11.1 Caples and pus on zero-coupon bonds) Show ha he formula 11.29) for he price of a European pu opion on a zero-coupon bond follows from he caple formula 11.19) and formula 2.12) on page 32.

Chaper 12 The measuremen and managemen of ineres rae risk 12.1 Inroducion The primary reason for he flucuaions in he values of bonds and oher fixed income securiies is changes in he erm srucure of ineres raes. Mos invesors wan o measure and compare he sensiiviies of differen securiies o erm srucure movemens. The ineres rae risk measures of he individual securiies are needed in order o obain an overview of he oal ineres rae risk of he invesors porfolio and o idenify he conribuion of each securiy o his oal risk. Many insiuional invesors are required o produce such risk measures for regulaory auhoriies and for publicaion in heir accouning repors. In addiion, such risk measures consiue an imporan inpu o he porfolio managemen. In his chaper we will discuss how o quanify he ineres rae risk of bonds and how hese risk measures can be used in he managemen of he ineres rae risk of porfolios. We will firs describe he radiional, bu sill widely used, duraion and convexiy measures and discuss heir relaions o he dynamics of he erm srucure of ineres raes. Then we will consider risk measures ha are more direcly linked o he dynamic erm srucure models we have analyzed in he previous chapers. Here we focus on diffusion models and emphasize models wih a single sae variable. We will compare he differen risk measures and heir use in he consrucion of so-called immunizaion sraegies. Finally, we will show how he duraion measure can be useful for he pricing of European opions on bonds and hence he pricing of European swapions. 12.2 Tradiional measures of ineres rae risk 12.2.1 Macaulay duraion and convexiy The Macaulay duraion of a bond was defined by Macaulay 1938) as a weighed average of he ime disance o he paymen daes of he bond, i.e. an effecive ime-o-mauriy. As shown by Hicks 1939), he Macaulay duraion also measures he sensiiviy of he bond value wih respec o changes in is own yield. Le us consider a bond wih paymen daes T 1,..., T n, where we assume ha T 1 < < T n. The paymen a ime T i is denoed by Y i. The ime value of he bond is denoed by B. We le y B denoe he yield of he bond a ime, compued using coninuous 272

12.2 Tradiional measures of ineres rae risk 273 compounding so ha B = T i> Y i e yb Ti ), where he sum is over all he fuure paymen daes of he bond. The Macaulay duraion D Mac of he bond is defined as 12.1) D Mac = 1 B db dy B = T i> T i )Y i e yb Ti ) B = w Mac, T i )T i ), T i> where w Mac, T i ) = Y i e yb Ti ) /B, which is he raio beween he value of he i h paymen and he oal value of he bond. Noe ha for a bond wih only one remaining paymen he Macaulay duraion is equal o he ime-o-mauriy. A simple manipulaion of he definiion of he Macaulay duraion yields db = D Mac dy B B so ha he relaive price change of he bond due o an insananeous, infiniesimal change in is yield is proporional o he Macaulay duraion of he bond. Frequenly, he Macaulay duraion is defined in erms of he bond s annually compued yield ŷ B. By definiion, B = T i> Y i 1 + ŷ B ) Ti ) so ha db dŷ B = T i>t i )Y i 1 + ŷ B ) Ti ) 1. The Macaulay duraion is hen ofen defined as 12.2) D Mac = 1 + ŷb B db dŷ B = T i> T i )Y i 1 + ŷ B ) Ti ) B = w Mac, T i )T i ), T i> where he weighs w Mac, T i ) are he same as before since e yb = 1 + ŷ B ). Therefore he wo definiions provide precisely he same value for he Macaulay duraion. Because dy B /dŷ B = 1/1 + ŷ B ), we have ha db B = D Mac dŷ B 1 + ŷ B For bulle bonds, annuiy bonds, and serial bonds an explici expression for he Macaulay duraion can be derived. 1 price of he bond. In many newspapers he Macaulay duraion of each bond is lised nex o he The Macaulay duraion is defined as a measure of he price change induced by an infiniesimal change in he yield of he bond. For a non-infiniesimal change, a firs-order approximaion gives ha and hence B db dy B y B, B B D Mac y B. 1 The formula for he Macaulay duraion of a bulle bond can be found in many exbooks, e.g. Fabozzi 2000) and van Horne 2001)..

12.2 Tradiional measures of ineres rae risk 274 An obvious way o obain a beer approximaion is o include a second-order erm: B db dy B Defining he Macaulay convexiy by y B + 1 d 2 B ) y B 2 2 dy B ) 2. 12.3) K Mac = 1 d 2 B 2B dy B ) = 1 w Mac, T 2 i )T i ) 2, 2 T i> we can wrie he second-order approximaion as B B D Mac y B + K Mac ) y B 2. Noe ha he approximaion only describes he price change induced by an insananeous change in he yield. In order o evaluae he price change over some ime inerval, he effec of he reducion in he ime-o-mauriy of he bond should be included, e.g. by adding he erm B on he righ-hand side. The Macaulay measures are no direcly informaive of how he price of a bond is affeced by a change in he zero-coupon yield curve and are herefore no a valid basis for comparing he ineres rae risk of differen bonds. The problem is ha he Macaulay measures are defined in erms of he bond s own yield, and a given change in he zero-coupon yield curve will generally resul in differen changes in he yields of differen bonds. I is easy o show see e.g. Ingersoll, Skelon, and Weil 1978, Thm. 1)) ha he changes in he yields of all bonds will be he same if and only if he zero-coupon yield curve is always fla. In paricular, he yield curve is only allowed o move by parallel shifs. Such an assumpion is no only unrealisic, i also conflics wih he no-arbirage principle, as we shall demonsrae in Secion 12.2.3. 12.2.2 The Fisher-Weil duraion and convexiy Macaulay 1938) defined an alernaive duraion measure based on he zero-coupon yield curve raher han he bond s own yield. Afer decades of neglec his duraion measure was revived by Fisher and Weil 1971), who demonsraed he relevance of he measure for consrucing immunizaion sraegies. We will refer o his duraion measure as he Fisher-Weil duraion. The precise definiion is 12.4) D FW = T i> w, T i )T i ), where w, T i ) = Y i e yt i Ti ) /B. Here, y Ti is he zero-coupon yield prevailing a ime for he period up o ime T i. Relaive o he Macaulay duraion, he weighs are differen. w, T i ) is compued using he rue presen value of he i h paymen since he paymen is muliplied by he marke discoun facor for ime T i paymens, B Ti = e yt i Ti ). In he weighs used in he compuaion of he Macaulay measures he paymens are discouned using he yield of he bond. However, for ypical yield curves he wo se of weighs and hence he wo duraion measures will be very close, see e.g. Table 12.1 on page 282. If we hink of he bond price as a funcion of he relevan zero-coupon yields y T1,..., y Tn, B = T i> Y i e yt i Ti ),

12.2 Tradiional measures of ineres rae risk 275 we can wrie he relaive price change induced by an insananeous change in he zero-coupon yields as db B = T i> 1 B B y Ti dy Ti = T i> w, T i )T i )dy Ti. If he changes in all he zero-coupon yields are idenical, he relaive price change is proporional o he Fisher-Weil duraion. Consequenly, he Fisher-Weil duraion represens he price sensiiviy owards infiniesimal parallel shifs of he zero-coupon yield curve. Noe ha an infiniesimal parallel shif of he curve of coninuously compounded yields corresponds o an infiniesimal proporional shif in he curve of yearly compounded yields. This follows from he relaion y Ti = ln1 + ŷ Ti ) beween he coninuously compounded zero-coupon rae y Ti and he yearly compounded zero-coupon rae ŷ Ti, which implies ha dy Ti = dŷ Ti /1 + ŷ Ti ). We can also define he Fisher-Weil convexiy as 12.5) K FW = 1 w, T i )T i ) 2. 2 T i> The relaive price change induced by a non-infiniesimal parallel shif of he yield curve can hen be approximaed by B D FW y + K FW y ) 2, B where y is he common change in all he zero-coupon yields. Again he reducion in he imeo-mauriy should be included o approximae he price change over a given period. 12.2.3 The no-arbirage principle and parallel shifs of he yield curve In his secion we will invesigae under which assumpions he zero-coupon yield curve can only change in he form of parallel shifs. The analysis follows Ingersoll, Skelon, and Weil 1978). If he yield curve only changes in form of infiniesimal parallel shifs, he curve mus have exacly he same shape a all poins in ime. Hence, we can wrie any zero-coupon yield y +τ as a sum of he curren shor rae and a funcion which only depends on he ime-o-mauriy of he yield, i.e. y T = r + ht ), where h0) = 0. In paricular, he evoluion of he yield curve can be described by a model where he shor rae is he only sae variable and follows a process of he ype in he real world and hence in a hypoheical risk-neural world. dr = αr, ) d + βr, ) dz dr = ˆαr, ) d + βr, ) dz Q In such a model he price of any fixed income securiy will be given by a funcion solving he fundamenal parial differenial equaion 7.3) on page 164. In paricular, he price funcion of any zero-coupon bond B T r, ) saisfies B T r, ) + ˆαr, ) BT r r, ) + 1 2 βr, )2 2 B T r 2 r, ) rbt r, ) = 0, r, ) S [0, T ),

12.3 Risk measures in one-facor diffusion models 276 and he erminal condiion B T r, T ) = 1. However, we know ha he zero-coupon bond price is of he form B T r, ) = e yt T ) = e r[t ] ht )[T ]. Subsiuing he relevan derivaives ino he parial differenial equaion, we ge ha h T )T ) + ht ) = ˆαr, )T ) 1 2 βr, )2 T ) 2, r, ) S [0, T ). Since his holds for all r, he righ-hand side mus be independen of r. This can only be he case for all if boh ˆα and β are independen of r. Consequenly, we ge ha h T )T ) + ht ) = ˆα)T ) 1 2 β)2 T ) 2, [0, T [. The lef-hand side depends only on he ime difference T so his mus also be he case for he righ-hand side. This will only be rue if neiher ˆα nor β depend on. Therefore ˆα and β have o be consans. I follows from he above argumens ha he dynamics of he shor rae is of he form dr = ˆα d + β dz Q, oherwise non-parallel yield curve shifs would be possible. This shor rae dynamics is he basic assumpion of he Meron model sudied in Secion 7.3. There we found ha he zero-coupon yields are given by y +τ = r + 1 2 ˆατ 1 6 β2 τ 2, which corresponds o hτ) = 1 2 ˆατ 1 6 β2 τ 2. We can herefore conclude ha all yield curve shifs will be infiniesimal parallel shifs if and only if he yield curve a any poin in ime is a parabola wih downward sloping branches and he shor-erm ineres rae follows he dynamics described in Meron s model. These assumpions are highly unrealisic. Furhermore, Ingersoll, Skelon, and Weil 1978) show ha non-infiniesimal parallel shifs of he yield curve conflic wih he no-arbirage principle. The boom line is herefore ha he Fisher-Weil risk measures do no measure he bond price sensiiviy owards realisic movemens of he yield curve. The Macaulay risk measures are no consisen wih any arbirage-free dynamic erm srucure model. 12.3 Risk measures in one-facor diffusion models 12.3.1 Definiions and relaions To obain measures of ineres rae risk ha are more in line wih a realisic evoluion of he erm srucure of ineres raes, i is naural o consider uncerain price movemens in reasonable dynamic erm srucure models. In a model wih one or more sae variables we focus on he sensiiviy of he prices wih respec o a change in he sae variables). consider he one-facor diffusion models sudied in Chapers 7 and 9. form In his secion we We assume ha he shor rae r is he only sae variable, and ha i follows a process of he dr = αr, ) d + βr, ) dz. For an asse wih price B = Br, ), Iô s Lemma implies ha B db = r, ) + αr, ) B r r, ) + 1 ) 2 βr, ) 2 2 B r 2 r, ) d + B r r, )βr, ) dz,

12.3 Risk measures in one-facor diffusion models 277 and hence db B = 1 B Br, ) r 1, ) + αr, ) Br, ) 1 B + Br, ) r r, )βr, ) dz. For a bond he derivaive B r B r r, ) + 1 2 βr, ) 2 1 2 ) B Br, ) r 2 r, ) d r, ) is negaive in he models we have considered so he volailiy B r r, )βr, ). I seems eviden o use he asse-specific par of of he bond is given by 2 1 Br,) he volailiy as a risk measure. Therefore we define he duraion of he asse as 12.6) Dr, ) = 1 Br, ) B r, ). r Noe he similariy o he definiion of he Macaulay duraion. The unexpeced reurn on he asse is equal o he produc of is duraion and he unexpeced change in he shor rae, i.e. βr, ) dz. Furhermore, we define he convexiy as 12.7) Kr, ) = and he ime value as 12.8) Θr, ) = 1 2Br, ) 1 Br, ) 2 B r, ) r2 B r, ). Consequenly, he rae of reurn on he asse over he nex infiniesimal period of ime can be wrien as db B = Θr, ) αr, )Dr, ) + βr, ) 2 Kr, ) ) d Dr, )βr, ) dz. The duraion of a porfolio of ineres rae dependen securiies is given by a value-weighed average of he duraions of he individual securiies. For example, le us consider a porfolio of wo securiies, namely N 1 unis of asse 1 wih a uni price of B 1 r, ) and N 2 unis of asse 2 wih a uni price of B 2 r, ). The value of he porfolio is Πr, ) = N 1 B 1 r, ) + N 2 B 2 r, ). The duraion D Π r, ) of he porfolio can be compued as 12.9) D Π r, ) = 1 Πr, ) = 1 Πr, ) = N 1B 1 r, ) Πr, ) Π r, ) r B 1 N 1 r r, ) + N B 2 2 r ) B 1 r, ) r 1 B 1 r, ) = η 1 r, )D 1 r, ) + η 2 r, )D 2 r, ), ) r, ) + N 2B 2 r, ) Πr, ) 1 B 2 r, ) ) B 2 r, ) r where η i r, ) = N i B i r, )/Πr, ) is he porfolio weigh of he i h asse, and D i r, ) is he duraion of he i h asse, i = 1, 2. Obviously, we have η 1 r, ) + η 2 r, ) = 1. Similarly for he convexiy and he ime value. In paricular, he duraion of a coupon bond is a value-weighed average of he duraions of he zero-coupon bonds mauring a he paymen daes of he coupon bond. 2 Recall ha he volailiy of an asse is defined as he sandard deviaion of he reurn on he asse over he nex insan.

12.3 Risk measures in one-facor diffusion models 278 From Chaper 7 we know ha he funcion Br, ) mus saisfy he parial differenial equaion B B r, ) + ˆαr, ) r r, ) + 1 2 βr, )2 2 B r, ) rbr, ) = 0, r2 where ˆαr, ) = αr, ) βr, )λr, ) and λr, ) is he marke price on risk. Afer a division by Br, ) and a subsiuion of he definiions, we obain he following relaion beween he duraion, he convexiy, and he ime value: 3 12.10) Θr, ) ˆαr, )Dr, ) + βr, ) 2 Kr, ) = r. This relaion beween he ime value, he duraion, and he convexiy holds for all ineres rae dependen securiies and hence also for all porfolios of ineres rae dependen securiies. Noe ha he rae of reurn on he securiy over he nex insan can be rewrien as db B = r λr, )βr, )Dr, )) d Dr, )βr, ) dz, which only involves he duraion, no he convexiy or he ime value. We also know ha in order o replicae a given fixed income securiy in a one-facor model, one mus form a porfolio ha, a any poin in ime, has he same volailiy and herefore he same duraion as ha securiy. This is a consequence of he proof of he fundamenal parial differenial equaion, cf. Theorem 5.10 on page 121 and he subsequen discussion of hedging on page 122. However, a perfec hedge requires coninuous rebalancing of he porfolio. Due o ransacion coss and oher pracical issues such a coninuous rebalancing is no implemenable. Sraighforward differeniaion implies ha 12.11) D r r, ) = Dr, )2 2Kr, ) so ha he convexiy can be seen as a measure of he ineres rae sensiiviy of he duraion. If, a each ime he porfolio is rebalanced, he convexiies of he porfolio and of he posiion o be hedged are mached, heir duraions will probably say close unil he following rebalancing of he porfolio. The convexiy is herefore also of pracical use in he ineres rae risk managemen. The duraion, he convexiy, and he ime value can also be used for speculaion, i.e. for seing up a porfolio which will provide a high reurn if some specific expecaions of he fuure erm srucure are realized. For example, by consrucing a porfolio wih a zero duraion and a large posiive convexiy, one will obain a high reurn over a period wih a large change posiive or negaive) in he shor rae. I follows from he relaion 12.10) ha for such a porfolio he ime value will be negaive. Consequenly, he porfolio will give a negaive reurn over a period where he shor rae does no change significanly. The Macaulay duraion, defined in 12.1), and he Fisher-Weil duraion, defined in 12.4), are measured in ime unis ypically years) and can be inerpreed as measures of he effecive imeo-mauriy of a bond. The duraion defined in 12.6) is no measured in ime unis, bu i can be ransformed ino a ime-denominaed duraion. Following Cox, Ingersoll, and Ross 1979), we 3 In he Black-Scholes-Meron model he ime value and he so-called and Γ values are relaed in a similar way, cf. Hull 2003, Secion 14.7). Apparenly, Chrisensen and Sørensen 1994) were he firs o discover his relaion in he conex of erm srucure models and he imporance of aking he ime value ino accoun in he consrucion of ineres rae risk hedging sraegies.

12.3 Risk measures in one-facor diffusion models 279 define he ime-denominaed duraion of a coupon bond as he ime-o-mauriy of he zerocoupon bond ha has he same duraion as he coupon bond. If we denoe he ime-denominaed duraion by D r, ), he defining relaion can be saed as 1 B Br, ) r r, ) = 1 B +D r,) r, ) B +D r,) r r, ). For bonds wih only one remaining paymen he ime-denominaed duraion is equal o he imeo-mauriy, jus as for he Macaulay-duraion and he Fisher-Weil duraion. Cox, Ingersoll, and Ross 1979) used he erm sochasic duraion for he ime-denominaed duraion D r, ) o indicae ha his duraion measure is based on he sochasic evoluion of he erm srucure. Oher auhors use he erm sochasic duraion for he original duraion Dr, ). Noe ha boh hese duraion conceps are defined in relaion o a specific erm srucure model, and he duraion measures herefore indicae he sensiiviy of he bond price o he yield curve movemens consisen wih he model. The radiional Macaulay and Fisher-Weil duraions can be compued wihou reference o a specific model, bu, on he oher hand, hey only measure he price sensiiviy o a paricular ype of yield curve movemens ha is no consisen wih any reasonable ineres rae dynamics. Anoher advanage of he risk measures inroduced in his secion is ha hey are well-defined for all ypes of ineres rae dependen securiies, whereas he Macaulay and Fisher-Weil risk measures are only meaningful for bonds. 4 For he risk managemen of porfolios of many differen fixed income securiies we need risk measures for all he individual securiies, e.g. fuures, caps/floors, and swapions. 12.3.2 Compuaion of he risk measures in affine models In he ime homogeneous affine one-facor diffusion models, e.g. he Vasicek model and he CIR model, he zero-coupon bond prices are of he form B Ti r, ) = e ati ) bti )r. The price of a coupon bond wih paymen Y i a ime T i, i = 1,..., n, is Br, ) = Y i B Ti r, ). T i> Consequenly, he duraion of he coupon bond is Dr, ) = 1 B Br, ) r r, ) = 1 bt i )Y i B Ti r, ) = wr,, T i )bt i ), Br, ) T i> T i> where wr,, T i ) = Y i B Ti r, )/Br, ) is he i h paymen s share of he oal presen value of he bond. Noe ha he duraion of a zero-coupon bond mauring a ime T is bt ), which is differen from T excep in he unrealisic Meron model). The convexiy can be compued as Kr, ) = 1 wr,, T i )bt i ) 2. 2 T i> 4 The duraion Dr, ) is well-defined by 12.6) for any securiy. Since he volailiies of zero-coupon bonds are bounded from above in many models, he ime-denominaed duraion can only be defined for securiies wih a volailiy below ha upper bound. This is always rue for coupon bonds, bu no for all derivaive securiies.

12.3 Risk measures in one-facor diffusion models 280 The ime value of he coupon bond is given by Θr, ) = T i> wr,, T i ) a T i ) + b T i )r) = T i> wr,, T i )f Ti r, ), where f Ti r, ) is he forward rae a ime for he mauriy dae T i. duraion D r, ) is he soluion o he equaion wr,, T i )bt i ) = bd r, )). T i> The ime-denominaed If b is inverible, we can wrie he ime-denominaed duraion of a coupon bond explicily as ) 12.12) D r, ) = b 1 wr,, T i )bt i ). T i> Example 12.1 In he Vasicek model we know from Secion 7.4 ha he b-funcion is given by so ha he duraion of a coupon bond is Dr, ) = T i> wr,, T i ) 1 κ bτ) = 1 κ 1 e κτ ) 1 e κ[ti ]) = 1 κ 1 T i> wr,, T i )e κ[ti ] ). Since we have ha 1 1 e κτ ) = y τ = 1 ln1 κy), κ κ b 1 y) = 1 ln1 κy), κ and hence he ime-denominaed duraion of a coupon bond is D r, ) = 1 κ ln 1 κ ) wr,, T i )bt i ) T i> = 1 κ ln = 1 κ ln T i> 1 T i> wr,, T i )1 e κ[ti ] ) wr,, T i )e κ[ti ] ). ) For he exended Vasicek model he Hull-Whie model) we ge he same expression since he b-funcion in ha model is he same as in he original Vasicek model. Example 12.2 In he CIR model sudied in Secion 7.5 he b-funcion is given by bτ) = so ha he duraion of a coupon bond is 2e γτ 1) γ + ˆκ)e γτ 1) + 2γ Dr, ) = 2e γ[ti ] 1) wr,, T i ) γ + ˆκ)e γ[ti ] 1) + 2γ. T i>

12.3 Risk measures in one-facor diffusion models 281 Since b 1 y) = 1 γ ln 1 + ) 2γy, 2 ˆκ + γ)y he ime-denominaed duraion of a coupon bond is [ ] 1 12.13) D r, ) = 1 γ ln 2 1 + 2γ T wr,, T ˆκ + γ). i> i)bt i ) 12.3.3 A comparison wih radiional duraions Munk 1999) shows analyically ha for any bond he ime-denominaed duraion in he Vasicek model is smaller han he Fisher-Weil duraion. This is also rue for he CIR model if he parameer ˆκ = κ + λ is posiive, which is consisen wih ypical parameer esimaes. Therefore he Fisher- Weil duraion over-esimaes he ineres rae risk of coupon bonds. Excep for exreme yield curves, he Macaulay duraion and he Fisher-Weil duraion will be very, very close so ha he above conclusion also applies o he Macaulay duraion. Table 12.1 shows he differen duraion measures for bulle bonds of differen mauriies under he assumpion ha he yield curve and is dynamics are consisen wih he CIR model wih given, realisic parameer values. I is clear from he able ha, for all bonds, he Macaulay duraion and he Fisher-Weil duraion are very close. For relaively shor-erm bonds he ime-denominaed duraion is close o he radiional duraions, bu for longer-erm bonds he ime-denominaed duraion is significanly lower han he Macaulay and Fisher-Weil duraions. In paricular, we see ha he ineres rae sensiiviy, and herefore also he ime-denominaed duraion, for bulle bonds firs increases and hen decreases as he ime-o-mauriy increases. Wha is he explanaion for he differences in he duraion measures? As discussed in Secion 12.2.3, he Fisher-Weil duraion is only a reasonable ineres rae risk measure if he yield curve evolves as in he Meron model where boh he drif and he volailiy of he shor rae are assumed o be consan. In he Meron model he volailiy of a zero-coupon bond is proporional o he ime-o-mauriy of he bond, cf. 7.17) and 7.37). On he oher hand, in he CIR model he volailiy of a zero-coupon bond wih ime-o-mauriy τ equals bτ)β r, where he b-funcion is given by 7.70) on page 190. I can be shown ha b is an increasing, concave funcion wih b τ) < 1 for all τ. Hence, he volailiy of he zero-coupon bonds increases wih he ime-o-mauriy, bu less han proporionally. I can also be shown ha he b-funcion in he CIR model is a decreasing funcion of he speed-of-adjusmen parameer κ so he sronger he meanreversion, he furher apar he bond volailiies in he wo models. Consequenly, he disance beween he ime-denominaed duraion D and he Fisher-Weil duraion will ypically increase wih he speed-of-adjusmen parameer, alhough his probably canno be proved analyically due o he complicaed expression for D in 12.13).

12.4 Immunizaion 282 ime-o-mauriy in years price yield D Mac D FW D D 1 100.48 4.50% 1.00 1.00 1.00 0.89 2 100.31 4.84% 1.95 1.95 1.95 1.56 3 99.70 5.11% 2.86 2.86 2.83 2.05 4 98.81 5.34% 3.72 3.72 3.63 2.41 5 97.75 5.53% 4.54 4.54 4.34 2.67 6 96.60 5.68% 5.32 5.31 4.95 2.86 8 94.24 5.93% 6.74 6.72 5.86 3.09 10 91.96 6.10% 8.01 7.97 6.40 3.21 12 89.87 6.22% 9.14 9.07 6.68 3.26 15 87.15 6.35% 10.57 10.45 6.83 3.28 20 83.63 6.48% 12.39 12.16 6.80 3.28 25 81.13 6.56% 13.65 13.30 6.71 3.26 Table 12.1: A comparison of duraion measures for differen bonds under he assumpion ha he CIR model wih he parameers κ = 0.36, θ = 0.05, β = 0.1185, and λ = 0.1302 provides a correc descripion of he yield curve and is dynamics. The curren shor rae is 0.04. The bonds are bulle bonds wih a coupon rae of 5%, a face value of 100, one annual paymen dae, and exacly one year unil he nex paymen dae. 12.4 Immunizaion 12.4.1 Consrucion of immunizaion sraegies In many siuaions an individual or corporae invesor will inves in he bond marke eiher in order o ensure ha some fuure liabiliies can be me or jus o obain some desired fuure cash flow. For example, a pension fund will ofen have a relaively precise esimae of he size and iming of he fuure pension paymens o is cusomers. For such an invesor i is imporan ha he value of he invesmen porfolio remains close o he value of he liabiliies. Some financial insiuions are even required by law o keep he value of he invesmen porfolio a any poin in ime above he value of he liabiliies by some percenage margin. A cash flow or porfolio is said o be immunized agains ineres rae risk) if he value of he cash flow or porfolio is no negaively affeced by any possible change in he erm srucure of ineres raes. An invesor who has o pay a given cash flow can obain an immunized oal posiion by invesing in a porfolio of ineres rae dependen securiies ha perfecly replicaes ha cash flow. For example, if an invesor has o pay 10 million dollars in 5 years, he can make sure ha his will be possible by invesing in 5-year zero-coupon bonds wih a oal face value of 10 million dollars. The presen value of his oal posiion will be compleely immune o ineres rae movemens. An invesor who has a desired cash flow consising of several fuure paymens can obain perfec immunizaion by invesing in a porfolio of zero-coupon bonds ha exacly replicaes he cash flow. In many cases, however, all he necessary zero-coupon bonds are neiher raded on he bond marke nor possible o consruc by a saic porfolio of raded coupon bonds. Therefore, he desired cash flow can only be mached by consrucing a dynamically rebalanced porfolio of raded securiies.

12.4 Immunizaion 283 We know from he discussion in Chaper 5 ha if he erm srucure follows a one-facor diffusion model, any ineres rae dependen securiy or porfolio) can be perfecly replicaed by a paricular porfolio of any wo oher ineres rae dependen securiies. The porfolio weighs have o be adjused coninuously so ha he volailiy of he porfolio value will always be idenical o he volailiy of he value of he cash flow which is o be replicaed. In oher words, he duraion of he porfolio mus mach he duraion of he desired cash flow a any poin in ime. If we le ηr, ) denoe he value weigh of he firs securiy in he immunizing porfolio, he second securiy will have a value weigh of 1 ηr, ). According o 12.9), he duraion of he porfolio is D Π r, ) = ηr, )D 1 r, ) + 1 ηr, ))D 2 r, ), where D 1 r, ) and D 2 r, ) are he duraions of each of he securiies in he porfolio. If Dr, ) denoes he duraion of he cash flow o be mached, we wan o make sure ha ηr, )D 1 r, ) + 1 ηr, ))D 2 r, ) = Dr, ) for all r and. This relaion will hold if he porfolio weigh ηr, ) is chosen so ha 12.14) ηr, ) = Dr, ) D 2 r, ) D 1 r, ) D 2 r, ). If i) he porfolio is iniially consruced wih hese relaive weighs and scaled so ha he oal amoun invesed is equal o he presen value of he cash flow o be mached, and ii) he porfolio is coninuously rebalanced so ha 12.14) holds a any poin in ime, hen he desired cash flow will be mached wih cerainy, i.e. he posiion is perfecly immunized agains ineres rae movemens. Of course, coninuous rebalancing of a porfolio is no pracically implemenable or desirable considering real-world ransacion coss). If he porfolio is only rebalanced periodically, a perfec immunizaion canno be guaraneed. The duraions may be mached each ime he porfolio is rebalanced, bu beween hese daes he duraions may diverge due o ineres rae movemens and he passage of ime. Wih differen duraions he porfolio and he desired cash flow will no have he same sensiiviy owards anoher ineres rae change. The convexiy can be inerpreed as a measure of he sensiiviy of he duraion owards changes in he erm srucure of ineres raes. If boh he duraions and he convexiies of he porfolio and he cash flow are mached each ime he porfolio is rebalanced, he duraions are likely o say close even afer several ineres rae changes. Therefore, maching he convexiies should improve he effeciveness of he immunizaion sraegy. Noe ha when boh duraions and convexiies are mached, i follows from 12.10) ha he ime values are also idenical. Maching boh duraions and convexiies requires a porfolio of hree securiies. Le us wrie he duraions and convexiies of he hree securiies in he porfolio as D i r, ) and K i r, ), respecively. The value weighs of he hree securiies are denoed by η i r, ), and he convexiy of he desired cash flow is denoed by Kr, ). Since η 3 r, ) = 1 η 1 r, ) η 2 r, ), duraions and convexiies will be mached if η 1 r, ) and η 2 r, ) are chosen such ha η 1 r, )D 1 r, ) + η 2 r, )D 2 r, ) + [1 η 1 r, ) η 2 r, )] D 3 r, ) = Dr, ), η 1 r, )K 1 r, ) + η 2 r, )K 2 r, ) + [1 η 1 r, ) η 2 r, )] K 3 r, ) = Kr, ).

12.4 Immunizaion 284 This equaion sysem has he unique soluion 12.15) η 1 r, ) = 12.16) η 2 r, ) = Dr, ) D 3 r, ))K 2 r, ) K 3 r, )) D 2 r, ) D 3 r, )) Kr, ) K 3 r, )) D 1 r, ) D 3 r, ))K 2 r, ) K 3 r, )) D 2 r, ) D 3 r, ))K 1 r, ) K 3 r, )), D 1r, ) D 3 r, )) Kr, ) K 3 r, )) Dr, ) D 3 r, ))K 1 r, ) K 3 r, )) D 1 r, ) D 3 r, ))K 2 r, ) K 3 r, )) D 2 r, ) D 3 r, ))K 1 r, ) K 3 r, )). If only duraions are mached, he convexiy of he porfolio and hence he effeciveness of he immunizaion sraegy will be highly dependen on which wo securiies he porfolio consiss of. If he convexiy of he invesmen porfolio is larger han he convexiy of he cash flow, a big change posiive or negaive) in he shor rae will induce an increase in he ne value of he oal posiion. On he oher hand, if he shor rae says almos consan, he ne value of he posiion will decrease since he ime value of he porfolio is hen lower han he ime value of he cash flow, cf. 12.10). The converse conclusions hold in case he convexiy of he porfolio is less han he convexiy of he cash flow. Tradiionally, immunizaion sraegies have been consruced on he basis of Macaulay duraions insead of he sochasic duraions as above. The Macaulay duraion of a porfolio is ypically very close o, bu no exacly equal o, he value-weighed average of he Macaulay duraions of he securiies in he porfolio. We will ignore he small errors induced by his approximaion, jus as praciioners seem o. The immunizaion sraegy based on Macaulay duraions is hen defined by Eq. 12.14) where Macaulay duraions are used on he righ-hand side. Immunizaion sraegies based on he Fisher-Weil duraion can be consruced in a similar manner. In earlier secions of his chaper we have argued ha he Macaulay and Fisher-Weil risk measures are inappropriae for realisic yield curve movemens. Consequenly, immunizaion sraegies based on hose measures are likely o be ineffecive. Below we perform an experimen ha illusraes how far off he mark he radiional immunizaion sraegies are. 12.4.2 An experimenal comparison of immunizaion sraegies For simpliciy, le us consider an invesor who seeks o mach a paymen of 1000 dollars exacly 10 years from now. We assume ha he CIR model dr = κ[θ r ] d + β r dz = κθ [κ + λ]r ) d + β r dz Q wih he parameer values κ = 0.3, θ = 0.05, β = 0.1, and λ = 0.1 provides a correc descripion of he evoluion of he erm srucure of ineres raes. The asympoic long-erm yield y is hen 6.74%. The zero-coupon yield curve will be increasing if he curren shor rae is below 6.12% and decreasing if he curren shor rae is above 7.50%. For inermediae values of he shor rae, he yield curve will have a small hump. Duraion maching immunizaion In he following we will compare he effeciveness of duraion maching immunizaion sraegies based on he Macaulay duraion, he Fisher-Weil duraion, and he sochasic duraion derived from he CIR model. In addiion o he duraion measure applied, he immunizaion sraegy

12.4 Immunizaion 285 is characerized by he rebalancing frequency and by he securiies ha consiue he porfolio. We will consider sraegies wih 2, 12, and 52 equally spaced annual porfolio adjusmens. We consider only porfolios of wo bulle bonds of differen mauriies. The bonds have a coupon rae of 5%, one annual paymen dae, and exacly one year o he nex paymen dae. We assume ha he invesor is free o pick wo such bonds among he bonds ha have ime-o-mauriies in he se {1, 2,... } a he ime when he sraegy is iniiaed. We apply wo crieria for he choice of mauriies. One crierion is o choose he mauriy of one of he bonds so ha he Macaulay duraion of he bond is less han, bu as close as possible o, he Macaulay duraion of he liabiliy o be mached. The oher bond is chosen o be he bond wih Macaulay duraion above, bu as close as possible o, he Macaulay duraion of he liabiliy. This crierion has he nice implicaion ha he Macaulay convexiies of he porfolio and he liabiliy will be close, which should improve he effeciveness of periodically adjused immunizaion porfolios. We will refer o his crierion as he Macaulay crierion. The oher crierion is o choose a shor-erm bond and a long-erm bond so ha he convexiy of he porfolio will be significanly higher han he convexiy of he liabiliy. The shor-erm bond has a ime-o-mauriy of one year a he mos, while he long bond maures five years afer he liabiliy is due. We will refer o his crierion as he shor-long crierion. Irrespecive of he crierion used, we assume ha one year before he liabiliy is due, he porfolio is replaced by a posiion in he bond wih only one year o mauriy. Consequenly, he sraegies are no affeced by ineres rae movemens in he final year. The effeciveness of he differen immunizaion sraegies is sudied by performing 30000 simulaions of he evoluion of he yield curve in he CIR model over he 10-year period o he due dae of he liabiliy. In he simulaions we use 360 ime seps per year. Table 12.2 illusraes he effeciveness of he differen immunizaion sraegies. The lef par of he able conains resuls based on he Macaulay bond selecion crierion, whereas he righ par is based on he shor-long bond second crierion. In order o explain he numbers in he able, le us ake he righ-mos column as an example. The numbers in his column are from an immunizaion sraegy based on maching CIR duraions using a porfolio of a shor-erm and a long-erm bond. For wo annual porfolio adjusmens he average of he 30000 simulaed erminal porfolio values was 1000.01, which is very close o he desired value of 1000. The average absolue deviaion was 0.124% of he desired porfolio value. In 29.1% of he 30000 simulaed oucomes he absolue deviaion was less han 0.05%. In 53.3% of he simulaed oucomes he absolue deviaion was less han 0.1%, ec. 5 The sraegies based on he Macaulay and he Fisher-Weil duraions generae very similar resuls due o he fac ha hese duraion measures ypically are very close. The effeciveness of hese sraegies seems independen of he rebalancing frequency. The choice of bonds applied in he sraegy is more imporan. The deviaions from he arge are generally significanly larger for a porfolio of a shor and a long bond a high convexiy porfolio) han for a porfolio of bonds wih very similar mauriies a low convexiy porfolio). The high convexiy sraegy deviaes by more han five percen in more han half of all cases. The CIR sraegy of maching sochasic duraions is far more effecive han maching Macaulay or Fisher-Weil duraions. This can be seen boh from he average erminal porfolio value, he 5 Even wih 30000 simulaions he specific fraciles are quie uncerain, bu he averages are quie reliable. Experimens wih oher sequences of random numbers and a larger number of simulaions have resuled in very similar fraciles.

12.4 Immunizaion 286 2 porfolio adjusmens per year Macaulay crierion shor-long crierion Macaulay Fisher-Weil CIR Macaulay Fisher-Weil CIR Avg. erminal value 994.41 994.42 999.99 968.32 968.41 1000.01 Avg. absolue deviaion 1.28% 1.27% 0.072% 5.65% 5.60% 0.124% Dev. < 0.05% 2.2% 2.2% 45.2% 0.4% 0.4% 29.1% Dev. < 0.1% 4.3% 4.3% 76.2% 0.9% 0.8% 53.3% Dev. < 0.5% 21.5% 21.5% 99.7% 4.5% 4.5% 98.6% Dev. < 1.0% 42.4% 42.6% 100.0% 8.9% 8.9% 99.9% Dev. < 5.0% 99.6% 99.6% 100.0% 45.9% 46.2% 100.0% 12 porfolio adjusmens per year Macaulay crierion shor-long crierion Macaulay Fisher-Weil CIR Macaulay Fisher-Weil CIR Avg. erminal value 994.54 994.43 1000.00 968.62 968.53 1000.01 Avg. absolue deviaion 1.28% 1.27% 0.032% 5.61% 5.59% 0.053% Dev. < 0.05% 2.2% 2.2% 80.5% 0.4% 0.4% 59.6% Dev. < 0.1% 4.5% 4.4% 97.1% 0.9% 0.8% 86.0% Dev. < 0.5% 21.7% 21.5% 100.0% 4.5% 4.5% 100.0% Dev. < 1.0% 42.7% 42.7% 100.0% 9.0% 9.0% 100.0% Dev. < 5.0% 99.6% 99.6% 100.0% 46.4% 46.3% 100.0% 52 porfolio adjusmens per year Macaulay crierion shor-long crierion Macaulay Fisher-Weil CIR Macaulay Fisher-Weil CIR Avg. erminal value 994.50 994.47 1000.00 968.55 968.53 1000.01 Avg. absolue deviaion 1.27% 1.26% 0.015% 5.61% 5.58% 0.026% Dev. < 0.05% 2.1% 2.2% 97.3% 0.4% 0.4% 87.1% Dev. < 0.1% 4.3% 4.4% 99.9% 0.9% 0.9% 98.6% Dev. < 0.5% 21.2% 21.3% 100.0% 4.3% 4.3% 100.0% Dev. < 1.0% 42.7% 42.8% 100.0% 8.7% 8.6% 100.0% Dev. < 5.0% 99.7% 99.7% 100.0% 46.8% 47.0% 100.0% Table 12.2: Resuls from he immunizaion of a 10-year liabiliy based on 30000 simulaions of he CIR model. The curren shor rae is 5%.

12.4 Immunizaion 287 average absolue deviaion, and he lised fraciles from he disribuion of he absolue deviaions. Even wih jus wo annual porfolio adjusmens he CIR sraegy will miss he arge by less han 0.5 percen in more han 98% of all oucomes, no maer which bond selecion crierion is used. The Macaulay and Fisher-Weil sraegies miss he mark by more han one percen in more han 50% of all oucomes, even when he bonds are seleced according o heir Macaulay duraions. Clearly, he effeciveness of he CIR sraegy increases wih he frequency of he porfolio adjusmens. In paricular, frequen rebalancing is advanageous if he immunizaion porfolio has a relaively high convexiy. However, he effeciveness of he CIR sraegy seems o depend less on he bonds chosen han does he effeciveness of he radiional sraegies. Simulaions using oher iniial shor raes and herefore differen iniial yield curves have shown ha he average erminal value of he Macaulay sraegy is highly dependen on he iniial shor rae. For a nearly fla iniial yield curve he average erminal value is very close o he argeed value of 1000, bu he average absolue deviaion is no smaller han for oher iniial yield curves. The CIR sraegy is far more effecive han he Macaulay sraegy, also for a nearly fla iniial yield curve. The effeciveness of he sraegies decreases wih he curren ineres rae level due o he fac ha he ineres rae volailiy is assumed o increase wih he level in he CIR model. Furhermore, he accuracy of he immunizaion sraegies will ypically be decreasing in β and θ and increasing in κ. Duraion and convexiy maching immunizaion sraegies In he following we consider he case where boh he duraion and he convexiy of he liabiliy are mached by he invesmen porfolio. In our experimen we assume ha he porfolio consiss of a bond wih a ime-o-mauriy of a mos one year, a bond mauring wo years afer he liabiliy is due, and a bond mauring en years afer he liabiliy is due. Table 12.3 illusraes he gain in efficiency by maching boh duraion and convexiy insead of jus maching duraion using a porfolio of he shor and he long bond. For he Macaulay sraegy he average deviaion is reduced by a facor 10. The Fisher-Weil sraegy generaes almos idenical resuls and is herefore omied. For he CIR sraegy he relaive improvemen is even more dramaic and in all of he 30000 simulaed oucomes he deviaion is less han 0.05% alhough he porfolio is only rebalanced once a monh! The numbers under he column heading Hull-Whie are explained below. Model uncerainy The resuls above clearly show ha if he CIR model gives a correc descripion of he erm srucure dynamics, an immunizaion sraegy based on he CIR risk measures is far more effecive han sraegies based on he radiional risk measures. However, if he CIR model does no provide a good descripion of he evoluion of he erm srucure, an immunizaion sraegy based on he sochasic duraions compued using he CIR model will be less successful. Since he CIR model in any case is closer o he rue dynamics of he shor rae han he Meron model underlying he Fisher-Weil duraion, he CIR sraegy is sill expeced o be more effecive han radiional sraegies. Our analysis indicaes ha for immunizaion purposes i is imporan o apply risk measures ha are relaed o he dynamics of he erm srucure. Therefore, i is imporan o idenify an empirically reasonable model and hen o implemen immunizaion sraegies and hedge sraegies

12.5 Risk measures in muli-facor diffusion models 288 12 porfolio adjusmens per year Idenical duraions Idenical duraions and convexiies Macaulay CIR Hull-Whie Macaulay CIR Hull-Whie Avg. erminal value 968.62 1000.01 1005.39 997.11 1000.00 999.78 Avg. absolue deviaion 5.61% 0.053% 0.98% 0.57% 0.0002% 0.33% Dev. < 0.05% 0.4% 59.6% 2.5% 4.7% 100.0% 10.1% Dev. < 0.1% 0.9% 86.0% 5.0% 9.5% 100.0% 19.7% Dev. < 0.5% 4.5% 100.0% 26.1% 47.6% 100.0% 77.6% Dev. < 1.0% 9.0% 100.0% 53.4% 86.9% 100.0% 97.5% Dev. < 5.0% 46.4% 100.0% 100.0% 100.0% 100.0% 100.0% Table 12.3: Resuls from he immunizaion of a 10-year liabiliy of 1000 dollars based on 30000 simulaions of he CIR model. The curren shor rae is 5%. in general) based on he relevan risk measures associaed wih he model. How effecive is an immunizaion sraegy based on risk measures associaed wih a model which does no give a correc descripion of he yield curve dynamics? To invesigae his issue, we assume ha he CIR model is correc, bu ha he immunizaion sraegy is consruced using risk measures from he Hull-Whie model he exended Vasicek model). Jus before each porfolio adjusmen he Hull-Whie model is calibraed o he rue yield curve, i.e. he yield curve of he CIR model. Table 12.3 shows he resuls of such an immunizaion sraegy. As expeced he sraegy is far less effecive han he sraegy based on he rue yield curve dynamics, bu he Hull- Whie sraegy is sill far beer han he radiional Macaulay sraegy. So even hough we base our immunizaion sraegy on a model which, in some sense, is far from he rue model, we sill obain a much more effecive immunizaion han we would by using he radiional immunizaion sraegy. 12.5 Risk measures in muli-facor diffusion models 12.5.1 Facor duraions, convexiies, and ime value In muli-facor diffusion models i is naural o measure he sensiiviy of a securiy price wih respec o changes in he differen sae variables. Le us consider a wo-facor diffusion model where he sae variables x 1 and x 2 are assumed o develop as 12.17) 12.18) dx 1 = α 1 x 1, x 2 ) d + β 11 x 1, x 2 ) dz 1 + β 12 x 1, x 2 ) dz 2, dx 2 = α 2 x 1, x 2 ) d + β 21 x 1, x 2 ) dz 1 + β 22 x 1, x 2 ) dz 2. For a securiy wih he price B = Bx 1, x 2, ), Iô s Lemma implies ha 12.19) db B =... d D 1 x 1, x 2, ) [β 11 x 1, x 2 ) dz 1 + β 12 x 1, x 2 ) dz 2 ] D 2 x 1, x 2, ) [β 21 x 1, x 2 ) dz 1 + β 22 x 1, x 2 ) dz 2 ],

12.5 Risk measures in muli-facor diffusion models 289 where we have omied he drif and inroduced he noaion 1 B D 1 x 1, x 2, ) = x 1, x 2, ), Bx 1, x 2, ) x 1 1 B D 2 x 1, x 2, ) = x 1, x 2, ). Bx 1, x 2, ) x 2 We will refer o D 1 and D 2 as he facor duraions of he securiy. In such a wo-facor model any ineres rae dependen securiy can be perfecly replicaed by a porfolio ha always has he same facor duraions as he given securiy. Again coninuous rebalancing is needed. In he pracical implemenaion of hedge sraegies i is relevan o include second-order derivaives jus as we did in he one-facor models above. In a wo-facor model we have hree relevan second-order derivaives ha lead o he following facor convexiies: Defining he ime value as we ge he following relaion: 1 2 B K 1 x 1, x 2, ) = 2Bx 1, x 2, ) x 2 x 1, x 2, ), 1 1 2 B K 2 x 1, x 2, ) = 2Bx 1, x 2, ) x 2 x 1, x 2, ), 2 1 2 B K 12 x 1, x 2, ) = x 1, x 2, ). Bx 1, x 2, ) x 1 x 2 Θx 1, x 2, ) = 1 B Bx 1, x 2, ) x 1, x 2, ), Θx 1, x 2, ) ˆα 1 x 1, x 2 )D 1 x 1, x 2, ) ˆα 2 x 1, x 2 )D 2 x 1, x 2, ) + γ 1 x 1, x 2 ) 2 K 1 x 1, x 2, ) + γ 2 x 1, x 2 ) 2 K 2 x 1, x 2, ) + γ 12 x 1, x 2 )K 12 x 1, x 2, ) = rx 1, x 2 ). Here he erms γ1 2 = β11 2 + β12 2 and γ2 2 = β21 2 + β22 2 are he variance raes of changes in he firs and he second sae variables, and γ 12 = β 11 β 21 + β 12 β 22 is he covariance rae beween hese changes. In a wo-facor affine model he prices of zero-coupon bonds are of he form B T x 1, x 2, ) = e at ) b1t )x1 b2t )x2. Therefore, he facor duraions of a zero-coupon bond are given by D j x 1, x 2, ) = b j T ) for j = 1, 2. For a coupon bond wih he price Bx 1, x 2, ) = T Y i> ib Ti x 1, x 2, ) he facor duraions are 1 B D j x 1, x 2, ) = x 1, x 2, ) = wx 1, x 2,, T i )b j T i ), Bx 1, x 2, ) x j T i> where wx 1, x 2,, T i ) = Y i B Ti x 1, x 2, )/Bx 1, x 2, ). The convexiies and he ime value are K j x 1, x 2, ) = wx 1, x 2,, T i )b j T i ) 2, j = 1, 2, T i> K 12 x 1, x 2, ) = T i> wx 1, x 2,, T i )b 1 T i )b 2 T i ), Θx 1, x 2, ) = T i> wx 1, x 2,, T i ) a T i ) + b 1T i )x 1 + b 2T i )x 2 ).

12.5 Risk measures in muli-facor diffusion models 290 The facor duraions defined above can be ransformed ino ime-denominaed facor duraions in he following manner. For each sae variable or facor j we define he ime-denominaed facor duraion D j = D j x 1, x 2, ) as he ime-o-mauriy of he zero-coupon bond wih he same price sensiiviy and hence he same facor duraion relaive o his sae variable: 1 B 1 x 1, x 2, ) = Bx 1, x 2, ) x j B +D j x1, x 2, ) In an affine model his equaion reduces o so ha B +D j x j wx 1, x 2,, T i )b j T i ) = b j Dj ) T i> D j = D j x 1, x 2, ) = b 1 j under he assumpion ha b j is inverible. x 1, x 2, ). ) wx 1, x 2,, T i )b j T i ) T i> 12.5.2 One-dimensional risk measures in muli-facor models For pracical purposes i may be relevan o summarize he risks of a given securiy in a single one-dimensional) risk measure. The volailiy of he securiy is he mos naural choice. By definiion he volailiy of a securiy is he sandard deviaion of he rae of reurn on he securiy over he nex insan. In he wo-facor model given by 12.17) and 12.18) he variance of he rae of reurn can be compued from 12.19): ) db Var = Var [D 1 β 11 + D 2 β 21 ] dz 1 + [D 1 β 12 + D 2 β 22 ] dz 2 ) B = [D 1 β 11 + D 2 β 21 ] 2 + [D 1 β 12 + D 2 β 22 ] 2) d = D 2 1γ 2 1 + D 2 2γ 2 2 + 2D 1 D 2 γ 12 ) d, where we for noaional simpliciy have omied he argumens of he D- and β-funcions. The volailiy is hus given by σ B x 1, x 2, ) = D 1 x 1, x 2, ) 2 γ 1 x 1, x 2 ) 2 + D 2 x 1, x 2, ) 2 γ 2 x 1, x 2 ) 2 + 2D 1 x 1, x 2, )D 2 x 1, x 2, )γ 12 x 1, x 2 )) 1/2. Also his risk measure can be ransformed ino a ime-denominaed risk measure, namely he ime-o-mauriy of he zero-coupon bond having he same volailiy as he securiy considered. Leing σ T x 1, x 2, ) denoe he volailiy of he zero-coupon bond mauring a ime T, he imedenominaed duraion D x 1, x 2, ) is given as he soluion D = D x 1, x 2, ) o he equaion σ B x 1, x 2, ) = σ +D x 1, x 2, ) or, equivalenly, σ B x 1, x 2, ) 2 = σ +D x 1, x 2, ) 2.

12.5 Risk measures in muli-facor diffusion models 291 mauriy, years price yield D Mac D FW D1 D2 D 1 99.83 5.18% 1.00 1.00 1.00 1.00 1.00 2 98.94 5.58% 1.95 1.95 1.94 1.21 1.94 3 97.60 5.90% 2.86 2.86 2.81 1.21 2.81 4 96.01 6.16% 3.72 3.71 3.59 1.21 3.59 5 94.30 6.37% 4.53 4.52 4.25 1.21 4.25 6 92.56 6.54% 5.30 5.29 4.79 1.21 4.79 8 89.20 6.79% 6.70 6.68 5.52 1.20 5.52 10 86.15 6.97% 7.94 7.89 5.87 1.20 5.87 12 83.45 7.09% 9.01 8.94 5.99 1.20 5.99 15 80.07 7.22% 10.36 10.23 6.00 1.20 6.00 20 75.88 7.34% 11.99 11.75 5.89 1.19 5.89 Table 12.4: Duraion measures for 5% bulle bonds wih one annual paymen dae assuming he Longsaff-Schwarz model wih he parameer values β1 2 = 0.005, β2 2 = 0.0814, κ 1 = 0.3299, ˆκ 2 = 14.4277, ϕ 1 = 0.020112, and ϕ 2 = 0.26075 provides a correc descripion of he yield curve dynamics. The curren shor rae is r = 0.05 wih an insananeous variance rae of v = 0.002. This equaion can only be solved numerically. For an affine wo-facor model he equaion is of he form ) 2 ) 2 ) ) w i b 1 T i ) γ1+ 2 w i b 2 T i ) γ2+2 2 w i b 1 T i ) w i b 2 T i ) T i> T i> T i> T i> = b 1 D ) 2 γ1 2 + b 2 D ) 2 γ2 2 + 2b 1 D )b 2 D )γ 12, where we again have simplified he noaion, e.g. w i represens wx 1, x 2,, T i ). Some basic properies of he ime-denominaed duraion were derived by Munk 1999). The ime-denominaed duraion is a heoreically beer founded one-dimensional risk measure han he radiional Macaulay and Fisher-Weil duraions. Furhermore, he ime-denominaed duraion is closely relaed o he volailiy concep, which mos invesors are familiar wih. Table 12.4 liss differen duraion measures based on he wo-facor model of Longsaff and Schwarz 1992a) sudied in Secion 8.5.2 on page 214. The parameers of he models are fixed a he values esimaed by Longsaff and Schwarz 1992b), which generaes a reasonable disribuion of he fuure values of he sae variables r and v. For 5% bulle bonds of differen mauriies he able shows he price, he yield, he Macaulay duraion D Mac, he Fisher-Weil duraion D FW, he ime-denominaed facor duraions D1 and D2, and he one-dimensional ime-denominaed duraion D. Also in his case he radiional duraion measures are overesimaing he risk of long-erm bonds. Also noe ha wih he parameer values applied in he compuaions, he firs ime-denominaed facor duraion D1 and he one-dimensional ime-denominaed duraion D are basically idenical. The reason is ha he sensiiviy o he second facor depends very lile on he ime-o-mauriy. This is no necessarily he case for oher parameer values. γ 12

12.6 Duraion-based pricing of opions on bonds 292 12.6 Duraion-based pricing of opions on bonds 12.6.1 The general idea In he framework of one-facor diffusion models Wei 1997) suggess ha he price of a European call opion on a coupon bond can be approximaed by he price of a European call opion on a paricular zero-coupon bond, namely he zero-coupon bond having he same sochasic) duraion as he coupon bond underlying he opion o be priced. According o Secion 2.9.2, his approximaion can also be applied o he pricing of European swapions. As usual, we le C K,T,S be he ime price of a European call opion wih expiraion ime T and exercise price K, wrien on a zero-coupon bond mauring a ime S > T. Furhermore, C K,T,cpn is he ime price of a European call opion wih expiraion ime T and exercise price K, wrien on a given coupon bond. We denoe by B he ime value of he paymens of he coupon bonds afer expiraion of he opion, i.e. B = T Y i>t ib Ti where Y i is he paymen a ime T i. Wei s approximaion is hen given by he following relaion: 12.20) C K,T,cpn K,T,cpn C = B C K,T,+D B +D, where K = KB +D /B, and where D denoes he ime-denominaed duraion of he cash flow of he underlying coupon bonds afer expiraion of he opion. Wei does no moivae he approximaion, bu shows by numerical examples in he one-facor models of Vasicek 1977) and Cox, Ingersoll, and Ross 1985b) ha he approximaion is very accurae. The advanage of using he approximaion in hese wo models is ha only he price of one call opion on a zero-coupon bond needs o be compued. To apply Jamshidian s rick see Secion 7.2.3 on page 169) we have o compue a zero-coupon bond opion price for each of he paymen daes of he coupon bond afer he expiraion dae of he opion. In addiion, one equaion in one unknown has o be solved numerically o deermine he criical ineres rae r. Neverheless, he exac price can be very quickly compued by Jamshidian s formula, bu if many opions on coupon bonds or swapions) have o priced, he slighly faser approximaion may be relevan o use. The inuiion behind he accuracy of he approximaion is ha he underlying zero-coupon bond of he approximaing opion is chosen o mach he volailiy of he underlying coupon bond for he opion we wan o price. Since we know ha he volailiy of he underlying asse is an exremely imporan facor for he price of an opion, his choice makes good sense. Munk 1999) sudies he approximaion in more deail, gives an analyical argumen for is accuracy, and illusraes he precision in muli-facor models in several numerical examples. Noe ha he compuaional advanage of using he approximaion is much bigger in muli-facor models han in one-facor models since no explici formula for European opions on coupon bonds has been found for any muli-facor model. Whereas he alernaive o he approximaion in he one-facor models is a slighly more complicaed explici expression, he alernaive in he mulifacor models is o use a numerical echnique, e.g. Mone Carlo simulaion or numerical soluion of he relevan muli-dimensional parial differenial equaion. Below we go hrough he analyical argumen for he applicabiliy of he approximaion. Afer ha we will illusrae he accuracy of he approximaion in numerical examples.

12.6 Duraion-based pricing of opions on bonds 293 I should be noed ha several oher echniques o approximaing prices of European opions on coupon bonds have been suggesed in he lieraure. For example, in he framework of affine models Collin-Dufresne and Goldsein 2001) and Singleon and Umansev 2002) inroduce wo approximaions ha appear o dominae wih respec o accuracy and compuaional speed) he duraion-based approach discussed here, bu hese approximaions are much harder o undersand. 12.6.2 A mahemaical analysis of he approximaion Le us firs sudy he error in using he approximaion 12.21) C K,T,cpn B B S C KS,T,S, where S is any given mauriy dae of he underlying zero-coupon bond of he approximaing opion, and where K S = KB S /B. Aferwards we will argue ha he error will be small when S = + D, which is exacly he approximaion 12.20). Boh he correc opion price and he price of he approximaing opion can be wrien in erms of expeced values under he S-forward maringale measure Q S. Under his measure he price of any asse relaive o he zero-coupon bond price B S is by definiion a maringale. Hence, he correc price of he opion can be wrien as C KS,T,S = B S E QS C K,T,cpn = B S E QS while he price of he approximaing opion is ) max BT S KBS B, 0 B S T [ maxbt K, 0) B S T = B S E QS ], [ )] max 1 KBS B BT S, 0. The dollar error incurred by using he approximaion 12.21) is herefore equal o C K,T,cpn B [ ] B S C KS,T,S = B S maxbt K, 0) E QS BT S B [ B S E QS max 1 KBS B BT S [ = B S BT E QS max BT S K ) B BT S, 0 max B S K )] 12.22) BT S, 0. From he definiion of he S-forward maringale measure i follows also ha [ ] BT 12.23) E QS = B B S B S T )]), 0 and ha 12.24) E QS [ ] K B S T = KBT B S. For deep-in-he-money call opions boh max-erms in 12.22) will wih a high probabiliy reurn he firs argumen, and i follows hen from 12.23) ha he dollar error will be close o zero. Since he opion price in his case is relaively high, he percenage error will be very close o zero. For deep-ou-of-he-money call opions boh max-erms will wih a high probabiliy reurn zero so ha he dollar error again is close o zero. The opion price will also be close o zero, so he percenage error may be subsanial.

12.6 Duraion-based pricing of opions on bonds 294 The error is due o he oucomes where only one and no boh max-erms is differen from zero. This will be he case when he realized values of B T and B S T are such ha he raio K/BS T lies beween B T /B S T and B /B S. As indicaed by 12.23) and 12.24) his affecs he value of forward near-he-money opions where B KB T. We will herefore expec he dollar pricing errors o be larges for such opions. The consideraions above are valid for any S. In order o reduce he probabiliy of ending up in he oucomes ha induce he error, we seek o choose S so ha B T /B and BT S/BS are likely o end up close o each oher. As a firs aemp o achieve his we could ry o pick S such ha [ ] he variance Var QS BT /B BT S/BS is minimized, bu his idea is no implemenable due o he ypically very complicaed expressions for BT S and, in paricular, for B T. Alernaively, we can choose S so ha he relaive changes in B and B S over he nex insan are close o each oher. This is exacly wha we achieve by using S = + D. Anoher promising choice is S = T mv which is he value of S ha minimizes [ he ] variance of he difference in he relaive price change over he nex insan, i.e. Var QS db B dbs. This idea B S also gives rise o an alernaive ime-denominaed duraion measure, D mv = T mv, which we could call he variance-minimizing duraion. I can be shown see Munk 1999)) ha for onefacor models he wo duraion measures are idenical, D = D mv. In muli-facor models he wo measures will ypically be close o each oher, and consequenly he accuracy of he approximaion will ypically be he same no maer which duraion measure is used o fix he mauriy of he zero-coupon bond. In he exreme cases where he measures differ significanly, he approximaion based on D seems o be more accurae. Noe ha he analysis in his subsecion applies o all erm srucure models. We have no assumed ha he evoluion of he erm srucure can be described by a one-facor diffusion model. Therefore we can expec he approximaion o be accurae in all models. Below we will invesigae he accuracy of he approximaion in a specific erm srucure model, namely he wo-facor model of Longsaff and Schwarz discussed in Secion 8.5.2. These resuls are aken from Munk 1999), who also presens similar resuls for a wo-facor Gaussian Heah-Jarrow-Moron model see Chaper 10 for an inroducion o hese models). Wei 1997) sudies he accuracy of he approximaion in he one-facor models of Vasicek and of Cox, Ingersoll, and Ross. 12.6.3 The accuracy of he approximaion in he Longsaff-Schwarz model According o 8.43), he price of a European call opion on a zero-coupon bond in he Longsaff- Schwarz model can be wrien as C K,T,S = B S χ 2 1 KB T χ 2 2, where χ 2 1 and χ 2 2 are wo probabiliies aken from he wo-dimensional non-cenral χ 2 -disribuion. No explici formula for he price of a European call opion on a coupon bond has been found. Consequenly, an approximaion like 12.20) will be very valuable if i is sufficienly accurae. To esimae he accuracy, we will compare he approximae price K,T,cpn C o a correc price C K,T,cpn compued using Mone Carlo simulaion. 6 Of course, in he pracical use of he 6 The resuls shown are based on simulaions of 10000 pairs of aniheic sample pahs of he wo sae variables r and v. The ime period unil he expiraion dae of he opion is divided ino approximaely 100 subinervals per year.

12.6 Duraion-based pricing of opions on bonds 295 wo-monh opions six-monh opions K appr. price abs. dev. rel. dev. sd. dev. K appr. price abs. dev. rel. dev. sd. dev. 86 5.08407 0.1 10 5 0.000% 1.8 10 4 91 4.45364 2.0 10 5 0.000% 4.8 10 4 87 4.10368 0.2 10 5 0.000% 1.7 10 4 92 3.53250 4.9 10 5 0.001% 4.0 10 4 88 3.12553 0.7 10 5 0.000% 1.5 10 4 93 2.63434 8.2 10 5 0.003% 3.4 10 4 89 2.16242 2.1 10 5 0.001% 1.2 10 4 94 1.79287 9.6 10 5 0.005% 3.2 10 4 90 1.26608 3.2 10 5 0.003% 1.0 10 4 95 1.06678 5.5 10 5 0.005% 3.1 10 4 91 0.56030 0.7 10 5 0.001% 0.9 10 4 96 0.52036-3.3 10 5-0.006% 2.4 10 4 92 0.15992-2.7 10 5-0.017% 0.7 10 4 97 0.19074-9.6 10 5-0.050% 2.0 10 4 93 0.02442-2.0 10 5-0.083% 0.7 10 4 98 0.04576-7.7 10 5-0.168% 2.0 10 4 94 0.00163-0.4 10 5-0.253% 0.4 10 4 99 0.00576-2.7 10 5-0.474% 1.5 10 4 95 0.00001-0.0 10 5-1.545% 0.1 10 4 100 0.00021-0.2 10 5-1.051% 0.5 10 4 Table 12.5: Prices of wo- and six-monh European call opions on a wo-year bulle 8% bond in he Longsaff-Schwarz model. The underlying bond has a curren price of 89.3400, a wo-monh forward price of 91.2042, a six-monh forward price of 95.7687, and a ime-denominaed sochasic duraion of 1.9086 years. approximaion he approximae price will be compued using he explici formula for he price of he opion on he zero-coupon bond. Bu o make a fair comparison, we will compue he approximae price using he same simulaed sample pahs as used for compuing he correc opion price. In his way our evaluaion of he approximaion is no sensiive o a possible bias in he correc price induced by he simulaion echnique. We will consider European call opions wih an expiraion ime of wo or six monhs wrien on an 8% bulle bond wih a single annual paymen dae and a ime-o-mauriy of wo or en years. The parameers in he dynamics of he sae variables, see 8.40) and 8.41), are aken o be β1 2 = 0.01, β2 2 = 0.08, ϕ 1 = 0.001, ϕ 2 = 1.28, κ 1 = 0.33, κ 2 = 14, and λ = 0. These values are close o he parameer values esimaed by Longsaff and Schwarz in heir original aricle. The curren shor rae is assumed o be r = 0.08 wih an insananeous variance of v = 0.002. The accuracy of he approximaion does no seem o depend on hese values in any sysemaic way. Table 12.5 liss resuls for opions on he wo-year bond for various exercise prices around he forward-a-he-money value of K, i.e. B /B T. The corresponding resuls for opions on he en-year bond are shown in Table 12.6. The absolue deviaion shown in he ables is defined as he approximae price minus he correc price, whereas he relaive deviaion is compued as he absolue deviaion divided by he correc price. The ables also show he sandard deviaion of he simulaed difference beween he correc and he approximae price. All he approximae prices are correc o hree decimals, and he percenage deviaions are also very, very small. In all cases he absolue deviaion is considerably smaller han he sandard deviaion of he Mone Carlo simulaed differences. Based on he mahemaical analysis of he approximaion we expec he errors o be smaller for shorer mauriies of he opion and he underlying bond han for longer mauriies. This expecaion is confirmed by our examples. Also in line wih our discussion, we see ha he absolue deviaion is larges for forward-near-he-money

12.7 Alernaive measures of ineres rae risk 296 wo-monh opions six-monh opions K appr. price abs. dev. rel. dev. sd. dev. K appr. price abs. dev. rel. dev. sd. dev. 74 4.42874 1.2 10 4 0.003% 1.9 10 3 78 4.27344 1.1 10 3 0.027% 4.4 10 3 75 3.46569 2.5 10 4 0.007% 1.6 10 3 79 3.42836 1.3 10 3 0.037% 4.3 10 3 76 2.53643 3.9 10 4 0.015% 1.4 10 3 80 2.64289 1.2 10 3 0.045% 4.3 10 3 77 1.69005 4.0 10 4 0.024% 1.4 10 3 81 1.93654 0.8 10 3 0.042% 4.2 10 3 78 0.98799 1.8 10 4 0.018% 1.3 10 3 82 1.33393 0.2 10 3 0.015% 3.8 10 3 79 0.48542-1.7 10 4-0.036% 1.0 10 3 83 0.85064-0.5 10 3-0.063% 3.1 10 3 80 0.19080-3.9 10 4-0.202% 0.9 10 3 84 0.49430-1.1 10 3-0.220% 2.8 10 3 81 0.05666-3.2 10 4-0.570% 0.9 10 3 85 0.25641-1.3 10 3-0.508% 2.6 10 3 82 0.01267-1.6 10 4-1.263% 0.8 10 3 86 0.11491-1.2 10 3-1.001% 2.7 10 3 83 0.00185-0.5 10 4-2.424% 0.5 10 3 87 0.04372-0.8 10 3-1.786% 2.6 10 3 Table 12.6: Prices on wo- and six-monh European call opions on a en-year bulle 8% bond in he Longsaff-Schwarz model. The underlying bond has a curren price of 76.9324, a wo-monh forward price of 78.5377, a six-monh forward price of 82.4682, and a ime-denominaed sochasic duraion of 4.8630 years. opions and smalles for deep-in- and deep-ou-of-he-money opions. Figure 12.1 illusraes how he precision of he approximaion depends on he exercise price for differen ime-o-mauriies of he zero-coupon bond underlying he approximaing opion. The figure is based on wo-monh opions on he wo-year bulle bond, bu a similar picure can be drawn for he oher opions considered. For deep-ou-of- and deep-in-he-money opions he approximaion is very accurae no maer which zero-coupon bond is used in he approximaion, bu for near-he-money opions i is imporan o choose he righ zero-coupon bond, namely he zero-coupon bond wih a ime-o-mauriy equal o he ime-denominaed sochasic duraion of he underlying coupon bond. Also hese resuls are consisen wih he analyical argumens and he discussion in he preceding subsecion. 12.7 Alernaive measures of ineres rae risk In his chaper we have focused on measures of ineres rae risk in arbirage-free dynamic diffusion models of he erm srucure. Similar risk measures can be defined in Heah-Jarrow- Moron HJM) models and marke models which, as discussed in Chapers 10 and 11, do no necessarily fi ino he diffusion seing. In an HJM model where all he insananeous forward raes are affeced by a single Brownian moion, df T = α, T, f s ) s ) d + β, T, f s ) s ) dz, he price of any fixed income securiy will have a dynamics of he form db B = µ B, f s ) s ) d + σ B, f s ) s ) dz.

12.7 Alernaive measures of ineres rae risk 297 0.08 0.06 absolue pricing error 0.04 0.02 0-0.02-0.04-0.06-0.08-0.1 84 86 88 90 92 94 96 98 exercise price 1.5 1.75 D*) 2.0 2.25 Figure 12.1: The absolue price errors for wo-monh opions on wo-year bulle bonds in he Longsaff-Schwarz-model for differen mauriies of he zero-coupon bond underlying he approximaing opion. The ime-denominaed sochasic duraion of he underlying coupon bond is D = 1.9086 years, and he wo-monh forward price of he coupon bond is 91.2042. Here he volailiy σ B is an obvious candidae for measuring he ineres rae risk of he securiy, and he ime-denominaed duraion D = D, f s ) s ) can be defined implicily by he equaion σ B, f s ) s ) = σ +D, f s ) s ), where σ T, f s ) s ) = T β, u, f s ) s ) du is he volailiy of he zero-coupon bond mauring a ime T, cf. Theorem 10.1 on page 242. 7 Similar risk measures can be defined for HJM models involving more han one Brownian moion. In he more pracically oriened par of he lieraure several alernaive measures of ineres rae risk have been suggesed. A seemingly popular approach is o use he so-called key rae duraions inroduced by Ho 1992). The basic idea is o selec a number of key ineres raes, i.e. zero-coupon yields for cerain represenaive mauriies, e.g. 1, 2, 5, 10, and 20 years. A change in one of hese key raes is assumed o affec he yields for nearby mauriies. For example, wih he key raes lised above, a change in he wo-year zero-coupon yield is assumed o affec all zero-coupon yields of mauriies beween one year and five years. The change in hose yields is assumed o be proporional o he mauriy disance o he key rae. For example, a change of 0.01 100 basis poins) in he wo-year rae is assumed o cause a change of 0.005 50 basis poins) in he 1.5-year rae since 1.5 years is halfway beween wo years and he preceding key rae mauriy of one year. Similarly, he change in he wo-year rae is assumed o cause a change of approximaely 0.0033 33 basis poins) in he four-year rae. A simulaneous change in several key raes will cause a piecewise linear change in he enire yield curve. Wih sufficienly many key raes, any yield curve change can be well approximaed in his way. I is relaively simple o measure he 7 If β is posiive, he volailiy of he zero-coupon bond is sricly speaking σ T.

12.7 Alernaive measures of ineres rae risk 298 sensiiviies of zero-coupon and coupon bonds wih respec o changes in he key raes. These sensiiviies are called he key rae duraions. When differen bonds and posiions in derivaive securiies are combined, he oal key rae duraions of a porfolio can be conrolled so ha he invesor can hedge agains or speculae in) specific yield curve movemens. The key rae duraions are easy o compue and relae o, bu here are several pracical and heoreical problems in applying hese duraions. The individual key raes do no move independenly, and hence we have o consider which combinaions of key rae changes ha are realisic and do no conflic wih he no-arbirage principle. Furhermore, o evaluae he ineres rae risk of a securiy or a porfolio, we mus specify he probabiliy disribuion of he possible key rae changes. Praciioners ofen assume ha he changes in he differen key raes can be described by a muli-variae normal disribuion and esimae he means, variances, and covariances of he disribuion from hisorical daa. While he normal disribuion is very racable, empirical sudies canno suppor such a disribuional assumpion. An invesor who believes ha he yield curve dynamics can be represened by he evoluion of some seleced key raes should use a heoreically beer founded model for pricing and risk managemen, e.g. an arbirage-free dynamic model using hese key raes as sae variables, cf. he shor discussion in Secion 8.6.4 on page 222. In such a model all yield curve movemens are consisen wih he no-arbirage principle. Furhermore, for he poins on he yield curve ha lie in beween he key rae mauriies, such a model will give a more reasonable descripion han does he simple linear inerpolaion assumed in he compuaion of he key rae duraions. Furhermore, he model can be specified using relaively few parameers and sill provide a good descripion of he covariance srucure of he key raes. Oher auhors sugges duraion measures ha represen he price sensiiviy owards changes in he level, he slope, and he curvaure of he yield curve, see e.g. Willner 1996) and Phoa and Shearer 1997). This seems like a good idea since hese facors empirically provide a good descripion of he shape and movemens of he yield curve, cf. he discussion in Secion 8.1. However, also hese duraion measures should be compued in he seing of a realisic, arbirage-free dynamics of hese characerisic variables. This can be ensured by consrucing a erm srucure model using hese facors as sae variables.

Chaper 13 Morgage-backed securiies Sandard morgage-backed bonds: Dunn and McConnell 1981), Schwarz and Torous 1989, 1992), Boudoukh, Whielaw, Richardson, and Sanon 1997), LeRoy 1996), Deng, Quigley, and Van Order 2000) Collaeralized Morgage Obligaions CMO s): McConnell and Singh 1994), Childs, O, and Riddiough 1996) 299

Chaper 14 Credi risky securiies Bonds issued by corporaions are credi risky. Two ypes of models are used for pricing corporae bonds. Srucural models ha are based on assumpions abou he specific issuing firm, e.g. an uncerain flow of earnings and a given or opimally derived capial srucure: Meron 1974), Black and Cox 1976), Shimko, Tejima, and van Devener 1993), Leland 1994), Longsaff and Schwarz 1995), Goldsein, Ju, and Leland 2001), Chrisensen, Flor, Lando, and Milersen 2002). Reduced form models: Jarrow, Lando, and Turnbull 1997), Lando 1998), Duffie and Singleon 1999). 300

Chaper 15 Sochasic ineres raes and he pricing of sock and currency derivaives 15.1 Inroducion In he preceding chapers we have focused on securiies wih paymens and values ha only depend on he erm srucure of ineres raes, no on any oher random variables. However, he shape and he dynamics of he yield curve will also affec he prices of securiies wih paymens ha depend on oher random variables, e.g. sock prices and currency raes. The reason is ha he presen value of a securiy involves he discouning of he fuure paymens, and he appropriae discoun facors depend on he ineres rae uncerainy and he correlaions beween ineres raes and he random variables ha deermine he paymens of he securiy. We will in his chaper firs consider he pricing of sock opions when we allow for he uncerain evoluion of ineres raes, in conras o he classical Black-Scholes-Meron model. We show ha for Gaussian erm srucure models he price of a European sock opion is given by a simple generalizaion of he Black-Scholes-Meron formula. This generalized formula corresponds o he way which praciioners ofen implemen he Black-Scholes-Meron formula. In he Gaussian ineres rae seing we will also derive similar pricing formulas for European opions on forwards and fuures on socks. Subsequenly, we consider securiies wih paymens relaed o a foreign exchange rae. Wih a lognormal foreign exchange rae and Gaussian ineres raes we obain simple expressions for currency fuures prices and European currency opion prices. Throughou he chaper we focus on European call opions. The prices of he corresponding European pu opions follow from he relevan version of he pu-call pariy. As always, o price American opions we generally have o resor o numerical mehods. 15.2 Sock opions 15.2.1 General analysis Le us look a a European call opion ha expires a ime T >, is wrien on a sock wih price process S ), and has an exercise price on K. We know from Chaper 5 ha he ime price of his opion is given by 15.1) C = B T E QT [max S T K, 0)], 301

15.2 Sock opions 302 where Q T is he T -forward maringale measure. For simpliciy we assume ha he underlying asse does no provide any paymens in he life of he opion. The forward price of he underlying asse for delivery a dae T is given by F T = S /B T. In paricular, F T T = S T so ha he opion price can be rewrien as C = B T E QT [ max F T T K, 0 )]. Recall ha, by definiion of he T -forward maringale measure, we have ha E QT [F T T ] = F T = S /B T. To compue he expeced value eiher in closed form or by simulaion, we have o know he disribuion of S T = FT T under he T -forward maringale measure. This disribuion will follow from he dynamics of he forward price F T. Bu firs we will se up a model for he price of he underlying sock and for he relevan discoun facors, i.e. he zero-coupon bond prices. As usual, we will sick o models where he basic uncerainy is represened by one or several sandard Brownian moions. In a model wih a single Brownian moion, all sochasic processes will be insananeously perfecly correlaed, cf. he discussion in he inroducion o Chaper 8. To price sock opions in a seing wih sochasic ineres raes, we have o model boh he sock price and he appropriae discoun facor. Since hese wo variables are no perfecly correlaed, we have o include more han one Brownian moion in our model. Under he spo maringale measure Q he drif of he price of any raded asse in ime inervals wih no dividend paymens) is equal o he shor-erm ineres rae, r. The dynamics of he price of he underlying asse is assumed o be of he form [ 15.2) ds = S r d + σ s ) dz Q where z Q is a muli-dimensional sandard Brownian moion under he spo maringale measure Q, and where σ s is a vecor represening he sensiiviy of he sock price wih respec o he exogenous shocks. We will refer o σ s as he sensiiviy vecor of he sock price. In general, σ s may iself be sochasic, e.g. depend on he level of he sock price, bu we will only derive explici opion prices in he case where σ s is a deerminisic funcion of ime and hen we will use he noaion σ s ). I is hard o imagine ha he volailiy of a sock will depend direcly on calendar ime, so he mos relevan example of a deerminisic volailiy is a consan sensiiviy vecor. We can also wrie 15.2) as ds = S r d + n σj s dz Q j j=1 where n is he number of independen one-dimensional Brownian moions in he model, and σ1 s,..., σn s are he componens of he sensiiviy vecor. Similarly, we will assume ha he price of he zero-coupon bond mauring a ime T will evolve according o 15.3) db T = B T ],, [ r d + σ T ) ] dz Q, where he sensiiviy vecor σ T of he bond may depend on he curren erm srucure of ineres raes and in heory also on previous erm srucures). Equivalenly, we can wrie he bond price dynamics as db T = B T r d + n σj T dz Q j. j=1

15.2 Sock opions 303 In he model given by 15.2) and 15.3) he variance of he insananeous rae of reurn on he sock is given by Var Q ds /S ) = Var Q n σj s dz Q j j=1 = n j=1 σ s j) 2 d n so ha he volailiy of he sock is equal o he lengh of he vecor σ s, i.e. σ s = j=1 σ s 2. j) Similarly, he volailiy of he zero-coupon bond is given by σ T. The covariance beween he rae of reurn on he sock and he rae of reurn on he zero-coupon bond is σ s ) σ T = n j=1 σs j σt j. Consequenly, he insananeous correlaion is σ s ) σ T / [ σ s σ T ]. Noe ha if we jus wan o model he prices of his paricular sock and his paricular bond, a model wih n = 2 is sufficien o capure he imperfec correlaion. For example, if we specify he dynamics of prices as 15.4) 15.5) [ ds = S r d + v s db T = B T dz Q 1 ], [ r d + ρv T dz Q 1 + 1 ρ 2 v T dz Q 2 he volailiies of he sock and he bond are given by v s and v T, respecively, while ρ [ 1, 1] is he insananeous correlaion. However, we will sick o he more general noaion inroduced earlier. Given he dynamics of he sock price and he bond price in Equaions 15.2) and 15.3), we obain he dynamics of he forward price F T = S /B T under he Q T probabiliy measure by an applicaion of Iô s Lemma for funcions of wo sochasic processes, cf. Theorem 3.5 on page 66. Knowing ha F T is a Q T -maringale so ha is drif is zero, we do no have o compue he drif erm from Iô s Lemma. Therefore, we jus have o find he sensiiviy vecor, which we know is he same under all he maringale measures. Wriing F T = gs, B T ), where gs, P ) = S/P, he relevan derivaives are g/ S = 1/P and g/ P = S/P 2 so ha we obain he following forward price dynamics: ], df T = g S S, B T ) ) S σ s dz T + g P S, B T ) B T = F T σ s σ T ) dz T. σ T ) dz T A sandard calculaion yields and hence dln F T ) = 1 2 σs 15.6) ln S T = ln F T T = ln F T 1 2 T σ T 2 d + σ s σ T ) dz T, T σu s σu T 2 du + σ s u σ T ) u dzu T. In general, σ s and σ T will be sochasic, in which case we canno idenify he disribuion of ln S T and hence S T, bu Equaion 15.6) provides he basis for Mone Carlo simulaions of S T and hus an approximaion of he opion price. Below, we discuss he case where σ s and σ T are deerminisic. In ha case we can obain an explici opion pricing formula.

15.2 Sock opions 304 15.2.2 Deerminisic volailiies If we assume ha boh σ s and σ T are deerminisic funcions of ime, i follows from 15.6) and Theorem 3.2 on page 54 ha ln S T = ln FT T is normally disribued, i.e. S T = FT T is lognormally disribued, under he T -forward maringale measure. Theorem A.4 in Appendix A implies ha he price of he sock opion given in Equaion 15.1) can be wrien in closed form as } C = B {E T QT [FT T ]Nd 1 ) KNd 2 ), where ) 1 E QT d 1 = v F, T ) ln [FT T ] + 1 K 2 v F, T ), d 2 = d 1 v F, T ), v F, T ) 2 Var QT [ln F T T ]. By he maringale propery, E QT [FT T ] = F T = S /B T. We can compue he variance as [ T v F, T ) 2 = Var QT [ln FT T ] = Var QT σ s u) σ T u) ) ] = Var QT = = = = = n Var QT j=1 T n j=1 T j=1 T T n T dz T u n σ s j u) σj T u) ) dzju T j=1 [ T σ s j u) σ T j u) ) dz T ju σ s j u) σ T j u) ) 2 du, σ s j u) σ T j u) ) 2 du, σ s u) σ T u) 2 du T T σ s u) 2 du + σ T u) 2 du 2 σ s u) σ T u) du, where he hird equaliy follows from he independence of he Brownian moions z T 1,..., z T n, and he fourh equaliy follows from Theorem 3.2. Clearly, he firs erm in he final expression for he variance is due o he uncerainy abou he fuure price of he underlying sock, he second erm is due o he uncerainy abou he discoun facor, and he hird erm is due o he covariance of he sock price and he discoun facor. The price of he opion can be rewrien as 15.7) C = S Nd 1 ) KB T Nd 2 ), ] where d 1 = 1 v F, T ) ln S KB T d 2 = d 1 v F, T ). ) + 1 2 v F, T ),

15.2 Sock opions 305 If he sensiiviy vecor of he sock is consan, we ge T T 15.8) v F, T ) 2 = σ s 2 T ) + σ T u) 2 du 2 σ s ) σ T u) du. The Black-Scholes-Meron model is he special case in which he shor-erm ineres rae r is consan, which implies a consan, fla yield curve and deerminisic zero-coupon bond prices of B T = e r[t ] wih σ T u) 0. Under hese addiional assumpions, he opion pricing formula 15.7) reduces o he famous Black-Scholes-Meron formula 15.9) C) = S Nd 1 ) Ke r[t ] Nd 2 ), where d 1 = 1 σ s T ln d 2 = d 1 σ s T. S Ke r[t ] ) + 1 2 σs T, The more general formula 15.7) was firs shown by Meron 1973). I holds for all Gaussian erm srucure models, e.g. in he Vasicek model and he Gaussian HJM models, because he sensiiviy vecor and hence he volailiy of he zero-coupon bonds are hen deerminisic funcions of ime. In a reduced equilibrium model as Vasicek s he bond price B T enering he opion pricing formula is given by he well-known expression for he zero-coupon bond price in he model, e.g. 7.51) on page 178 in he one-facor Vasicek model. For he exended Vasicek model and he Gaussian HJM models he currenly observed zero-coupon bond price is used in he opion pricing formula. This laer approach is consisen wih praciioners use of he Black-Scholes-Meron formula since, insead of a fixed ineres rae r for opions of all mauriies, hey use he observed zero-coupon yield y T unil he mauriy dae of he opion. However, also he relevan variance v F, T ) 2 in Meron s formula 15.9) differs from he Black-Scholes-Meron formula. The firs of he hree erms in 15.8) is exacly he variance expression ha eners he Black-Scholes-Meron formula. The oher wo erms have o be added o ake ino accoun he variaion of ineres raes and he covariaion of ineres raes and he sock price. Praciioners seem o disregard hese wo erms. For ypical parameer values he wo laer erms will be much smaller han he firs erm so ha he errors implied by neglecing he wo las erms will be insignifican. Therefore Meron s generalizaion suppors praciioners use of he Black-Scholes-Meron formula. However, he assumpions underlying Meron s exension are problemaic since Gaussian erm srucure models are highly unrealisic. For oher erm srucure models one mus resor o numerical mehods for he compuaion of he sock opion prices. One possibiliy is o approximae he expeced value in 15.1) by an average of payoffs generaed by Mone Carlo simulaions of he erminal sock price under he T -forward maringale measure, e.g. based on 15.6). Noe ha if, for example, σu T depends on he shor rae r u, he evoluion in he shor rae over he ime period [, T ] has o be simulaed ogeher wih he sock price. Alernaively, he fundamenal parial differenial equaion can be solved numerically. Apparenly, he effecs of sochasic ineres raes on sock opion pricing and hedge raios have no been subjec o much research. Based on he analysis of a model allowing for boh a sochasic sock price volailiy and sochasic ineres raes, Bakshi, Cao, and Chen 1997) conclude ha ypical opion prices are more sensiive o flucuaions in sock price volailiies han

15.3 Opions on forwards and fuures 306 o flucuaions in ineres raes. Wheher his conclusion generalizes o oher model specificaions and sock opions wih oher conracual erms remains an unanswered quesion. 15.3 Opions on forwards and fuures In his secion we will discuss he pricing of opions on forwards and fuures on a securiy raded a he price S. As before, we assume ha his underlying asse has no dividend paymens. We will derive explici formulas for European call opions in he case where all price volailiies are deerminisic. Amin and Jarrow 1992) obain similar resuls for he special case where he dynamics of he erm srucure is given by a Gaussian HJM model, which implies ha he volailiies of he zero-coupon bonds are deerminisic. We will le T denoe he expiry ime of he opion and le T denoe he ime of delivery or final selemen) of he forward or he fuures conrac. Here, T T. 15.3.1 Forward and fuures prices As shown in Secion 5.4, he forward price for delivery a ime T is given by F T = S /B T, while he fuures price Φ T for final selemen a ime T is characerized by [ ] Φ T = E Q [S T ] = E Q F T T. is As in he preceding secion, we assume ha he dynamics under he spo maringale measure Q [ ds = S r d + σ s ) ] dz Q for he price of he securiy underlying he forward and he fuures and ) ] db T = B T [r d + σ T dz Q for he price of he zero-coupon bond mauring a T. Iô s Lemma yields firs ha ) 15.10) df T = F T [ σ T ) ) ] σ s σ T d + σ s σ T dz Q, and, subsequenly, ha [ dln F T ) = 1 ) 2 σs σ T 2 σ T ) ] ) σ s σ T d + σ s σ T dz Q. I follows ha ln F T T = ln F T + Equivalenly, S T = F T T = F T exp T { T [ 1 ) 2 σs u σ T u 2 σ T ) ] T ) u σu s σ T u du + σu s σ T u dz Q u. [ 1 ) 2 σs u σ T u 2 σ T ) ] T ) u σu s σ T u du + σu s σ T u dz Q u }. Therefore he fuures price can be wrien as [ { T [ Φ T = F T E Q exp 1 ) 2 σs u σ T u 2 σ T ) ] T ) u σu s σ T u du + σu s σ T u dz Q u }].

15.3 Opions on forwards and fuures 307 For he case where he volailiies σ s and σ T closed form as 15.11) Φ T = F T exp E Q [ = F T exp { T exp { { T are deerminisic, he fuures price is given in [ 12 σs u) σ T u) 2 σ T u) σ s u) σ u)) ] } T du T }] ) σ s u) σ T u) dz Q u } ) σ T u) σ s u) σ T u) du, where he las equaliy is a consequence of Theorem 3.2 and Theorem A.2. If he volailiies are no deerminisic, no explici expression for he fuures price is available. 15.3.2 Opions on forwards is From he analysis in Chaper 5 we know ha he price of a European call opion on a forward 15.12) C = B T E QT [ ] maxf T T K, 0), where T is he expiry ime and K is he exercise price. Le us find he dynamics in he forward price F T under he T -forward maringale measure Q T. From 5.7) and 5.20) we can shif he probabiliy measure from Q o Q T by applying he relaion 15.13) dz T = dz Q σ T d. Subsiuing his ino 15.10), we can wrie he dynamics in F T under Q T as [ ) df T = F T σ T σ T ) ) ] σ s σ T d + σ s σ T dz T. Noe ha only if T = T, he drif will be zero and F T will be a Q T -maringale. I follows ha T ) ln F T T = ln F T + σu T σ T ) u σu s σ T u du 1 2 T σ s u σ T u 2 du + T σ s u σ T u ) dz T u. Under he assumpion ha σu s, σu T, and σ T u are all deerminisic funcions of ime, we have ha ln F T T given F T ) is normally disribued under Q T wih mean value and variance µ F E QT [ln F T T ] T = ln F T + 1 2 v 2 F Var QT T ) σ T u) σ T ) u) σ s u) σ T u) du σ s u) σ T u) 2 du ] [ln F T T = T σ s u) σ T u) 2 du. Applying Theorem A.4) in Appendix A, we can compue he opion price from 15.12) as { } C = B T e µf + 1 2 v2 F Nd1 ) KNd 2 ),

15.3 Opions on forwards and fuures 308 where d 1 = µ F ln K v F d 2 = d 1 v F. + v F = µ F + 1 2 v2 F ln K v F + 1 2 v F, Since we can replace e µf + 1 2 v2 F µ F + 1 T ) 2 v2 F = ln F T + σ T u) σ T ) u) σ s u) σ T u) du, by F T e ξ, where ξ = T Hence, he opion price can be rewrien as ) σ T u) σ T ) u) σ s u) σ T u) du. 15.14) C = B T F T e ξ Nd 1 ) KB T Nd 2 ), and d 1 can be rewrien as d 1 = lnf T /K) + ξ v F + 1 2 v F. 15.3.3 Opions on fuures A European call opion on a fuures has a value of 15.15) C = B T E QT [ ] maxφ T T K, 0) Wih deerminisic volailiies we can apply 15.11) and inser Φ T T = F T T e ψt, T, T ), where we have inroduced he noaion ψt, U, T ) = U T ) σ T u) σ s u) σ T u) du. Consequenly, he opion price can be wrien as [ ] C = B T E QT maxf T T e ψt, T, T ) K, 0) = B T e ψt, T, T ) E QT [ maxf T T Ke ψt, T, T ), 0) We see ha, under hese assumpions, a call opion on a fuures wih he exercise price K is equivalen o e ψt, T, T ) call opions on a forward wih he exercise price Ke ψt, T, T ). From 15.14) i follows ha he price of he fuures opion is [ ] C = e ψt, T, T ) B T F T e ξ Nd 1 ) Ke ψt, T, T ) B T Nd 2 ), which can be rewrien as 15.16) C = F T B T e ξ ψt, T, T ) Nd 1 ) KB T Nd 2 ),. ]. where d 1 = lnf T /K) + ξ ψt, T, T ) v F + 1 2 v F, d 2 = d 1 v F, T 1/2 v F = σ s u) σ T u) du) 2.

15.4 Currency derivaives 309 Applying F T price as = Φ T e ψ, T, T ) and ψ, T, T ) ψt, T, T ) = ψ, T, T ), we can also express he opion 15.17) C = Φ T B T e ξ ψ,t, T ) Nd 1 ) KB T Nd 2 ) wih d 1 = lnφ T /K) + ξ ψ, T, T ) v F + 1 2 v F. 15.4 Currency derivaives Corporaions and individuals who operae inernaionally are exposed o currency risk since mos foreign exchange raes flucuae in an unpredicable manner. The exposure can be reduced or eliminaed by invesmens in suiable financial conracs. Boh on organized exchanges and in he OTC markes numerous conracs wih currency dependen payoffs are raded. Some of hese conracs also depend on oher economic variables, e.g. ineres raes or sock prices. However, we will focus on currency derivaives whose paymens only depend on a single forward exchange rae. This is he case for sandard currency forwards, fuures, and opions. Before we go ino he valuaion of he currency derivaives, we will inroduce some noaion. The ime spo price of one uni of he foreign currency is denoed by ε. This is he number of unis of he domesic currency ha can be exchanged for one uni of he foreign currency. As before, r denoes he shor-erm domesic ineres rae and B T denoes he price in he domesic currency) of a zero-coupon bond ha delivers one uni of he domesic currency a ime T. By ˇP T we will denoe he price in unis of he foreign currency of a zero-coupon bond ha delivers one uni of he foreign currency a ime T. Similarly, ř and ˇy T foreign zero-coupon yield for mauriy T, respecively. denoe he foreign shor rae and he 15.4.1 Currency forwards The simples currency derivaive is a forward conrac on one uni of he foreign currency. This is a binding conrac of delivery of one uni of he foreign currency a ime T a a prespecified exchange rae K so ha he payoff a ime T is ε T K. The no-arbirage value a ime < T of his payoff is ˇP T ε B T K since his is he value of a porfolio ha provides he same payoff as he forward, namely a porfolio of one uni of he foreign zero-coupon bond mauring a T and a shor posiion in K unis of he domesic zero-coupon bond mauring a ime T. The forward exchange rae a ime for delivery a ime T is denoed by F T and i is defined as he value of he delivery price K ha makes he presen value equal o zero, i.e. 15.18) F T = ˇP T B T This relaion is consisen wih he resuls on forward prices derived in Theorem 2.1 and in Secion 5.4.1. The forward exchange rae can be expressed as ε. F T = ε e yt ˇyT )T ), where y T and ˇy T denoe he domesic and he foreign zero-coupon raes for mauriy dae T, respecively. If y T > ˇy T, he forward exchange rae will be higher han he spo exchange rae,

15.4 Currency derivaives 310 oherwise an arbirage will exis. Conversely, if y T < ˇy T, he forward exchange rae will be lower han he spo exchange rae. The saed expressions for he forward exchange rae are based only on he no-arbirage principle and hold independenly of he dynamics in he spo exchange rae and he ineres rae of he wo counries. 15.4.2 A model for he exchange rae In order o be able o price currency derivaive securiies oher han currency forwards, assumpions abou he evoluion of he spo exchange rae are necessary. As always we focus on models where he fundamenal uncerainy is represened by Brownian moions. Since we have o model he evoluion in boh he exchange rae and he erm srucures of he wo counries, and hese objecs are no perfecly correlaed, he model has o involve a muli-dimensional Brownian moion. Foreign currency can be held in a deposi accoun earning he foreign shor-erm ineres rae. Therefore, we can hink of foreign currency as an asse providing a coninuous dividend a a rae equal o he foreign shor rae, ř. Under he domesic spo maringale measure Q he oal expeced rae of reurn on any asse will equal he domesic shor rae. Since foreign currency provides a cash rae of reurn of ř, he expeced percenage increase in he price of foreign currency, i.e. he exchange rae, mus equal r ř. The dynamics of he spo exchange rae will herefore be of he form 15.19) dε = ε [r ř ) d + σ ε ) dz Q where z Q is a muli-dimensional sandard Brownian moion under he spo maringale measure Q, and where σ ε is a vecor of he sensiiviies of he spo exchange rae owards he changes in he individual Brownian moions. processes. ], Noe ha, in general, r, ř, and σ e in 15.19) will be sochasic Define Y = ˇP T ε, i.e. Y is he price of he foreign zero-coupon bond measured in unis of he domesic currency. If we le ˇσ T denoe he sensiiviy vecor of he foreign zero-coupon bond, i follows from Iô s Lemma ha he sensiiviy vecor for Y can be wrien as σ ε + ˇσ T. Furhermore, we know ha, measured in he domesic currency, he expeced reurn on any asse under he riskneural probabiliy measure Q will equal he domesic shor rae, r. Hence, we have dy = Y [r d + σ ε + ˇσ T ) ] dz Q. For he domesic zero-coupon bond he price dynamics is of he form [ db T = B T r d + σ T ) ] dz Q. According o 15.18), he forward exchange rae is given by F T = Y /B T. An applicaion of Iô s Lemma yields ha he dynamics of he forward exchange rae is [ 15.20) df T = F T σ T ) σ ε + ˇσ T σ T ) d + σ ε + ˇσ T σ T ) ] dz Q. This is idenical o he dynamics of he forward price on a sock, excep ha he sock price sensiiviy vecor σ s has been replaced by he sensiiviy vecor for Y, which is σ ε + ˇσ T. I follows

15.4 Currency derivaives 311 ha 15.21) { T ) σ ε T = FT T = F T ) T [ exp u σ ε u + ˇσ u T σu T ] 1 σu ε + ˇσ u T σu T 2 du 2 } T [ + σ ε u + ˇσ u T σu T ] dzu Q. Here σ ε, ˇσ T, and σ T will generally be sochasic processes. As menioned above, we may hink of F T as he forward price of a raded asse he foreign zero-coupon bond) wih no paymens before mauriy. I follows from he analysis in Chaper 5 ha he forward price process F T ) is a Q T -maringale. In paricular, E QT [FT T ] = F T, and he drif in F T is zero under he Q T measure. Consequenly, he dynamics of he forward price F T under he T -forward maringale measure Q T is 15.22) df T = F T σ ε + ˇσ T σ T ) dz T. This can also be seen by subsiuing 15.13) ino 15.20). In order o obain explici expressions for he prices on currency derivaive securiies we will in he following wo subsecions focus on he case where σ ε, ˇσ T, and σ T are all deerminisic funcions of ime. As discussed earlier, deerminisic volailiies on zero-coupon bonds are obained only in Gaussian erm srucure models, e.g. he one- or wo-facor Vasicek models and Gaussian HJM models. 15.4.3 Currency fuures Le Φ T denoe he fuures price of he foreign currency wih final selemen a ime T. From 5.27) on page 117 we have ha Φ T = E Q [ε T ], where we can inser 15.21). In general he expecaion canno be compued explicily, bu if we assume ha σ ε, ˇσ T, and σ T are all deerminisic, we ge { Φ T = F T exp T σ T u) σ ε u) + ˇσ T u) σ T u) ) du. Amin and Jarrow 1991) demonsrae his under he assumpion ha boh he domesic and he foreign erm srucure are correcly described by Gaussian HJM models. In paricular, we recover he well-known resul ha Φ T = F T when σ T u) = 0, i.e. when he domesic erm srucure is non-sochasic. } 15.4.4 Currency opions Le us consider a European call opion on one uni of foreign currency. Le T denoe he expiry dae of he opion and K he exercise price expressed in he domesic currency). The opion grans is owner he righ o obain one uni of he foreign currency a ime T in reurn for a paymen of K unis of he domesic currency, i.e. he opion payoff is maxε T K, 0). According o he analysis in Chaper 5, he value of his opion a ime < T is given by C = B T E QT [max ε T K, 0)],

15.4 Currency derivaives 312 where Q T is he T -forward maringale measure. This relaion can be used for approximaing he opion price by Mone Carlo simulaions of he erminal exchange rae ε T under he Q T measure. Subsiuing he relaion 15.13) ino 15.19), we obain [ dε = ε r ř + σ ε ) σ T ) d + σ ε ) dz T ]. Therefore, in he general case, we have o simulae no jus he exchange rae, bu also he shorerm ineres raes in boh counries. Le us now assume ha σ ε, ˇσ T, and σ T are all deerminisic funcions of ime. By definiion, he forward price wih immediae delivery is equal o he spo price so ha ε T can be replaced by F T T : 15.23) C = B T E QT [ max F T T K, 0 )]. I follows from 15.22) ha he fuure forward exchange rae F T T is lognormally disribued wih v F, T ) 2 Var QT [ ln F T T ] = T σ ε u) + ˇσ T u) σ T u) 2 du. Noe ha he fuure spo exchange rae under hese volailiy assumpions is also lognormally disribued boh under spo maringale measure Q and under T -forward maringale measure Q T, bu no necessarily under he real-world probabiliy measure. In line wih earlier compuaions, he opion price becomes 15.24) C = B T F T Nd 1 ) KB T Nd 2 ), where d 1 = lnf T /K) v F, T ) d 2 = d 1 v F, T ). + 1 2 v F, T ), We can also inser F T = ˇP T ε /B T and wrie he opion price as 15.25) C = ε ˇP T Nd 1 ) KB T Nd 2 ), where d 1 can be expressed as d 1 = T 1 v F, T ) ln ε ˇP KB T ) + 1 2 v F, T ). Anoher alernaive is obained by subsiuing in B T = e yt T ) and ˇP T = e ˇyT T ), which yields 15.26) C = ε e ˇyT T ) Nd 1 ) Ke yt T ) Nd 2 ), where d 1 can be wrien as d 1 = lnε /K) + [y T ˇy T ]T ) v F, T ) + 1 2 v F, T ). Similar formulas were firs derived by Grabbe 1983). Amin and Jarrow 1991) demonsrae he resul for he case where boh he domesic and he foreign erm srucure of ineres raes can be described by Gaussian HJM models.

15.5 Final remarks 313 In he bes known model for currency opion pricing, Garman and Kohlhagen 1983) assume ha he shor rae in boh counries is consan, which implies a consan and fla yield curve in boh counries. In ha case we have B T = e r[t ], ˇP T = e ř[t ], and σ T ) = ˇσ T ) = 0. In addiion, he sensiiviy vecor of he exchange rae, i.e. σ ε ), is assumed o be a consan. Hence, he model can be viewed as a simple variaion of he Black-Scholes-Meron model for sock opions. Under hese resricive assumpions, he opion pricing formula saed above will simplify o 15.27) C = ε e ř[t ] Nd 1 ) Ke r[t ] Nd 2 ), where d 1 = lnε /K) + r ř)t ) σ ε T d 2 = d 1 σ ε T. + 1 2 σε T, This opion pricing formula is called he Garman-Kohlhagen formula. If we compare wih Equaion 15.25), we see ha he exension from consan ineres raes o Gaussian ineres raes implies jus as for sock opions) ha he ineres raes r and ř in he Garman-Kohlhagen formula 15.27) mus be replaced by he zero-coupon yields y T and ˇy T. Furhermore, he relevan variance has o reflec he flucuaions in boh he exchange rae and he discoun facors. As discussed earlier, he exra erms in he variance end o be insignifican for sock opions, bu for currency opions he exra erms are ypically no negligible. 15.4.5 Alernaive exchange rae models For exchange raes ha are no freely floaing, he above model for he exchange rae dynamics is inappropriae. For counries paricipaing in a so-called arge zone, he exchange raes are only allowed o flucuae in a fixed band around some cenral pariy. The cenral banks of he counries are commied o inervening in he financial markes in order o keep he exchange rae wihin he band. If a arge zone is perfecly credible, he exchange rae model has o assign zero probabiliy o fuure exchange raes ouside he band. 1 Clearly, his is no he case when he exchange rae is lognormally disribued. Krugman 1991) suggess a more appropriae model for he dynamics of exchange raes wihin a credible arge zone. However, mos arge zones are no perfecly credible in he sense ha he cenral pariies and he bands may be changed by he counries involved. The possibiliy of hese so-called realignmens may have large effecs on he pricing of currency derivaives. Chrisensen, Lando, and Milersen 1997) propose a model for exchange raes in a arge zone wih possible realignmens and show how currency opions may be priced numerically wihin ha model. See also Dumas, Jennergren, and Näslund 1995) for a differen, bu relaed, model specificaion. 15.5 Final remarks In his chaper we have focused on he pricing of forwards, fuures, and European opions on socks and foreign exchange, when we ake he sochasic naure of ineres raes ino accoun. 1 This mus hold under he real-world probabiliy measure, and since he maringale measures are equivalen o he real-world measure, i will also hold under he maringale measures.

15.5 Final remarks 314 Under raher resricive assumpions we have derived Black-Scholes-Meron-ype formulas for opion prices. Explici pricing formulas for oher securiies can be derived under similar assumpions. For example, Milersen and Schwarz 1998) sudy he pricing of opions on commodiy forwards and fuures under sochasic ineres raes. In conras o socks, bonds, exchange raes, ec., commodiies will ypically be valuable as consumpion goods or producion inpus. This value is modeled in erms of a convenience yield, cf. Hull 2003, Chap. 3). In order o be able o price opions on commodiy forwards and fuures he dynamics of boh he commodiy price and he convenience yield has o be modeled. Milersen and Schwarz obain Black-Scholes-Meronype pricing formulas for such opions under assumpions similar o hose we have applied in his chaper, e.g. a Gaussian process for he convenience yield of he underlying commodiy. Anoher class of securiies raded in he inernaional OTC markes is opions on foreign securiies, e.g. an opion ha pays off in euro, bu he size of he payoff is deermined by a U.S. sock index. The payoff is ransformed ino euro eiher by using he dollar/euro exchange rae prevailing a he expiraion of he opion or a prespecified exchange rae in ha case he opion is called a quano). Under paricular assumpions on he dynamics of he relevan variables, Black-Scholes- Meron-ype pricing formulas can be obained. Consul Musiela and Rukowski 1997, Chap. 17) for examples. In he OTC markes some securiies are raded which involve boh he exchange rae beween wo currencies and he yield curves of boh counries. A simple example is a currency swap where he wo paries exchange wo cash flows of ineres rae paymens, one cash flow deermined by a floaing ineres rae in he firs counry and he oher cash flow deermined by a floaing ineres rae in he oher counry. Many variaions of such currency swaps and also opions on hese swaps are raded on a large scale. Some of hese securiies are described in more deail in Musiela and Rukowski 1997, Chap. 17), who also provide pricing formulas for he case of deerminisic volailiies.

Chaper 16 Numerical echniques Numerical soluion of PDE s. See Ames 1977), Johnson 1990), Wilmo, Dewynne, and Howison 1993), Chrisiansen 1995), Thomas 1995), and Dydensborg 1999). Schwarz 1977) and Brennan and Schwarz 1977) were he firs papers applying a finie difference mehod o price opions. Tree models. Firs applicaion in finance: Cox, Ross, and Rubinsein 1979), Rendleman and Barer 1979). Born ree-models of he erm srucure of ineres raes: Ho and Lee 1986), Pedersen, Shiu, and Thorlacius 1989), Black, Derman, and Toy 1990). Tree-models as approximaions o one-facor coninuous-ime erm srucure models: Tian 1993), Hull and Whie 1990b, 1993, 1994b, 1996), Hull 2003, Ch. 23). Tree-models as approximaions o muli-facor coninuous-ime erm srucure models: Hull and Whie 1994a). See He 1990) for a general approach o approximaing an n-facor diffusion model by an n + 1)-nomial discree-ime model, i.e. a ree wih n + 1 branches from each node. Mone Carlo simulaion. For models which canno be formulaed as diffusion models wih relaively few sae variables neiher approximaing rees or numerical soluions o PDE s are usually applicable in pracice. Bu many problems can be solved using Mone Carlo simulaion. Inroducion o Mone Carlo simulaion: Hull 2003, Sec. 18.6-18.7). Firs applicaion o opion pricing: Boyle 1977). Mone Carlo simulaion for asses wih more han one exercise opporuniy, e.g. American opions and Bermuda swapions: Tilley 1993), Barraquand and Marineau 1995), Boyle, Broadie, and Glasserman 1997), Broadie and Glasserman 1997a, 1997b), Broadie, Glasserman, and Jain 1997), Carr and Yang 1997), Andersen 2000), Longsaff and Schwarz 2001). 315

Appendix A Resuls on he lognormal disribuion A random variable Y is said o be lognormally disribued if he random variable X = ln Y is normally disribued. In he following we le m be he mean of X and s 2 be he variance of X, so ha X = ln Y Nm, s 2 ). The probabiliy densiy funcion for X is given by { 1 f X x) = exp 2πs 2 x m)2 2s 2 }, x R. Theorem A.1 The probabiliy densiy funcion for Y is given by { 1 f Y y) = 2πs2 y exp and f Y y) = 0 for y 0. ln y m)2 2s 2 }, y > 0, This resul follows from he general resul on he disribuion of a random variable which is given as a funcion of anoher random variable; see any inroducory ex book on probabiliy heory and disribuions. Theorem A.2 For X Nm, s 2 ) and γ R we have E [ e γx] = exp { γm + 12 } γ2 s 2. Proof: Per definiion we have E [ e γx] = Manipulaing he exponen we ge + E [ e γx] = e γm+ 1 2 γ2 s 2 + = e γm+ 1 2 γ2 s 2 + = e γm+ 1 2 γ2 s 2, e γx 1 x m)2 e 2s 2 dx. 2πs 2 1 2πs 2 e 1 2s 2 [x m) 2 2γms 2 +γ 2 s 4 ] dx 1 x [m+γs2 ]) 2 2πs 2 e 2s 2 dx 316

317 where he las equaliy is due o he fac ha he funcion x 1 x [m+γs2 ]) 2 2πs 2 e 2s 2 is a probabiliy densiy funcion, namely he densiy funcion for an Nm + γs 2, s 2 ) disribued random variable. Using his heorem, we can easily compue he mean and he variance of he lognormally disribued random variable Y = e X. The mean is le γ = 1) A.1) E[Y ] = E [ e X] = exp {m + 12 s2 }. Wih γ = 2 we ge E [ Y 2] = E [ e 2X] = e 2m+s2), so ha he variance of Y is Var[Y ] = E [ Y 2] E[Y ]) 2 A.2) = e 2m+s2) e 2m+s2 ) = e 2m+s2 e s2 1. The nex heorem provides an expression of he runcaed mean of a lognormally disribued random variable, i.e. he mean of he par of he disribuion ha lies above some level. We define he indicaor variable 1 {Y >K} o be equal o 1 if he oucome of he random variable Y is greaer han he consan K and equal o 0 oherwise. Theorem A.3 If X = ln Y Nm, s 2 ) and K > 0, hen we have E [ ) ] Y 1 {Y >K} = e m+ 1 m ln K 2 s2 N + s s ) m ln K = E [Y ] N + s. s Proof: Because Y > K X > ln K, i follows from he definiion of he expecaion of a random variable ha where E [ Y 1 {Y >K} ] = E [ e X 1 {X>ln K} ] = = + ln K + ln K e x 1 x m)2 e 2s 2 dx 2πs 2 1 x [m+s2 ]) 2 2πs 2 e 2s 2 e 2ms2 +s 4 2s 2 dx + = e m+ 1 2 s2 f Xx) dx, f Xx) = ln K 1 x [m+s2 ]) 2 2πs 2 e 2s 2

318 is he probabiliy densiy funcion for an Nm + s 2, s 2 ) disribued random variable. The calculaions + ln K f Xx) dx = Prob X > ln K) X [m + s 2 ] = Prob > ln K [m + ) s2 ] s s X [m + s 2 ] = Prob < ln K [m + ) s2 ] s s = N ln K [m + ) s2 ] s ) m ln K = N + s s complee he proof. Theorem A.4 If X = ln Y Nm, s 2 ) and K > 0, we have ) ) E [max 0, Y K)] = e m+ 1 m ln K m ln K 2 s2 N + s KN s s ) ) m ln K m ln K = E [Y ] N + s KN. s s Proof: Noe ha E [max 0, Y K)] = E [ Y K)1 {Y >K} ] = E [ Y 1 {Y >K} ] KProb Y > K). The firs erm is known from Theorem A.3. The second erm can be rewrien as Prob Y > K) = Prob X > ln K) X m = Prob > ln K m ) s s X m = Prob < ln K m ) s s = N ln K m ) s ) m ln K = N. s The claim now follows immediaely.

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