Introduction to Arbitrage Pricing

Transcription

1 Inroducion o Arbirage Pricing Marek Musiela 1 School of Mahemaics, Universiy of New Souh Wales, 252 Sydney, Ausralia Marek Rukowski 2 Insiue of Mahemaics, Poliechnika Warszawska, -661 Warszawa, Poland 1 [email protected] 2 [email protected]

2 Conens I Spo and Fuures Markes 5 1 An Inroducion o Financial Derivaives Derivaive Markes Opions Fuures Conracs and Opions Forward Conracs Call and Pu Spo Opions One-period Spo Marke Replicaing Porfolios Maringale Measure for a Spo Marke Absence of Arbirage Opimaliy of Replicaion Pu Opion Fuures Call and Pu Opions Fuures Conracs and Fuures Prices One-period Fuures Marke Maringale Measure for a Fuures Marke Absence of Arbirage One-period Spo/Fuures Marke Forward Conracs Opions of American Syle General No-arbirage Inequaliies The Cox-Ross-Rubinsein Model The CRR Model of a Sock Price The CRR Opion Pricing Formula The Black-Scholes Opion Pricing Formula Probabilisic Approach Condiional Expecaions Maringale Measure Risk-neural Valuaion Formula Valuaion of American Opions American Pu Opion Opions on a Dividend-paying Sock The Black-Scholes Model Iô Sochasic Calculus Iô s Lemma Predicable Represenaion Propery Girsanov s heorem The Black-Scholes Opion Valuaion Formula

3 2 CONTENTS Sock Price Self-financing Sraegies Maringale Measure for he Spo Marke The Pu-Call Pariy for Spo Opions The Black-Scholes PDE Sensiiviy Analysis Opion on a Dividend-paying Sock Hisorical Volailiy Implied Volailiy Numerical Mehods Fuures Marke Self-financing Sraegies Maringale Measure for he Fuures Marke The Black Fuures Opion Formula Opions on Forward Conracs Foreign Marke Derivaives Cross-currency Marke Model Domesic Maringale Measure Foreign Maringale Measure Currency Forward Conracs and Opions Forward Exchange Rae Currency Opion Valuaion Formula Foreign Equiy Forward Conracs Forward Price of a Foreign Sock Quano Forward Conracs Foreign Equiy Opions Opions Sruck in a Foreign Currency Opions Sruck in Domesic Currency Quano Opions Equiy-linked Foreign Exchange Opions American Opions Valuaion of American Claims American Call and Pu Opions Early Exercise Represenaion of an American Pu Free Boundary Problem Approximaions of he American Pu Price Opion on a Dividend-paying Sock Exoic Opions Packages Collars Break Forwards Range Forwards Forward-sar Opions Chooser Opions Compound Opions Digial Opions Barrier Opions Asian Opions Baske Opions Lookback Opions

4 CONTENTS 3 II Fixed-income Markes Ineres Raes and Relaed Conracs Zero-coupon Bonds Term Srucure of Ineres Raes Forward Ineres Raes Shor-erm Ineres Rae Coupon-bearing Bonds Ineres Rae Fuures Treasury Bond Fuures Bond Opions Treasury Bill Fuures Eurodollar Fuures Ineres Rae Swaps Models of he Shor-erm Rae Arbirage-free Family of Bond Prices Expecaions Hypoheses Case of Iô Processes Single-facor Models American Bond Opions Opions on Coupon-bearing Bonds Models of Forward Raes HJMModel Absence of Arbirage Forward Measure Approach Forward Price Forward Maringale Measure Gaussian HJM Model Model of LIBOR Raes Discree-enor Case Coninuous-enor Case Model of Forward Swap Raes Opion Valuaion in Gaussian Models European Spo Opions Bond Opions Sock Opions Opion on a Coupon-bearing Bond Pricing of General Coningen Claims Replicaion of Opions Fuures Prices Fuures Opions PDE Approach o Ineres Rae Derivaives PDEs for Spo Derivaives PDEs for Fuures Derivaives Valuaion of Swap Derivaives Ineres Rae Swaps Gaussian Model Forward Caps and Floors Capions

5 4 CONTENTS Swapions Model of Forward LIBOR Raes Caps Swapions Model of Forward Swap Raes Bibliography 187

6 Par I Spo and Fuures Markes 5

7

8 Chaper 1 An Inroducion o Financial Derivaives We shall firs review briefly he mos imporan kinds of financial conracs, raded eiher on exchanges or over-he-couner OTC, beween financial insiuions and heir cliens. For a deailed accoun of he fundamenal feaures of spo i.e., cash and fuures financial markes he reader is referred, for insance, o Cox and Rubinsein 1985, Richken 1987, Chance 1989, Duffie 1989, Merrick 199, Kolb 1991, Dubofsky 1992, Edwards and Ma 1992, Sucliffe 1993, Hull 1994, 1997 or Redhead Derivaive Markes We sar his secion by describing he spo opions marke, afer which we shall focus on fuures conracs and fuures opions. Finally, he basic feaures of forward conracs will be discussed Opions Opions are examples of exchange-raded derivaive securiies ha is, securiies whose value depends on he prices of oher more basic securiies so called primary securiies or asses such as socks or bonds. By socks we mean common socks ha is, shares in he ne asse value no bearing fixed ineres. They give he righ o dividends according o profis, afer paymens on preferred socks. The preferred socks give some special righs o he sockholder, ypically a guaraneed fixed dividend. A bond is a cerificae issued by a governmen or a public company promising o repay borrowed money a a fixed rae of ineres a a specified ime. Basically, a call opion a pu opion, respecively is he righ o buy o sell, respecively he opion s underlying asse a some fuure dae for a prespecified price. Opions in paricular, warrans 1 have been raded for cenuries; unprecedened expansion of he opions marke sared, however, quie recenly wih he inroducion in 1973 of exchange-raded opions on socks in he Unied Saes. I should be emphasized ha mos of he raded opions are of American ype ha is, he holder has he righ o exercise an opion a any insan before he opion s expiry. When an invesor noifies his broker of he inenion o exercise an opion, he broker in urn noifies he OCC 2 member who clears he invesor s rade. This member hen places an exercise order wih he OCC. The OCC randomly selecs a member wih an ousanding shor posiion in he same 1 A warran is a call opion issued by a company or a financial insiuion. Warrans are frequenly issued by companies on heir own socks; new shares are issued when warrans are exercised. In some cases,he warrans are subsequenly raded on an exchange. Warrans are bough and sold in much he same way as socks. 2 OCC sands for he Opions Clearing Corporaion. The OCC keeps he record of all long and shor posiions. The OCC guaranees ha he opion wrier will fulfil obligaions under he erms of he opion conrac. The OCC has a number of members, and all opion rades mus be cleared hrough a member. 7

9 8 CHAPTER 1. AN INTRODUCTION TO FINANCIAL DERIVATIVES opion. The chosen member, in urn, selecs a paricular invesor who has wrien he opion such an invesor is said o be assigned. If he opion is a call, his invesor is required o sell sock a he so-called srike price or exercise price if i is a pu, he is required o buy sock a he srike price. When he opion is exercised, he open ineres ha is, he number of opions ousanding goes down by one. In addiion o opions on paricular socks, a large variey of oher opion conracs are raded nowadays on exchanges: foreign currency opions such as, e.g., Briish pound, German mark or Japanese yen opion conracs raded on he Philadelphia Exchange, index opions e.g., hose on S&P1 and S&P5 raded on he CBOE, and fuures opions e.g., he Treasury bond fuures opion raded on he Chicago Board of Trade CBOT. Ineres rae opions are also implici in several oher ineres rae insrumens, such as caps or floors hese are, however, over-he-couner raded conracs. Derivaive financial insrumens involving opions are also widely raded ouside he exchanges by financial insiuions and heir cliens. We may idenify here such conracs as swapions ha is, opions on an ineres rae swap, or a large variey of exoic opions. Finally, opions are implici in several financial insrumens, for example in some bond or sock issues callable bonds, savings bonds or converible bonds, o menion a few. One of he mos appealing feaures of opions apar from he obvious chance of making exraordinary reurns is he possibiliy of easy speculaion on he fuure behavior of a sock price. Usually his is done by means of so called combinaions ha is, combined posiions in several opions, and possibly he underlying asse. For insance, a bull spread is porfolio creaed by buying a call opion on a sock wih a cerain srike price and selling a call opion on he same sock wih a higher srike price boh opions have he same expiry dae. Equivalenly, bull spreads can be creaed by buying a pu wih a low srike price and selling a pu wih a high srike price. An invesor enering a bull spread is hoping ha he sock price will increase. Like a bull spread, a bear spread can be creaed by buying a call wih one srike price and selling a call wih anoher srike price. The srike price of he opion purchased is now greaer han he srike price of he opion sold, however. An invesor who eners a bear spread is hoping ha he sock price will decline. A buerfly spread involves posiions in opions wih hree differen srike prices. I can be creaed by buying a call opion wih a relaively low srike price, buying anoher call opion wih a relaively high srike price, and selling wo call opions wih a srike price halfway beween he oher wo srike prices. The buerfly spread leads o a profi if he sock price says close o he srike price of he call opions sold, bu gives rise o a small loss if here is a significan sock price move in eiher direcion. A porfolio creaed by selling a call opion wih a cerain srike price and buying a longer-mauriy call opion wih he same srike price is commonly known as a calendar spread. A sraddle involves buying a call and pu wih he same srike price and expiry dae. If he sock price is close o his srike price a expiry of he opion, he sraddle leads o a loss. A sraddle is appropriae when an invesor is expecing a large move in sock price bu does no know in which direcion he move will be. Relaed ypes of rading sraegies are commonly known as srips, sraps and srangles Fuures Conracs and Opions Anoher imporan class of exchange-raded derivaive securiies comprises fuures conracs, and opions on fuures conracs, commonly known as fuures opions. Fuures conracs apply o a wide range of commodiies e.g., sugar, wool, gold and financial asses e.g., currencies, bonds, sock indices; he larges exchanges on which fuures conracs are raded are he Chicago Board of Trade and he Chicago Mercanile Exchange CME. In wha follows, we resric our aenion o financial fuures as opposed o commodiy fuures. To make rading possible, he exchange specifies cerain sandardized feaures of he conrac. Fuures prices are regularly repored in he financial press. They are deermined on he floor in he same way as oher prices ha is, by he law of supply and demand. If more invesors wan o go long han o go shor, he price goes up; if he reverse is rue, he price falls. Posiions in fuures conracs are governed by a specific daily selemen procedure commonly referred o as marking o marke. An invesor s iniial deposi,

10 1.1. DERIVATIVE MARKETS 9 known as he iniial margin, is adjused daily o reflec he gains or losses ha are due o he fuures price movemens. Le us consider, for insance, a pary assuming a long posiion he pary who agreed o buy. When here is a decrease in he fuures price, her margin accoun is reduced by an appropriae amoun of money, her broker has o pay his sum o he exchange and he exchange passes he money on o he broker of he pary who assumes he shor posiion. Similarly, when he fuures price rises, brokers for paries wih shor posiions pay money o he exchange, and brokers of paries wih long posiions receive money from he exchange. This way, he rade is marked o marke a he close of each rading day. Finally, if he delivery period is reached and delivery is made by a pary wih a shor posiion, he price received is generally he fuures price a he ime he conrac was las marked o marke. In a fuures opion, he underlying asse is a fuures conrac. The fuures conrac normally maures shorly afer he expiry of he opion. When he holder of a call fuures opion exercises he opion, she acquires from he wrier a long posiion in he underlying fuures conrac plus a cash amoun equal o he excess of he curren fuures price over he opion s srike price. Since fuures conracs have zero value and can be closed ou immediaely, he payoff from a fuures opion is he same as he payoff from a sock opion, wih he sock price replaced by he fuures price. Fuures opions are now available for mos of he insrumens on which fuures conracs are raded. The mos acively raded fuures opion is he Treasury bond fuures opion raded on he Chicago Board of Trade. On some markes for insance, on he Ausralian marke, fuures opions have he same feaures as fuures conracs hemselves ha is, hey are no paid up-fron as classic opions, bu are raded a he margin. Unless oherwise saed, by a fuures opion we mean here a sandard opion wrien on a fuures conrac Forward Conracs A forward conrac is an agreemen o buy or sell an asse a a cerain fuure ime for a cerain price. One of he paries o a forward conrac assumes a long posiion and agrees o buy he underlying asse on a cerain specified fuure dae for a delivery price; he oher pary assumes a shor posiion and agrees o sell he asse on he same dae for he same price. A he ime he conrac is enered ino, he delivery price is deermined so ha he value of he forward conrac o boh paries is zero. I is hus clear ha some feaures of forward conracs resemble hose of fuures conracs. However, unlike fuures conracs, forward conracs do no rade on exchanges. Also, a forward conrac is seled only once, a he mauriy dae. The holder of he shor posiion delivers he asse o he holder of he long posiion in reurn for a cash amoun equal o he delivery price. The following lis summarizes he main differences beween forward and fuures conracs. 1. Conrac specificaion and delivery Fuures conracs. The conrac precisely specifies he underlying insrumen and price. Delivery daes and delivery procedures are sandardized o a limied number of specific daes per year, a approved locaions. Delivery is no, however, he objecive of he ransacion, and less han 2% are delivered. Forward conracs. There is an almos unlimied range of insrumens, wih individually negoiaed prices. Delivery can ake place on any individual negoiaed dae and locaion. Delivery is he objec of he ransacion, wih over 9% of forward conracs seled by delivery. 2. Prices Fuures conracs. The price is he same for all paricipans, regardless of ransacion size. Typically, here is a daily price limi alhough, for insance, on he FT-SE 1 index, fuures prices are unlimied. Trading is usually by open oucry aucion on he rading floor of he exchange. Prices are disseminaed publicly. Each ransacion is conduced a he bes price available a he ime. Forward conracs. The price varies wih he size of he ransacion, he credi risk, ec. There are no daily price limis. Trading akes place by elephone and fax beween individual buyers and sellers. Prices are no disseminaed publicly. Hence, here is no guaranee ha he price is he bes available.

11 1 CHAPTER 1. AN INTRODUCTION TO FINANCIAL DERIVATIVES 3. Markeplace and rading hours Fuures conracs. Trading is cenralized on he exchange floor, wih worldwide communicaions, during hours fixed by he exchange. Forward conracs. Trading akes place by elephone and elex beween individual buyers and sellers. Trading is over-he-couner world-wide, 24 hours per day, wih elephone and elex access. 4. Securiy deposi and margin Fuures conracs. The exchange rules require an iniial margin and he daily selemen of variaion margins. A cenral clearing house is associaed wih each exchange o handle he daily revaluaion of open posiions, cash paymens and delivery procedures. The clearing house assumes he credi risk. Forward conracs. The collaeral level is negoiable, wih no adjusmen for daily price flucuaions. There is no separae clearing house funcion. Thus, he marke paricipan bears he risk of he couner-pary defauling. 5. Volume and marke liquidiy Fuures conracs. Volume and open ineres informaion is published. There is very high liquidiy and ease of offse wih any oher marke paricipan due o sandardized conracs. Forward conracs. Volume informaion is no available. The limied liquidiy and offse is due o he variable conrac erms. Offse is usually wih he original couner-pary. 1.2 Call and Pu Spo Opions Le us firs describe briefly he se of general assumpions imposed on our models of financial markes. We consider hroughou, unless explicily saed oherwise, he case of a so-called fricionless marke, meaning ha: all invesors are price-akers, all paries have he same access o he relevan informaion, here are no ransacion coss or commissions, and all asses are assumed o be perfecly divisible and liquid. There is no resricion whasoever on he size of a bank credi, and he lending and borrowing raes are equal. Finally, individuals are allowed o sell shor any securiy and receive full use of he proceeds of course, resiuion is required for payoffs made o securiies held shor. Unless oherwise specified, by an opion we shall mean hroughou a European opion, giving he righ o exercise he opion only a he expiry dae. In mahemaical erms, he problem of pricing of American opions is closely relaed o opimal sopping problems. Unforunaely, closed-form expressions for he prices of American opions are rarely available; for insance, no closed-form soluion is available for he price of an American pu opion in he now classic framework of he Black-Scholes opion pricing model. A European call opion wrien on a common sockis a financial securiy ha gives is holder he righ bu no he obligaion o buy he underlying sock on a prespecified dae and for a prespecified price. The ac of making his ransacion is referred o as exercising he opion. If an opion is no exercised, we say i is abandoned. Anoher class of opions comprises so-called American opions. These may be exercised a any ime on or before he prespecified dae. The prespecified fixed price, say K, is ermed he srike or exercise price; he erminal dae, denoed by T in wha follows, is called he expiry dae or mauriy. I should be emphasized ha an opion gives he holder he righ o do somehing; however, he holder is no obliged o exercise his righ. In order o purchase an opion conrac, an invesor needs o pay an opion s price or premium o a second pary a he iniial dae when he conrac is enered ino. Le us denoe by S T he sock price a he erminal dae T. I is naural o assume ha S T is no known a ime, hence S T gives rise o uncerainy in our model. We argue ha from he perspecive of he opion holder, he payoff g a expiry dae T from a European call opion is given by he formula gs T =S T K + def = max {S T K, }, 1.1 ha is o say gs T = { ST K if S T >K opion is exercised, if S T K opion is abandoned.

12 1.2. CALL AND PUT SPOT OPTIONS 11 In fac, if a he expiry dae T he sock price is lower han he srike price, he holder of he call opion can purchase an underlying sock direcly on a spo i.e., cash marke, paying less han K. In oher words, i would be irraional o exercise he opion, a leas for an invesor who prefers more wealh o less. On he oher hand, if a he expiry dae he sock price is greaer han K, an invesor should exercise his righ o buy he underlying sock a he srike price K. Indeed, by selling he sock immediaely a he spo marke, he holder of he call opion is able o realize an insananeous ne profi S T K noe ha ransacion coss and/or commissions are ignored here. In conras o a call opion, a pu opion gives is holder he righ o sell he underlying asse by a cerain dae for a prespecified price. Using he same noaion as above, we arrive a he following expression for he payoff h a mauriy T from a European pu opion or more explicily hs T =K S T + def = max {K S T, }, 1.2 { if ST K opion is abandoned, hs T = K S T if S T <K opion is exercised. I follows immediaely ha he payoffs of call and pu opions saisfy he following simple bu useful equaliy gs T hs T =S T K + K S T + = S T K. 1.3 The las equaliy can be used, in paricular, o derive he so-called pu-call pariy relaionship for opion prices. Basically, pu-call pariy means ha he price of a European pu opion is deermined by he price of a European call opion wih he same srike and expiry dae, he curren price of he underlying asse, and he properly discouned value of he srike price One-period Spo Marke Le us sar by considering an elemenary example of an opion conrac. Example Assume ha he curren sock price is $28, and afer hree monhs he sock price may eiher rise o $32or decline o $26. We shall find he raional price of a 3-monh European call opion wih srike price K = $28, provided ha he simple risk-free ineres rae r for 3-monh deposis and loansis r = 5%. Suppose ha he subjecive probabiliy of he price rise is.2, and ha of he fall is.8; hese assumpions correspond, loosely, o a so-called bear marke. Noe ha he word subjecive means ha we ake he poin of view of a paricular individual. Generally speaking, he wo paries involved in an opion conrac may have and usually do have differing assessmens of hese probabiliies. To model a bull marke one may assume, for example, ha he firs probabiliy is.8, so ha he second is.2. Le us focus firs on he bear marke case. The erminal sock price S T may be seen as a random variable on a probabiliy space Ω = {ω 1,ω 2 } wih a probabiliy measure P given by P{ω 1 } =.2 =1 P{ω 2 }. Formally, S T is a funcion S T :Ω R + given by he following formula { S S T ω = u = 32, if ω = ω 1, S d = 26, if ω = ω 2. Consequenly, he erminal opion s payoff X = C T =S T K + saisfies { C C T ω = u =4, if ω = ω 1, C d =, if ω = ω 2.

13 12 CHAPTER 1. AN INTRODUCTION TO FINANCIAL DERIVATIVES Noe ha he expeced value under P of he discouned opion s payoff equals E P 1 + r 1 C T = =7.62. I is clear ha he above expecaion depends on he choice of he probabiliy measure P; ha is, i depends on he invesor s assessmen of he marke. For a call opion, he expecaion corresponding o he case of a bull marke would be greaer han ha which assumes a bear marke. In our example, he expeced value of he discouned payoff from he opion under he bull marke hypohesis equals Sill, o consruc a reliable model of a financial marke, one has o guaranee he uniqueness of he price of any derivaive securiy. This can be done by applying he concep of he so-called replicaing porfolio, which we will now inroduce Replicaing Porfolios The wo-sae opion pricing model presened below was developed independenly by Sharpe 1978 and Rendleman and Barer 1979 a poin worh menioning is ha he ground-breaking papers of Black and Scholes 1973 and Meron 1973, who examined he arbirage pricing of opions in a coninuous-ime framework, were published five years earlier. The idea is o consruc a porfolio a ime which replicaes exacly he opion s erminal payoff a ime T. Le φ = φ =α,β R 2 denoe a porfolio of an invesor wih a shor posiion in one call opion. More precisely, le α sand for he number of shares of sock held a ime, and β be he amoun of money deposied on a bank accoun or borrowed from a bank. By V φ we denoe he wealh of his porfolio a daes = and = T ; ha is, he payoff from he porfolio φ a given daes. I should be emphasized ha once he porfolio is se up a ime, i remains fixed unil he erminal dae T. Therefore, for is wealh process V φ we have V φ =α S + β and V T φ =α S T + β 1 + r. 1.4 We say ha a porfolio φ replicaes he opion s erminal payoff whenever V T φ =C T, ha is, if { V V T φω = u φ =α S u +1+rβ = C u, if ω = ω 1, V d φ =α S d +1+rβ = C d, if ω = ω 2. For he daa of Example 1.2.1, he porfolio φ is deermined by he following sysem of linear equaions { 32 α +1.5 β =4, 26 α +1.5 β =, wih unique soluion α =2/3 and β = Observe ha for every call we are shor, we hold α of sock 3 and he dollar amoun β in riskless bonds in he hedging porfolio. Pu anoher way, by purchasing shares and borrowing agains hem in he righ proporion, we are able o replicae an opion posiion. Acually, one can easily check ha his propery holds for any coningen claim X which seles a ime T. I is naural o define he manufacuring cos C of a call opion as he iniial invesmen needed o consruc a replicaing porfolio, i.e., C = V φ =α S + β =2/ = I should be sressed ha in order o deermine he manufacuring cos of a call we did no need o know he probabiliy of he rise or fall of he sock price. In oher words, i appears ha he manufacuring cos is invarian wih respec o individual assessmens of marke behavior. In paricular, i is idenical under he bull and bear marke hypoheses. To deermine he raional price of a call we have used he opion s srike price, he curren value of he sock price, he range of flucuaions in he sock price ha is, he fuure levels of he sock price, and he risk-free rae of 3 We shall refer o he number of shares held for each call sold as he hedge raio. Basically,o hedge means o reduce risk by making ransacions ha reduce exposure o marke flucuaions.

14 1.2. CALL AND PUT SPOT OPTIONS 13 ineres. The invesor s ransacions and he corresponding cash flows may be summarized by he following wo exhibis one wrien call opion C, a ime = α shares purchased α S, amoun of cash borrowed β, and a ime = T payoff from he call opion C T, α shares sold α S T, loan paid back ˆrβ, where ˆr =1+r. I should be observed ha no ne iniial invesmen is needed o esablish he above porfolio; ha is, he porfolio is cosless. On he oher hand, for each possible level of sock price a ime T, he hedge exacly breaks even on he opion s expiry dae. Also, i is easy o verify ha if he call were no priced a $21.59, i would be possible for a sure profi o be gained, eiher by he opion s wrier if he opion s price were greaer han is manufacuring cos or by is buyer in he opposie case. Sill, he manufacuring cos canno be seen as a fair price of a claim X, unless he marke model is arbirage-free, in a sense examined below. Indeed, i may happen ha he manufacuring cos of a non-negaive claim is a sricly negaive number. Such a phenomenon conradics he usual assumpion ha i is no possible o make riskless profis Maringale Measure for a Spo Marke Alhough, as shown above, subjecive probabiliies are useless when pricing an opion, probabilisic mehods play an imporan role in coningen claims valuaion. They rely on he noion of a maringale, which is, inuiively, a probabilisic model of a fair game. In order o apply he so-called maringale mehod of derivaive pricing, one has o find firs a probabiliy measure P equivalen o P, and such ha he discouned or relaive sock price process S, which is defined by he formula S = S, S T =1+r 1 S T, follows a P -maringale; ha is, he equaliy S = E P ST holds. Such a probabiliy measure P is called a maringale measure for he discouned sock price process S. In he case of a wo-sae model, he probabiliy measure P is easily seen o be uniquely deermined provided i exiss by he following linear equaion S =1+r 1 p S u +1 p S d, 1.5 where p = P {ω 1 } and 1 p = P {ω 2 }. Solving his equaion for p yields P {ω 1 } = 1 + rs S d S u S d, P {ω 2 } = Su 1 + rs S u S d. 1.6 Le us now check ha he price C coincides wih C, where we wrie C o denoe he expeced value under P of an opion s discouned erminal payoff ha is C def = E P 1 + r 1 C T = E P 1 + r 1 S T K +. Indeed, using he daa of Example we find p =17/3, so ha C =1+r 1 p C u +1 p C d =21.59 = C. Remarks. Observe ha since he process S follows a P -maringale, we may say ha he discouned sock price process may be seen as a fair game model in a risk-neural economy ha is, in he sochasic economy in which he probabiliies of fuure sock price flucuaions are deermined by

15 14 CHAPTER 1. AN INTRODUCTION TO FINANCIAL DERIVATIVES he maringale measure P. I should be sressed, however, ha he fundamenal idea of arbirage pricing is based solely on he exisence of a porfolio ha hedges perfecly he risk exposure relaed o uncerain fuure prices of risky securiies. Therefore, he probabilisic properies of he model are no essenial. In paricular, we do no assume ha he real-world economy is acually risk-neural. On he conrary, he noion of a risk-neural economy should be seen raher as a echnical ool. The aim of inroducing he maringale measure is wofold: firsly, i simplifies he explici evaluaion of arbirage prices of derivaive securiies; secondly, i describes he arbirage-free propery of a given pricing model for primary securiies in erms of he behavior of relaive prices. This approach is frequenly referred o as he parial equilibrium approach, as opposed o he general equilibrium approach. Le us sress ha in he laer heory he invesors preferences, usually described in sochasic models by means of heir expeced uiliy funcions, play an imporan role. To summarize, he noion of an arbirage price for a derivaive securiy does no depend on he choice of a probabiliy measure in a paricular pricing model for primary securiies. More precisely, using sandard probabilisic erminology, his means ha he arbirage price depends on he suppor of a subjecive probabiliy measure P, bu is invarian wih respec o he choice of a paricular probabiliy measure from he class of muually equivalen probabiliy measures. In financial erminology, his can be resaed as follows: all invesors agree on he range of fuure price flucuaions of primary securiies; hey may have differen assessmens of he corresponding subjecive probabiliies, however Absence of Arbirage Le us consider a simple wo-sae, one-period, wo-securiy marke model defined on a probabiliy space Ω = {ω 1,ω 2 } equipped wih he σ-fields F = {, Ω}, F T =2 Ω i.e., F T conains all subses of Ω, and a probabiliy measure P on Ω, F T such ha P{ω 1 } and P{ω 2 } are sricly posiive numbers. The firs securiy is a sock whose price process is modelled as a sricly posiive discreeime process S =S {,T }. We assume ha he process S is F -adaped, i.e., ha he random variables S are F -measurable for {,T}. This means ha S is a real number, and { S u if ω = ω S T ω = 1, S d if ω = ω 2, where, wihou loss of generaliy, S u >S d. The second securiy is a riskless bond whose price process is B =1,B T =1+r for some real r. Le Φ sand for he linear space of all sock-bond porfolios φ = φ =α,β, where α and β are real numbers clearly, he class Φ may be hus idenified wih R 2. We shall consider he pricing of coningen claims in a securiy marke model M =S, B, Φ. We shall now check ha an arbirary coningen claim X which seles a ime T i.e., any F T -measurable real-valued random variable admis a unique replicaing porfolio in our marke model. In oher words, an arbirary coningen claim X is aainable in he marke model M. Indeed, if { X u if ω = ω Xω = 1, X d if ω = ω 2, hen he replicaing porfolio φ is deermined by he following sysem of linear equaions { α S u +1+rβ = X u, α S d +1+rβ = X d 1.7, which admis a unique soluion α = Xu X d S u S d, β = Xd S u X u S d 1 + rs u S d, 1.8 for arbirary values of X u and X d. Consequenly, an arbirary coningen claim X admis a unique manufacuring cos π X inm which is given by he formula π X def = V φ =α S + β = Xu X d S u S d S + Xd S u X u S d 1 + rs u S d. 1.9

16 1.2. CALL AND PUT SPOT OPTIONS 15 As already menioned, he manufacuring cos of a sricly posiive coningen claim may appear o be a negaive number, in general. If his were he case, here would be a profiable riskless rading sraegy so-called arbirage opporuniy involving only he sock and riskless borrowing and lending. To exclude such siuaions, which are clearly inconsisen wih any broad noion of a raional marke equilibrium as i is common o assume ha invesors are non-saiaed, meaning ha hey prefer more wealh o less, we have o impose furher essenial resricions on our marke model. Definiion We say ha a securiy pricing model M is arbirage-free if here is no porfolio φ Φ for which V φ =,V T φ and P{V T φ > } >. 1.1 A porfolio φ for which he se 1.1 of condiions is saisfied is called an arbirage opporuniy. A srong arbirage opporuniy is a porfolio φ for which V φ < and V T φ I is cusomary o ake eiher 1.1 or 1.11 as he definiion of an arbirage opporuniy. Noe, however, ha boh noions are no necessarily equivalen. We are in a posiion o inroduce he noion of an arbirage price; ha is, he price derived using he no-arbirage argumens. Definiion Suppose ha he securiy marke M is arbirage-free. Then he manufacuring cos π X is called he arbirage price of X a ime in securiy marke M. As he nex resul shows, under he absence of arbirage in a marke model, he manufacuring cos may be seen as he unique raional price of a given coningen claim ha is, he unique price compaible wih any raional marke equilibrium. Proposiion Suppose ha he spo marke M =S, B, Φ is arbirage-free. Le H sand for he raional price process of some aainable coningen claim X; more explicily, H R and H T = X. Le Φ H denoe he class of all porfolios in sock, bond and derivaive securiy H. Then he spo marke S, B, H, Φ H is arbirage-free if and only if H = π X. Proof. Since he proof is sraighforward, i is lef o he reader Opimaliy of Replicaion Le us show ha replicaion is, in a sense, an opimal way of hedging. Firsly, we say ha a porfolio φ perfecly hedges agains X if V T φ X, ha is, whenever { α S u +1+rβ X u, α S d +1+rβ X d The minimal iniial cos of a perfec hedging porfolio agains X is called he seller s price of X, and i is denoed by πx. s Le us check ha πx s =π X. By denoing c = V φ, we may rewrie 1.12 as follows { α S u S 1 + r + c1 + r X u, α S d S 1 + r + c1 + r X d I is rivial o check ha he minimal c R for which 1.13 holds is acually ha value of c for which inequaliies in 1.13 become equaliies. This means ha he replicaion appears o be he leas expensive way of perfec hedging for he seller of X. Le us now consider he oher pary of he conrac, i.e., he buyer of X. Since he buyer of X can be seen as he seller of X, he associaed problem is o minimize c R, subjec o he following consrains { α S u S 1 + r + c1 + r X u, α S d S 1 + r + c1 + r X d.

17 16 CHAPTER 1. AN INTRODUCTION TO FINANCIAL DERIVATIVES I is clear ha he soluion o his problem is π s X = πx =π X, so ha replicaion appears o be opimal for he buyer also. We conclude ha he leas price he seller is ready o accep for X equals he maximal amoun he buyer is ready o pay for i. If we define he buyer s price of X, denoed by π b X, by seing π b X = π s X, hen π s X =π b X =π X; ha is, all prices coincide. This shows ha in a wo-sae, arbirage-free model, he arbirage price of any coningen claim can be defined using he opimaliy crierion. I appears ha such an approach o arbirage pricing can be exended o oher models; we prefer, however, o define he arbirage price as ha value of he price which excludes arbirage opporuniies. Indeed, he fac ha observed marke prices are close o arbirage prices prediced by a suiable sochasic model should be explained by he presence of he raders known as arbirageurs 4 on financial markes, raher han by he clever invesmen decisions of mos marke paricipans. The nex proposiion explains he role of he so-called risk-neural economy in arbirage pricing of derivaive securiies. Observe ha he imporan role of risk preferences in classic equilibrium asse pricing heory is lef aside in he presen conex. Noice, however, ha he use of a maringale measure P in arbirage pricing corresponds o he assumpion ha all invesors are risk-neural, meaning ha hey do no differeniae beween all riskless and risky invesmens wih he same expeced rae of reurn. The arbirage valuaion of derivaive securiies is hus done as if an economy acually were risk-neural. Formula 1.14 shows ha he arbirage price of a coningen claim X can be found by firs modifying he model so ha he sock earns a he riskless rae, and hen compuing he expeced value of he discouned claim o he bes of our knowledge, his mehod of compuing he price was discovered in Cox and Ross Proposiion The spo marke M =S, B, Φ is arbirage-free if and only if he discouned sock price process S admis a maringale measure P equivalen o P. In his case, he arbirage price a ime of any coningen claim X which seles a ime T is given by he risk-neural valuaion formula π X =E P 1 + r 1 X, 1.14 or explicily π X = S 1 + r S d S u S d X u 1+r + Su S 1 + r S u S d X d 1+r Proof. We know already ha he maringale measure for S equivalen o P exiss if and only if he unique soluion p of equaion 1.5 saisfies <p < 1. Suppose here is no equivalen maringale measure for S ; for insance, assume ha p 1. Our aim is o consruc explicily an arbirage opporuniy in he marke model S, B, Φ. To his end, observe ha he inequaliy p 1 is equivalen o 1+rS S u recall ha S u is always greaer han S d. The porfolio φ = 1,S saisfies V φ = and { S V T φ = u +1+rS if ω = ω 1, S d +1+rS > if ω = ω 2, so ha φ is indeed an arbirage opporuniy. On he oher hand, if p, hen he inequaliy S d 1 + rs holds, and i is easily seen ha in his case he porfolio ψ =1, S = φ is an arbirage opporuniy. Finally, if <p < 1 for any porfolio φ saisfying V φ =, hen by virue of 1.9 and 1.6 we ge p V u φ+1 p V d φ = so ha V d φ < when V u φ > and V d φ > if V u φ <. This shows ha here are no arbirage opporuniies in M when <p < 1. To prove formula 1.14 i is enough o compare i 4 An arbirageur is ha marke paricipan who consisenly uses he price discrepancies o make almos risk-free profis. Arbirageurs are relaively few,bu hey are far more acive han mos long-erm invesors.

18 1.2. CALL AND PUT SPOT OPTIONS 17 wih 1.9. Alernaively, we may observe ha for he unique porfolio φ =α,β which replicaes he claim X, we have E P 1 + r 1 X = E P 1 + r 1 V T φ = E P α ST + β = α S + β = V φ =π X, so ha we are done. Remarks. The choice of he bond price process as a discoun facor is no essenial. Suppose, on he conrary, ha we have chosen he sock price S as a numeraire. In oher words, we now consider he bond price B discouned by he sock price S B = B /S for {,T}. The maringale measure P for he process B is deermined by he equaliy B = E PB T, or explicily p 1+r 1+r + q Su S d = 1, 1.16 S where q =1 p. One finds ha P{ω 1 } = p = 1 S d rs S u S d S u S d 1.17 and 1 P{ω 2 } = q = S u 1 S u S d 1 + rs S d S u I is easy o show ha he properly modified version of he risk-neural valuaion formula has he following form π X =S E P S 1 T X, 1.19 where X is a coningen claim which seles a ime T. I appears ha in some circumsances he choice of he sock price as a numeraire is more convenien han ha of he savings accoun. Le us apply his approach o he call opion of Example One finds easily ha p =.62, and hus formula 1.19 gives as expeced. Ĉ = S E P S 1 T S T K + =21.59 = C, Pu Opion We refer once again o Example However, we shall now focus on a European pu opion insead of a call opion. Since he buyer of a pu opion has he righ o sell a sock a a given dae T, he erminal payoff from he opion is now P T =K S T +, i.e., { P P T ω = u =, if ω = ω 1, P d =2, if ω = ω 2, where we have aken, as before, K= $28. The porfolio φ =α,β which replicaes he European pu opion is hus deermined by he following sysem of linear equaions { 32 α +1.5 β =, 26 α +1.5 β =2, so ha α = 1/3 and β = Consequenly, he arbirage price P of he European pu opion equals P = 1/ =8.25.

19 18 CHAPTER 1. AN INTRODUCTION TO FINANCIAL DERIVATIVES Noice ha he number of shares in a replicaing porfolio is negaive. This means ha an opion wrier who wishes o hedge risk exposure should sell shor a ime he number α =1/3 shares of sock for each sold pu opion. The proceeds from he shor-selling of shares, as well as he opion s premium, are invesed in an ineres-earning accoun. To find he arbirage price of he pu opion we may alernaively apply Proposiion By virue of 1.14, wih X = P T, we ge P = E P 1 + r 1 P T =8.25. Finally, he pu opion value can also be found by applying he following relaionship beween he prices of call and pu opions. Corollary The following pu-call pariy relaionship is valid C P = S 1 + r 1 K. 1.2 Proof. The formula is an immediae consequence of equaliy 1.3 and he pricing formula 1.14 applied o he claim S T K. I is worhwhile o menion ha relaionship 1.2 is universal ha is, i does no depend on he choice of he model he only assumpion we need o make is he addiiviy of he price. Using he pu-call pariy, we can calculae once again he arbirage price of he pu opion. Formula 1.2 yields immediaely P = C S +1+r 1 K =8.25. For ease of furher reference, we shall wrie down explici formulae for he call and pu price in he one-period, wo-sae model. We assume, as usual, ha S u >K>S d. Then and C = S 1 + r S d S u S d P = Su S 1 + r S u S d S u K 1+r, 1.21 K S d 1+r Fuures Call and Pu Opions We will firs describe very succincly he main feaures of fuures conracs, which are refleced in sochasic models of fuures markes o be developed laer. As in he previous secion, we will focus mainly on he arbirage pricing of European call and pu opions; clearly, insead of he spo price of he underlying asse, we will now consider is fuures price. The model of fuures prices we adop here is quie similar o he one used o describe spo prices. Sill, due o he specific feaures of fuures conracs used o se up a replicaing sraegy, one has o modify significanly he way in which he payoff from a porfolio is defined Fuures Conracs and Fuures Prices A fuures conrac is an agreemen o buy or sell an asse a a cerain dae in he fuure for a cerain price. The imporan feaure of hese conracs is ha hey are raded on exchanges. Consequenly, he auhoriies need o define precisely all he characerisics of each fuures conrac in order o make rading possible. More imporanly, he fuures price he price a which a given fuures conrac is enered ino is deermined on a given fuures exchange by he usual law of demand and supply in a similar way as for spo prices of lised socks. Fuures prices are herefore seled daily and he quoaions are repored in he financial press. A fuures conrac is referred o by is delivery monh, however an exchange specifies he period wihin ha monh when delivery mus be made. The exchange specifies he amoun of he asse o be delivered for one conrac, as well as some addiional deails when necessary e.g., he qualiy of a given commodiy or he mauriy

20 1.3. FUTURES CALL AND PUT OPTIONS 19 of a bond. From our perspecive, he mos fundamenal feaure of a fuures conrac is he way he conrac is seled. The procedure of daily selemen of fuures conracs is called marking o marke. A fuures conrac is worh zero when i is enered ino; however, each invesor is required o deposi funds ino a margin accoun. The amoun ha should be deposied when he conrac is enered ino is known as he iniial margin. A he end of each rading day, he balance of he invesor s margin accoun is adjused in a way ha reflecs daily movemens of fuures prices. To be more specific, if an invesor assumes a long posiion, and on a given day he fuures price rises, he balance of he margin accoun will also increase. Conversely, he balance of he margin accoun of any pary wih a shor posiion in his fuures conrac will be properly reduced. Inuiively, i is hus possible o argue ha fuures conracs are acually closed ou afer each rading day, and hen sar afresh he nex rading day. Obviously, o offse a posiion in a fuures conrac, an invesor eners ino he opposie rade o he original one. Finally, if he delivery period is reached, he delivery is made by he pary wih a shor posiion One-period Fuures Marke I will be convenien o sar his secion wih a simple example which, in fac, is a sraighforward modificaion of Example o a fuures marke. Example Le f = f S, T be a one-period process which models he fuures price of a cerain asse S, for he selemen dae T T. We assume ha f = 28, and { f f T ω = u = 32, if ω = ω 1, f d = 26, if ω = ω 2, where T = 3 monhs. 5 We consider a 3-monh European fuures call opion wih srike price K = $28. As before, we assume ha he simple risk-free ineres rae for 3-monh deposis and loans is r = 5%. The payoff from he fuures call opion C f T =f T K + equals C f T ω = { C fu =4, if ω = ω 1, C fd =, if ω = ω 2. A porfolio φ which replicaes he opion is composed of α fuures conracs and β unis of cash invesed in riskless bonds or borrowed. The wealh process V f φ, {, T}, of his porfolio equals V f φ =β, since fuures conracs are worhless when hey are firs enered ino. Furhermore, he erminal wealh of φ is V f T φ =α f T f +1+rβ, 1.23 where he firs erm on he righ-hand side represens gains or losses from he fuures conrac, and he second corresponds o a savings accoun or loan. Noe ha 1.23 reflecs he fac ha fuures conracs are marked o marke daily ha is, afer each period in our model. A porfolio φ =α,β is said o replicae he opion when V f T = Cf T, or more explicily, if he equaliies V f T ω = { α f u f rβ = C fu, if ω = ω 1, α f d f +1+rβ = C fd, if ω = ω 2 are saisfied. For Example 1.3.1, his gives he following sysem of linear equaions { 4 α +1.5 β =4, 2 α +1.5 β =, 5 Noice ha in he presen conex,he knowledge of he selemen dae T of a fuures conrac is no essenial. I is sufficien o assume ha T T.

21 2 CHAPTER 1. AN INTRODUCTION TO FINANCIAL DERIVATIVES yielding α =2/3 and β =12.7. The manufacuring cos of a fuures call opion is hus C f = V f φ =β =12.7. Similarly, he unique porfolio replicaing a sold pu opion is deermined by he following condiions { 4 α +1.5 β =, 2 α +1.5 β =2, so ha α = 1/3 and β =12.7in his case. Consequenly, he manufacuring coss of pu and call fuures opions are equal in our example. As we shall see soon, his is no a pure coincidence; in fac, by virue of formula 1.29 below, he prices of call and pu fuures opions are equal when he opion s srike price coincides wih he iniial fuures price of he underlying asse. The above consideraions may be summarized by means of he following exhibis noe ha β is a posiive number one sold fuures opion C f, a ime = fuures conracs, cash deposied in a bank β = C f, and a ime = T where, as before, ˆr =1+r. opion s payoff C f T, profis/losses from fuures α f T f, cash wihdrawal ˆrβ, Maringale Measure for a Fuures Marke We are looking now for a probabiliy measure P which makes he fuures price process wih no discouning follow a P-maringale. A probabiliy P, if i exiss, is hus deermined by he equaliy I is easily seen ha f = E P ft = pf u +1 p f d P{ω 1 } = p = f f d f u f d, P{ω2 } =1 p = f u f f u f d Using he daa of Example 1.3.1, one finds easily ha p = 1/3. Consequenly, he expeced value under he probabiliy P of he discouned payoff from he fuures call opion equals C f = E P 1 + r 1 f T K + =12.7= C f. This illusraes he fac ha he maringale approach may be used also in he case of fuures markes, wih a suiable modificaion of he noion of a maringale measure. Remarks. Using he radiional erminology of mahemaical finance, we may conclude ha he risk-neural fuures economy is characerized by he fair-game propery of he process of a fuures price. Remember ha he risk-neural spo economy is he one in which he discouned sock price as opposed o he sock price iself models a fair game Absence of Arbirage In his subsecion, we shall sudy a general wo-sae, one-period model of a fuures price. We consider he filered probabiliy space Ω, F {,T }, P inroduced in Sec The firs process, which inends o model he dynamics of he fuures price of a cerain asse for he fixed selemen dae T T, is an adaped and sricly posiive process f = f S, T,=,T. More specifically, f is assumed o be a real number, and f T is he following random variable { f f T ω = u, if ω = ω 1, f d, if ω = ω 2,

22 1.3. FUTURES CALL AND PUT OPTIONS 21 where, by convenion, f u >f d. The second securiy is, as in he case of a spo marke, a riskless bond whose price process is B =1, B T =1+r for some real r. Le Φ f sand for he linear space of all fuures conracs-bonds porfolios φ = φ =α,β ; i may be, of course, idenified wih he linear space R 2. The wealh process V f φ of any porfolio equals V φ =β, and V f T φ =α f T f +1+rβ 1.26 i is useful o compare hese formulae wih 1.4. We shall sudy he valuaion of derivaives in he fuures marke model M f =f,b,φ f. I is easily seen ha an arbirary coningen claim X which seles a ime T admis a unique replicaing porfolio φ Φ f. Pu anoher way, all coningen claims which sele a ime T are aainable in he marke model M f. In fac, if X is given by he formula { X u if ω = ω Xω = 1, X d if ω = ω 2, hen is replicaing porfolio φ Φ f may be found by solving he following sysem of linear equaions { α f u f +1+rβ = X u, α f d f +1+rβ = X d The unique soluion of 1.27 is α = Xu X d f u f d, β = Xu f f d +X d f u f 1 + rf u f d Consequenly, he manufacuring cos π f X inmf equals π f def X = V f φ =β = Xu f f d +X d f u f 1 + rf u f d We say ha a model M f of he fuures marke is arbirage-free if here are no arbirage opporuniies in he class Φ f of rading sraegies. The following simple resul provides necessary and sufficien condiions for he arbirage-free propery of M f. Proposiion The fuures marke M f =f,b,φ f is arbirage-free if and only if he process f ha models he fuures price admis a unique maringale measure P equivalen o P. In his case, he arbirage price a ime of any coningen claim X which seles a ime T equals π f X =E P 1 + r 1 X, 1.3 or explicily π f X = f f d f u f d X u 1+r + f u f f u f d X d 1+r Proof. If here is no maringale measure for f which is equivalen o P, we have eiher p 1or p. In he firs case, we have f f d f u f d and hus f f u >f d. Consequenly, a porfolio φ = 1, is an arbirage opporuniy. Similarly, when p he inequaliies f f d <f u are valid. Therefore he porfolio φ =1, is an arbirage opporuniy. Finally, if < p < 1 and for some φ Φ f we have V f φ =, hen i follows from 1.29 ha f f d f u f d V fu + f u f f u f d V fd = so ha V fd < if V fu >, and V fu < when V fd >. This shows ha he marke model M f is arbirage-free if and only if he process f admis a maringale measure equivalen o P. The valuaion formula 1.3 now follows by

23 22 CHAPTER 1. AN INTRODUCTION TO FINANCIAL DERIVATIVES When he price of he fuures call opion is already known, in order o find he price of he corresponding pu opion one may use he following relaion, which is an immediae consequence of equaliy 1.3 and he pricing formula 1.3 C f P f =1+r 1 f K I is now obvious ha he equaliy C f = P f is valid if and only if f = K; ha is, when he curren fuures price and he srike price of he opion are equal. Equaliy 1.32 is referred o as he pu-call pariy relaionship for fuures opions One-period Spo/Fuures Marke Consider an arbirage-free, one-period spo marke S, B, Φ described in Sec Moreover, le f = f S, T, {,T} be he process of fuures prices wih he underlying asse S and for he mauriy dae T. In order o preserve consisency wih he financial inerpreaion of he fuures price, we have o assume ha f T = S T. Our aim is o find he righ value f of he fuures price a ime ; ha is, ha level of he price f which excludes arbirage opporuniies in he combined spo/fuures marke. In such a marke, rading in socks, bonds, as well as enering ino fuures conracs is allowed. Corollary The fuures price a ime for he delivery dae T of he underlying asse S which makes he spo/fuures marke arbirage-free equals f =1+rS. Proof. Suppose an invesor eners a ime ino one fuures conrac. The payoff of his posiion a ime T corresponds o a ime T coningen claim X = f T f = S T f. Since i coss nohing o ener a fuures conrac we should have π X =π S T f =, or equivalenly π X =S 1 + r 1 f =. This proves he assered formula. Alernaively, one can check ha if he fuures price f were differen from 1 + rs, his would lead o arbirage opporuniies in he spo/fuures marke. 1.4 Forward Conracs A forward conrac is an agreemen, signed a he iniial dae, o buy or sell an asse a a cerain fuure ime T called delivery dae or mauriy in wha follows for a prespecified price K, referred o as he delivery price. In conras o sock opions and fuures conracs, forward conracs are no raded on exchanges. By convenion, he pary who agrees o buy he underlying asse a ime T for he delivery price K is said o assume a long posiion in a given conrac. Consequenly, he oher pary, who is obliged o sell he asse a he same dae for he price K, is said o assume a shor posiion. Since a forward conrac is seled a mauriy and a pary in a long posiion is obliged o buy an asse worh S T a mauriy for K, i is clear ha he payoff from he long posiion from he shor posiion, respecively in a given forward conrac wih a sock S being he underlying asse corresponds o he ime T coningen claim X X, respecively, where X = S T K I should be emphasized ha here is no cash flow a he ime he forward conrac is enered ino. In oher words, he price or value of a forward conrac a is iniiaion is zero. Noice, however, ha for >, he value of a forward conrac may be negaive or posiive. As we shall now see, a forward conrac is worhless a ime provided ha a judicious choice of he delivery price K is made.

24 1.5. OPTIONS OF AMERICAN STYLE 23 Before we end his secion, we shall find he raional delivery price for a forward conrac. To his end, le us inroduce firs he following definiion which is, of course, consisen wih ypical feaures of a forward conrac. Recall ha, ypically, here is no cash flow a he iniiaion of a forward conrac. Definiion The delivery price K ha makes a forward conrac worhless a iniiaion is called he forward price of an underlying financial asse S for he selemen dae T. Noe ha we use here he adjecive financial in order o emphasize ha he sorage coss, which have o be aken ino accoun when sudying forward conracs on commodiies, are negleced. In he case of a dividend-paying sock, in he calculaion of he forward price, i is enough o subsiue S wih S Î, where Î is he presen value of all fuure dividend paymens during he conrac s lifeime cf. Sec Proposiion Assume ha he one-period, wo-sae securiy marke model S, B, Φ is arbiragefree. Then he forward price a ime for he selemen dae T of one share of sock S equals F S,T = 1 + rs. Proof. We shall apply he maringale mehod of Proposiion By applying formulae 1.14 and 1.33, we ge π X =E P ˆr 1 X = E P S T ˆr 1 K = S ˆr 1 K =, 1.34 where ˆr =1+r. I is now apparen ha F S,T = 1 + rs. By combining Corollary wih he above proposiion, we conclude ha in a one-period model of a spo marke, he fuures and forward prices of financial asses for he same selemen dae are equal. 1.5 Opions of American Syle An opion of American syle or briefly, an American opion is an opion conrac in which no only he decision wheher o exercise he opion or no, bu also he choice of he exercise ime, is a he discreion of he opion s holder. The exercise ime canno be chosen afer he opion s expiry dae T. Hence, in our simple one-period model, he srike price can eiher coincide wih he iniial dae, or wih he erminal dae T. Noice ha he value or he price a he erminal dae of he American call or pu opion wrien on any asse equals he value of he corresponding European opion wih he same srike price K. Therefore, he only unknown quaniy is he price of he American opion a ime. In view of he early exercise feaure of he American opion, he concep of perfec replicaion of he erminal opion s payoff is no adequae for valuaion purposes. To deermine his value, we shall make use of he general rule of absence of arbirage in he marke model. By definiion, he arbirage price a ime of he American opion should be se in such a way ha rading in American opions would no desroy he arbirage-free feaure he marke. We will firs show ha he American call wrien on a sock ha pays no dividends during he opion s lifeime is always equivalen o he European call; ha is, ha boh opions necessarily have idenical prices a ime. As we shall see in wha follows, such a propery is no always rue in he case of American pu opions; ha is, American and European pus are no equivalen, in general. We place ourselves once again wihin he framework of a one-period spo marke M =S, B, Φ, as specified in Sec I will be convenien o assume ha European opions are raded securiies in our marke. For =,T, le us denoe by C a and P a he arbirage price a ime of he American call and pu, respecively. I is obvious ha C a T = C T and P a T = P T. As menioned earlier, boh arbirage prices C a and P a will be deermined using he following propery: if he marke M =S, B, Φ is arbirage-free, hen he marke wih rading in socks, bonds and American

25 24 CHAPTER 1. AN INTRODUCTION TO FINANCIAL DERIVATIVES opions should remain arbirage-free. I should be noed ha i is no eviden a priori ha he las propery deermines in a unique way he values of C a and P a. We assume hroughou ha he inequaliies S d <S 1 + r <S u hold and he srike price saisfies S d <K<S u. Oherwise, eiher he marke model would no be arbirage-free, or valuaion of he opion would be a rivial maer. Proposiion Assume ha he risk-free ineres rae r is a non-negaive real number. Then he arbirage price C a of an American call opion in he arbirage-free marke model M =S, B, Φ coincides wih he price C of he European call opion wih he same srike price K. Proof. Assume, on he conrary, ha C a C. Suppose firs ha C a >C. Noice ha he arbirage price C saisfies C = p S u K 1+r = 1 + rs S d S u S d S u K 1+r >S K, 1.35 if r. I is now sraighforward o check ha here exiss an arbirage opporuniy in he marke. In fac, o creae a riskless profi, i is sufficien o sell he American call opion a C a, and simulaneously buy he European call opion a C. If European opions are no raded, one may, of course, creae a replicaing porfolio for he European call a iniial invesmen C. The above porfolio is easily seen o lead o a riskless profi, independenly from he decision regarding he exercise ime made by he holder of he American call. If, on he conrary, he price C a were sricly smaller han C, hen by selling European calls and buying American calls, one would be able o creae a profiable riskless porfolio. I is worhwhile o observe ha inequaliy 1.35 is valid in a more general seup. Indeed, if r, S >K,and S T is a P -inegrable random variable, hen we have always + E P 1 + r 1 S T K + E P 1 + r 1 S T 1 + r 1 K = S 1 + r 1 K + S K, where he firs inequaliy follows by Jensen s inequaliy. Noice ha in he case of he pu opion we ge merely + E P 1 + r 1 K S T + E P 1 + r 1 K 1 + r 1 S T = 1 + r 1 K S + >K S, where he las inequaliy holds provided ha 1 <r<. If r =, we obain E P 1 + r 1 K S T + =K S. Finally, if r>, no obvious relaionship beween P and S K is available. This feaure suggess ha he counerpar of Proposiion he case of American pu should be more ineresing. Proposiion Assume ha r>. Then P a = P if and only if he inequaliy K S Su 1 + rs S u S d K S d 1+r is valid. Oherwise, P a = K S >P. If r =, hen invariably P a = P Proof. In view of 1.22, i is clear ha inequaliy 1.36 is equivalen o P K S. Suppose firs ha he las inequaliy holds. If, in addiion, P a >P P a <P, respecively, by selling he American pu and buying he European pu by buying he American pu and selling he European pu, respecively one creaes a profiable riskless sraegy. Hence, P a = P in his case. 6 Suppose 6 To be formal,we need o check ha no arbirage opporuniies are presen if P a = P and 1.36 holds.

26 1.5. OPTIONS OF AMERICAN STYLE 25 now ha 1.36 fails o hold ha is, P <K S, and assume ha P a K S. We wish o show ha P a should be se o be K S, oherwise arbirage opporuniies arise. Acually, if P a were sricly greaer ha K S, he seller of an American pu would be able o lock in a profi by perfecly hedging exposure using he European pu acquired a a sricly lower cos P. If, on he conrary, inequaliy P a <K S were rue, i would be profiable o buy he American pu and exercise i immediaely. Summarizing, if 1.36 fails o hold, he arbirage price of he American pu is sricly greaer han he price of he European pu. Finally, one verifies easily ha if he holder of he American pu fails o exercise i a ime, he opion s wrier is sill able o lock in a profi. Hence, if 1.36 fails o hold, he American pu should be exercised immediaely, oherwise arbirage opporuniies would arise in he marke. For he las saemen, observe ha if r =, hen inequaliy 1.36, which now reads K S Su S S u S d K Sd, is easily seen o be valid i is enough o ake K = S d and K = S u. The above resuls sugges he following general raional exercise rule in a discree-ime framework: a any ime before he opion s expiry, find he maximal expeced payoff over all admissible exercise rules and compare he oucome wih he payoff obained by exercising he opion immediaely. If he laer value is greaer, exercise he opion immediaely, oherwise go one sep furher. In fac, one checks easily ha he price a ime of an American call or pu opion may be compued as he maximum expeced value of he payoff over all exercises, provided ha he expecaion in quesion is aken under he maringale probabiliy measure. The las feaure disinguishes arbirage pricing of American opions from he ypical opimal sopping problems, in which maximizaion of expeced payoffs akes place under a subjecive or acual probabiliy measure raher han under an arificial maringale measure. We conclude ha a simple argumen ha he raional opion s holder will always ry o maximize he expeced payoff of he opion a exercise is no sufficien o deermine arbirage prices of American claims. A more precise saemen would read: he American pu opion should be exercised by is holder a he same dae as i is exercised by a risk-neural individual whose objecive is o maximize he discouned expeced payoff of he opion; oherwise arbirage opporuniies would arise in he marke. I will be useful o formalize he concep of an American coningen claim. Definiion A coningen claim of American syle or shorly, American claim is a pair X a = X,X T, where X is a real number and X T is a random variable. We inerpre X and X T as he payoffs received by he holder of he American claim X a if he chooses o exercise i a ime and a ime T, respecively. Noice ha in our presen seup, he only admissible exercise imes are he iniial dae and he expiry dae, say τ = and τ 1 = T. By convenion, we say ha an opion is exercised a expiry dae T if i is no exercised prior o ha dae, even when is erminal payoff equals zero so ha in fac he opion is abandoned. We assume also, for simpliciy, ha T = 1. Then we may formulae he following corollary o Proposiions , whose proof is lef as exercise. Corollary The arbirage prices of an American call and an American pu opion in he arbirage-free marke model M =S, B, Φ are given by C a = max E P τ T 1 + r τ S τ K + and P a = max E P τ T 1 + r τ K S τ + respecively, where T denoes he class of all exercise imes. More generally, if X a =X,X T is an arbirary coningen claim of American syle, hen is arbirage price πx a in M =S, B, Φ equals π X a = max E P τ T 1 + r τ X τ, πt X a =X T.

27 26 CHAPTER 1. AN INTRODUCTION TO FINANCIAL DERIVATIVES General No-arbirage Inequaliies I is clear ha he following propery is valid in any discree- or coninuous-ime, arbirage-free marke. Price monooniciy rule. In any model of an arbirage-free marke, if X T and Y T are wo European coningen claims, where X T Y T, hen π X T π Y T for every [,T], where π X T and π Y T denoe he arbirage prices a ime of X T and Y T, respecively. Moreover, if X T >Y T, hen π X T >π Y T for every [,T]. For he sake of noaional convenience, a consan non-negaive rae r will now be inerpreed as a coninuously compounded rae of ineres. Hence, he price a ime of one dollar o be received a ime T equals e rt ; in oher words, he savings accoun process equals B = e r for every [,T]. Proposiion Le C and P C a and P a, respecively sand for he arbirage prices a ime of European American, respecively call and pu opions, wih srike price K and expiry dae T. Then he following inequaliies are valid for every [,T] S Ke rt + C = C a S, 1.37 Ke rt S + P K, 1.38 and K S + P a K The pu-call pariy relaionship, which in he case of European opions reads C P = S Ke rt, 1.4 akes, in he case of American opions, he form of he following inequaliies S K C a P a S Ke rt Proof. All inequaliies may be derived by consrucing appropriae porfolios a ime and holding hem o he erminal dae. Le us consider, for insance, he firs one. Consider he following porfolios, A and B. Porfolio A consiss of one European call and Ke rt of cash; porfolio B conains only one share of sock. The value of he firs porfolio a ime T equals C T + K =S T K + + K = max{s T,K} S T, while he value of porfolio B is exacly S T. Hence, he arbirage price of porfolio A a ime dominaes he price of porfolio B ha is, C + Ke rt S, for every [,T]. Since he price of he opion is non-negaive, his proves he firs inequaliy in All remaining inequaliies in may be verified by means of similar argumens. To check ha C a = C, we consider he following porfolios: porfolio A one American call opion and Ke rt of cash; and porfolio B one share of sock. If he call opion is exercised a some dae [, T ], hen he value of porfolio A a ime equals S K + Ke rt <S, while he value of B is S. On he oher hand, he value of porfolio A a he erminal dae T is max{s T,K}, hence i dominaes he value of porfolio B, which is S T. This means ha early exercise of he call opion would conradic our general price monooniciy rule. A jusificaion of relaionship 1.4 is sraighforward, as C T P T = S T K. To jusify he second inequaliy in 1.41, noice ha in view of 1.4 and he obvious inequaliy P a P, we ge P a P = C a + Ke rt S, [,T]. The proof of he firs inequaliy in 1.41 goes as follows. Take he wo following porfolios: porfolio A one American call and K unis of cash; and porfolio B one American pu and one share of sock. If he pu opion is exercised a ime [, T ], hen he value of porfolio B a ime is K. On he oher hand, he value of porfolio A a his dae equals C + Ke r K. Therefore, porfolio A is more valuable a ime han porfolio B; ha is C a +K P a +S for every [,T].

28 Chaper 2 The Cox-Ross-Rubinsein Model A European call opion wrien on one share of a sock S, which pays no dividends during he opion s lifeime, is formally equivalen o he claim X whose payoff a ime T is coningen on he sock price S T, and equals X =S T K + def = max {S T K, }. 2.1 Therefore, he call opion value or price C T a he expiry dae T equals simply C T =S T K +. We assume here ha he erminal dae T is represened by a naural number. Our firs aim is o evaluae he opion price C a any insan =,...,T, when he price of a risky asse a sock is modelled by he Cox e al muliplicaive binomial laice his will be referred o as he CRR model of a sock price hereafer. 2.1 The CRR Model of a Sock Price We consider a discree-ime model of a financial marke wih he se of daes, 1,...,T, and wih wo primary raded securiies: a risky asse, referred o as a sock, and a risk-free invesmen, called a savings accoun or a bond. The savings accoun yields a consan rae of reurn r over each ime period [, +1], meaning ha is price process B equals by convenion B =1 where ˆr =1+r. The sock price process S saisfies B =1+r =ˆr, T, 2.2 ξ +1 = S +1 /S {u, d} 2.3 for =,...,T 1, where d<1+r<u are given real numbers and S is a sricly posiive consan. To provide a simple probabilisic model of he sock price, we assume ha ξ,=1,...,t are muually independen random variables on a common probabiliy space Ω, F, P, wih idenical probabiliy law P{ξ = u} = p =1 P{ξ = d} for every =1,...,T. The sock price process S can be modelled by seing Equivalenly, S = S ξ j, T. 2.4 j=1 S = S exp ζ j, T, 2.5 j=1 27

29 28 CHAPTER 2. THE COX-ROSS-RUBINSTEIN MODEL where ζ j s are independen, idenically disribued random variables such ha P{ζ j =lnu} = p =1 P{ζ j =lnd }, T. Due o represenaion 2.5, he sock price S given by 2.4 is frequenly referred o as an exponenial random walk. I should be sressed, however, ha he assumpion ha he random variables ξ,= 1,...,T, are muually independen is no essenial for our furher purposes; we may make his assumpion, wihou loss of generaliy, for mahemaical convenience. As we will see in wha follows, he arbirage price of any European or American coningen claim in he binomial model of a financial marke is independen of he choice of he probabiliy of upward and downward movemens of he sock price a any paricular node The CRR Opion Pricing Formula Le us inroduce some noaion. For any T, we wrie α o denoe he number of shares held during he period [, +1, while β sands for he dollar invesmen in he savings accoun during his period. To deermine he arbirage price of a call opion we shall show, using backward inducion, ha by adjusing his or her porfolio φ =α,β,=,...,t 1, a he beginning of each period, an invesor is able o mimic he payoff of an opion a ime T for every sae. We shall refer o his fac by saying ha he coningen claim X =S T K + admis a unique, dynamic, replicaing, self-financing sraegy. Though for concreeness we concenrae on European opions, i will soon become clear ha his propery remains valid for any European coningen claim X of he form X = gs T. Le us menion ha i is common o refer o such claims as pah-independen as opposed o pah-dependen claims, which have he form X = hs,s 1,...,S T ; ha is, hey may depend on he whole sample pah. The valuaion of a pu opion can be reduced, hrough he pu-call pariy relaionship, o ha of a call opion. Given a fixed mauriy dae 1 T T, we sar our analysis by considering he las period before he expiry dae, [T 1,T]. We assume ha a porfolio which replicaes he erminal payoff of a call opion is esablished a ime T 1, and remains fixed unil he expiry dae T. In oher words, we need o find he composiion of a porfolio φ T 1 =α T 1,β T 1 a he beginning of he las period in such a way ha is erminal wealh V T φ, which saisfies V T φ =α T 1 S T + β T 1ˆr, 2.6 replicaes he opion payoff C T ; ha is, V T φ = C T. Combining 2.1 wih 2.6, we ge he following equaliy α T 1 S T + β T 1ˆr =S T K By virue of our assumpions, we have S T = S T 1 ξ T 1 ; herefore, we may rewrie 2.7 in a more explici form { αt 1 us T 1 + β T 1ˆr =us T 1 K +, α T 1 ds T 1 + β T 1ˆr =ds T 1 K +. Such a sysem of linear equaions can be solved easily, yielding α T 1 = us T 1 K + ds T 1 K +, 2.8 S T 1 u d and β T 1 = uds T 1 K + dus T 1 K ˆru d Furhermore, he wealh V T 1 φ of his porfolio a ime T 1 equals V T 1 φ = α T 1 S T 1 + β T 1 = ˆr 1 p us T 1 K + +1 p ds T 1 K +,

30 2.1. THE CRR MODEL OF A STOCK PRICE 29 where p =ˆr d/u d. Assuming he absence of arbirage in he marke model, 1 he wealh V T 1 φ agrees wih he value ha is, he arbirage price of a call opion a ime T 1. Pu anoher way, he equaliy C T 1 = V T 1 φ is valid. We will coninue he above procedure by considering he ime-period [T 2,T 1]. In his sep, we are searching for a porfolio φ T 2 =α T 2,β T 2 which is creaed a ime T 2 in such a way ha is wealh a ime T 1 replicaes opion value C T 1 ; ha is α T 2 S T 1 + β T 2ˆr = C T Noice ha since C T 1 = V T 1 φ, he dynamic rading sraegy φ consruced in his way will possess he self-financing propery a ime T 1 α T 2 S T 1 + β T 2ˆr = α T 1 S T 1 + β T 1. Basically, he self-financing feaure means ha he porfolio is adjused a ime T 1 and more generally, a any rading dae in such a way ha no wihdrawals or inpus of funds ake place. Since S T 1 = S T 2 ξ T 2 and ξ T 2 {u, d}, we ge he following equivalen form of equaliy 2.1 { αt 2 us T 2 + β T 2ˆr = C u T 1, α T 2 ds T 2 + β T 2ˆr = C d T 1, 2.11 where and C u T 1 = 1ˆr C d T 1 = 1ˆr p u 2 S T 2 K + +1 p uds T 2 K + p uds T 2 K + +1 p d 2 S T 2 K +. In view of 2.11, i is eviden ha α T 2 = Cu T 1 Cd T 1 S T 2 u d, β T 2 = ucd T 1 dcu T 1. ˆru d Consequenly, he wealh V T 2 φ of he porfolio φ T 2 =α T 2,β T 2 a ime T 2 equals V T 2 φ =α T 2 S T 2 + β T 2 = 1ˆr p CT u 1 +1 p CT d 1 = 1ˆr 2 p 2 u 2 S T 2 K + +2p q uds T 2 K + + q 2 d 2 S T 2 K +. Using he same arbirage argumens as in he firs sep, we argue ha he wealh V T 2 φ of he porfolio φ a ime T 2 gives he arbirage price a ime T 2, i.e., C T 2 = V T 2 φ. I is eviden ha by repeaing he above procedure, one can compleely deermine he opion price a any dae T, as well as he unique rading sraegy φ ha replicaes he opion. Summarizing, he above reasoning provides a recursive procedure for finding he value of a call wih any number of periods o go noe ha i exends o he case of any claim X of he form X = gs T. I is worhwhile o noe ha in order o value he opion a a given dae and for a given level of he curren sock price S, i is enough o consider a sub-laice of he CRR binomial laice, which sars from S and involves T periods. Before we proceed furher, le us commen briefly on he informaion srucure of he CRR model. Le us denoe by F S he σ-field of all evens of F generaed by he observaions of he sock price S up o he dae, formally F S = σs,...,s for every T, where σs,...,s denoes he leas σ-field wih respec o which he random variables S,...,S are measurable. By consrucion of he replicaing sraegy, i is eviden ha for any fixed he random variables α,β which define he porfolio a ime, as well as he wealh V φ of his porfolio, are measurable wih respec o he σ-field F S. 1 We will reurn o his poin laer in his chaper. Le us only menion here ha he necessary and sufficien condiion for he absence of arbirage has he same form as in he case of he one-period model; ha is, d<1+r<u.

31 3 CHAPTER 2. THE COX-ROSS-RUBINSTEIN MODEL For any fixed m, le he funcion a m : R N be given by he formula N sands hereafer for he se of all non-negaive inegers a m x = min { j N xu j d m j >K}, where, by convenion, min =. To simplify he noaion, we wrie m x, j = j=a m j p j 1 p m j u j d m j x K Proposiion The arbirage price of a European call opion a ime = T m is given by he Cox-Ross-Rubinsein valuaion formula m m C T m = S T m p j 1 p m j Kˆr m m j m p j j 1 p m j 2.13 for m =1,...,T, where a = a m S T m,p =ˆr d/u d and p = p u/ˆr. A ime = T m 1, he unique replicaing sraegy has he form φ T m 1 =α T m 1,β T m 1, where m m α T m 1 = p j 1 p m j + δ mus T m 1,a u, j S T m 1 u d j=a d β T m 1 = K ˆr m+1 m j=a d j=a m p j j 1 p m j dδ mus T m 1,a u, ˆru d where a d = a m ds T m 1,a u = a m us T m 1 and δ =if a d = a u oherwise, δ =1. Proof. Sraighforward calculaions yield 1 p = d1 p /ˆr, and hus p j 1 p m j = p j 1 p m j u j d m j /ˆr m. Therefore, formula 2.13 is equivalen o he following C T m = 1 m m ˆr m p j j 1 p m j u j d m j S T m K = 1 ˆr m j=a m j= m j p j 1 p m j u j d m j S T m K +. We will now proceed by inducion wih respec o m. For m =, he above formula is manifesly rue. Assume now ha C T m is he arbirage price of a European call opion a ime T m. We have o selec a porfolio φ T m 1 =α T m 1,β T m 1 for he period [T m 1,T m ha is, esablished a ime T m 1 a each node of he binomial laice in such a way ha he porfolio s wealh a ime T m replicaes he value C T m of he opion. Formally, he wealh of he porfolio α T m 1,β T m 1 needs o saisfy he relaionship α T m 1 S T m + β T m 1ˆr = C T m, 2.14 which in urn is equivalen o he following pair of equaions { αt m 1 us T m 1 + β T m 1ˆr = C u T m, α T m 1 ds T m 1 + β T m 1ˆr = C d T m, where C u T m = 1 ˆr m = 1 ˆr m m m p j j 1 p m j + u j+1 d m j S T m 1 K j= m m p j j 1 p m j u j+1 d m j S T m 1 K j=a u

32 2.1. THE CRR MODEL OF A STOCK PRICE 31 and Consequenly, we have C d T m = 1 ˆr m = 1 ˆr m m m p j j 1 p m j + u j d m j+1 S T m 1 K j= m m p j j 1 p m j u j d m j+1 S T m 1 K. j=a d α T m 1 = Cu T m Cd T m S T m 1 u d 1 m = ˆr m u d = m j=a d j=a d m j where we wrie q =1 p. Similarly, p j q m j m p j 1 p m j + δ mus T m 1,a u. j S T m 1 u d u j+1 d m j u j d m j+1 + δ mus T m 1,a u S T m 1 u d β T m 1 = ucd T m dcu T m ˆru d 1 m m = ˆr m+1 p j u d j 1 p m j dk uk δ mus T m 1,a u ˆru d = K ˆr m+1 m j=a d j=a d m p j j 1 p m j dδ mus T m 1,a u. ˆru d The wealh of his porfolio a ime T m 1 equals noe ha jus esablished explici formulas for he replicaing porfolio are no employed here Finally, C T m 1 = α T m 1 S T m 1 + β T m 1 = u d 1 CT u m CT d m +ˆr 1 uct d m dct u m = ˆr 1 p CT u m +1 p CT d m 1 { m m = ˆr m+1 p j+1 q m j u j+1 d m j S T m 1 K + j j= m m + p j j q m+1 j u j d m+1 j S T m 1 K } + = C T m 1 = j= 1 { m+1 m ˆr m+1 p j j 1 q m+1 j u j d m+1 j S T m 1 K + j=1 m m + p j j q m+1 j u j d m+1 j S T m 1 K } +. j= 1 { ˆr m+1 m + j=1 p m+1 u m+1 S T m 1 K + m j p j q m+1 j u j d m+1 j S T m 1 K +

33 32 CHAPTER 2. THE COX-ROSS-RUBINSTEIN MODEL = m m + p j j 1 q m+1 j u j d m+1 j S T m 1 K + j=1 + q m+1 d m+1 S T m 1 K } + 1 ˆr m+1 m+1 j= m +1 j p j q m+1 j u j d m+1 j S T m 1 K +, since m j 1 + m j = m+1 j. This ends he proof of Proposiion I is imporan o noice ha he CRR valuaion formula 2.13 makes no reference o he subjecive probabiliy p. Inuiively, he pricing formula does no depend on he invesor s aiudes oward risk. The only assumpion made abou he behavior of an individual is ha all invesors prefer more wealh o less wealh, and hus have an incenive o ake advanage of riskless profiable invesmens. Consequenly, if arbirage opporuniies were presen in he marke, no marke equilibrium would be possible. This feaure of arbirage-free markes explains he erm parial equilibrium approach, frequenly used in economic lieraure in relaion o arbirage pricing of derivaive securiies. To summarize, no maer wheher invesors are risk-averse or risk-preferring, 2 we obain idenical arbirage prices for derivaive securiies. In his conex, i is worhwhile o poin ou ha he value p =ˆr d/u d corresponds o he risk-neural world ha is, a model of an economy in which all invesors are indifferen wih respec o risky invesmens whose discouned wealh is represened by maringales a he inuiive level, a maringale may be seen as a formalizaion of he concep of a fair game The Black-Scholes Opion Pricing Formula We will now show ha he classic Black-Scholes opion valuaion formula 2.22 can be obained from he CRR opion valuaion resul by an asympoic procedure, using a properly chosen sequence of binomial models. To his end, we need o examine he asympoic properies of he CRR model when he number of seps goes o infiniy and, simulaneously, he size of ime and space seps ends o zero in an appropriae way. Le T>be a fixed, bu arbirary, real number. For any n of he form n =2 k, we divide he inerval [,T]inon equal subinervals I j of lengh n = T/n, namely I j =[j n, j + 1 n ] for j =,...,n 1 noe ha n corresponds o T. Le us firs inroduce he modified accumulaion facor. We wrie r n o denoe he riskless rae of reurn over each inerval I j =[j n, j + 1 n ], hence B j n =1+r n j, j =,...,n. I is clear ha we deal in fac wih a sequence of processes, B n say; however, for he sake of noaional simpliciy, we shall usually omi he superscrip n in wha follows. The same remark applies o he sequence S = S n of binomial laices and random variables ξ n,j inroduced below. For every n, we assume ha he sock price can appreciae over he period I j by u n or decline by d n ; ha is S j+1 n = ξ n,j+1 S j n 2.15 for j =1,...,n 1, where for every fixed n, ξ n,j s are random variables wih values in he wo-elemen se {u n,d n }. In view of Proposiion 2.1.1, we may assume, wihou loss of generaliy, ha for any n he random variables ξ n,j,j=1,...,n are defined on a common probabiliy space Ω n, F n, P n, are muually independen, and P n {ξ n,j = u n } = p =1 P n {ξ n,j = d n }, j =1,...,n, 2 An ineresed reader may consul,for insance,huang and Lizenberger 1988 for he sudy of he noion of risk preferences under uncerainy.

34 2.1. THE CRR MODEL OF A STOCK PRICE 33 for some p, 1. Noe ha he choice of he parameer p, 1 is arbirary; for insance, we may assume ha p =1/2 for every n. In order o guaranee he convergence of he CRR opion valuaion formula o he Black-Scholes one, we need o impose, in addiion, specific resricions on he asympoic behavior of he quaniies r n,u n and d n. Le us pu 1+r n = e r n, u n = e σ n, d n = u 1 n, 2.16 where r and σ > are given real numbers. As menioned earlier, we wish o calculae he asympoic value of he call opion price when he number of seps, T n =n, ends o infiniy. Assume ha = j n = jt/2 k for some naural j and k; ha is, is an arbirary dyadic number from he inerval [,T]. Given any such number, we inroduce he sequence m n by seing m n =nt /T, n N I is apparen ha he sequence m n has naural values in he se for n sufficienly large. On he oher hand, T = m n n for every n N. Noice also ha lim 1 + r n mn = lim n + n + e r nmn = e rt Furhermore, for every n>r 2 σ 2 T we have d n = u 1 n < ˆr 1 n ˆr n <u n, where ˆr n =1+r n. Also, i is no difficul o check ha lim n + p,n = lim n + e r n e σ n e σ n e σ n =1/2, 2.19 and lim p n = lim n + n + ˆr 1 n p,n u n =1/ For a generic value of sock price a ime, S = S T mn n, we define a n = min { j N S u j nd mn j n >K} The nex proposiion provides he derivaion of he classic Black-Scholes opion valuaion formula by means of an asympoic procedure. I should be emphasized ha he limi of he CRR opion price depends essenially on he choice of sequences u n and d n. For he choice of u n s and d n s ha we have made here, he asympoic dynamic of he sock price is ha of he geomeric Brownian moion known also as he geomeric Wiener process. This means, in paricular, ha he asympoic evoluion of he sock price may be described by a sochasic process whose sample pahs almos all follow coninuous funcions; furhermore, he probabiliy law of he coninuous-ime sock price a any ime is lognormal. A sraighforward analysis of he coninuous-ime Black-Scholes model, based on he Iô sochasic calculus, is presened in Sec The proof of he nex resul is lef o he reader. Proposiion The following convergence is valid for any dyadic [,T] lim m n n + j=a n mn j { S p j n q n mn j Kˆr mn n where q n =1 p n, and C is given by he Black-Scholes formula } p j,nq,n mn j = C, C = S Nd 1 S,T Ke rt Nd 2 S,T, 2.22

35 34 CHAPTER 2. THE COX-ROSS-RUBINSTEIN MODEL where d 1 s, = lns/k+r σ2 σ, 2.23 d 2 s, = d 1 s, σ = lns/k+r 1 2 σ2 σ, 2.24 and N sands for he sandard Gaussian cumulaive disribuion funcion, Nx = 1 2π x e u2 /2 du, x R. Remarks. Le us sress ha for a differen choice of he sequences u n,d n and r n, he sock price may asympoically follow a sochasic process wih disconinuous sample pahs. For insance, if we pu u n = u and d n = e c/n hen he sock process will follow asympoically a log-poisson process, examined by Cox and Ross Essenially, his ineresing feaure is due o he fac ha we deal here wih a riangular array of random variables as opposed o an infinie sequence of independen random variables. For his reason, he class of asympoic probabiliy laws is much larger han in he case of he classic cenral limi heorem. 2.2 Probabilisic Approach The aim of his secion is o give a purely probabilisic inerpreaion of he CRR opion valuaion formula. We will proceed along he lines suggesed in remarks preceding Proposiion We sar by inroducing a finie probabiliy space Ω; namely, for a fixed naural number T, we ake Ω={ ω =a 1,...,a T a j =1ora j =}. In he presen conex, i will be sufficien o consider a specific class P of probabiliy measures on he measurable space Ω, F, where F is he σ-field of all subses of Ω, i.e., F =2 Ω. For any elemenary even ω =a 1,...,a T, we define is probabiliy Pω by seing T P{ω} = p j=1 aj 1 p T T j=1 aj, where <p<1 is a fixed real number. We wrie P o denoe he class of all probabiliy measures of his form on Ω, F. I is clear ha any elemen P Pis uniquely deermined by he value of he parameer p. Noe ha for every P P, he probabiliy of any even A Fequals P{A} = ω A P{ω}. For any j =1,...,T, le us denoe by A j he even A j = { ω Ω a j =1}. I is easily seen ha he evens A j,j=1,...,t, are muually independen; moreover, PA j =p for every j. We are in a posiion o define a sequence of random variables ξ j,j=1,...,t by seing ξ j ω =ua j + d1 a j, ω Ω, 2.25 where, wihou loss of generaliy, <d<u. The random variables ξ j are easily seen o be independen and idenically disribued, wih he following probabiliy law under P P{ξ j = u} = p =1 P{ξ j = d}, T As anicipaed, he sequence ξ j will be used o model sock price flucuaions in a probabilisic version of he CRR binomial laice.

36 2.2. PROBABILISTIC APPROACH Condiional Expecaions Le us sar by considering a finie decomposiion of he underlying probabiliy space. We say ha a finie collecion D = {D 1,...,D k } of non-empy subses of Ω is a decomposiion ofωif he ses D 1,...,D k are pairwise disjoin; ha is, if D i D j = for every i j, and he equaliy D 1 D 2... D k = Ω holds. A random variable ψ on Ω is called simple if i admis a represenaion ψω = m x i I Diψω, 2.27 i=1 where D i ψ ={ ω Ω ψω =x i } and x i,i=1,...,m are real numbers saisfying x i x j for i j. For a simple random variable ψ, we denoe by Dψ he decomposiion {D 1 ψ,...,d m ψ} generaed by ψ. I is clear ha if Ω is a finie se, hen any random variable ψ :Ω R is simple. Definiion For any decomposiion D of Ω and any even A F, he condiional probabiliy of A wih respec o D is defined by he formula PA D= k PA D j I Dj j=1 Moreover, if ψ is a simple random variable wih he represenaion 2.27, is condiional expecaion given D equals m m E P ψ D= x i E P I Diψ D= i=1 i=1 j=1 k x i PD i ψ D j I Dj Observe ha he condiional expecaion E P ψ D is consan on each se D j from D. Le η be anoher simple random variable on Ω, i.e., ηω = r y l I Dl ηω, 2.3 l=1 where D l η ={ ω Ω ηω =y l } and y i y j for i j. Suppose ha η and ψ are simple random variables given from 2.27 and 2.3, respecively. Then, by he definiion of condiional expecaion, E P ψ η coincides wih E P ψ Dη, and hus E P ψ η =E P ψ Dη = m x i E P I Dk ψ Dη, 2.31 i=1 where he second equaliy follows by Consequenly, E P ψ η = m i=1 l=1 r x i PD i ψ D l η I Dl η = r c l I Dl η, 2.32 where c l = m i=1 x i PD i ψ D l η for l =1,...,r. Noe ha E P ψ η does no depend on he paricular values of η. More precisely, if for wo random variables η 1 and η 2 we have Dη 1 =Dη 2, hen E P ψ η 1 =E P ψ η 2. On he oher hand, however, by virue of 2.32 i is clear ha E P ψ η =gη, where he funcion g : {y 1,...,y r } R is given by he formula gy l =c l for l =1, 2,...,r. We shall now define he condiional expecaion of a random variable wih respec o an arbirary σ-field G. Definiion Suppose ha ψ is an inegrable random variable on Ω, F, P i.e., E P ψ <. For an arbirary σ-field G of subses of Ω saisfying G Fi.e., a sub-σ-field of F, he condiional l=1

37 36 CHAPTER 2. THE COX-ROSS-RUBINSTEIN MODEL expecaion E P ψ Gofψ wih respec o G is defined by he following condiions: i E P ψ Gis G-measurable, i.e., {ω Ω E P ψ Gω a} Gfor any a R; ii for an arbirary even A G, we have E P ψ G dp = ψdp A I is well known ha he condiional expecaion exiss and is unique up o he P-a.s. equivalence of random variables. Remarks. If D is a finie decomposiion of Ω, he family of all unions of ses from D, ogeher wih an empy se, forms a σ-field of subses of Ω. We denoe i by σd and we call i he σ-field generaed by he decomposiion 3 D. I can be easily checked ha if G = σd, hen PA G=PA D, so ha he condiional expecaion wih respec o he σ-field σd coincides wih he condiional expecaion wih respec o he decomposiion D. Definiion For an arbirary simple random variable η, he σ-field Fη = σdη is called he σ-field generaed by η, or briefly, he naural σ-field of η. More generally, for any random variable η, he σ-field generaed by η is he leas σ-field of subses of Ω wih respec o which η is measurable. I is denoed by eiher ση or Fη. In he case of a real-valued random variable η, i can be shown ha he σ-field Fη ishe smalles σ-field ha conains all evens of he form {ω Ω ηω x}, where x is an arbirary real number. More generally, if η =η 1,...,η d isad-dimensional random variable, hen Fη =σ {ω Ω η 1 x 1,...,η d x d } x 1,...,x d R. For any P-inegrable random variable ψ and any random variable η, we define he condiional expecaion of ψ wih respec o η by seing E P ψ η =E P ψ Fη. I is possible o show ha for arbirary real-valued random variables ψ and η, here exiss a Borel measurable funcion g : R R such ha E P ψ η =gη. The following resul is sandard. Lemma Le ψ and η be inegrable random variables on Ω, F, P. Also le G and H be some sub-σ-fields of F. Then: i if ψ is G-measurable or equivalenly, if Fψ G, hen E P ψ G=ψ; ii for arbirary real numbers c, d, we have iii if H G, hen E P cψ + dη G=cE P ψ G+dE P η G; E P E P ψ G H = E P E P ψ H G = E P ψ H; iv if ψ is independen of G, i.e., he σ-fields Fψ and G are independen 4 under P, hen E P ψ G= E P ψ; v if ψ is G-measurable and η is independen of G, hen for any Borel measurable funcion h : R 2 R we have E P hψ, η G=Hψ, where Hx =E P hx, η, provided ha he inequaliy E P hψ, η < holds. Absrac Bayes formula. The las resul refers o he siuaion where wo muually equivalen probabiliy measures, P and Q say, are defined on a common measurable space Ω, F. Suppose ha he Radon-Nikodým derivaive of Q wih respec o P equals dq = η, P-a.s dp Noe ha he random variable η is sricly posiive P-a.s., moreover η is P-inegrable, wih E P η = 1. Finally, by virue of 2.34, i is clear ha equaliy E Q ψ = E P ψη holds for any Q-inegrable random variable ψ. 3 In he case of a finie Ω, any σ-field G of subses of Ω is of his form; ha is, G = σd for some decomposiion D of Ω. 4 We say ha wo σ-fields, G and H, are independen under P whenever PA B =PAPB for every A G and every B H. A

38 2.2. PROBABILISTIC APPROACH 37 Lemma Le G be a sub-σ-field of he σ-field F, and le ψ be a random variable inegrable wih respec o Q. Then he following absrac version of he Bayes formula holds E Q ψ G= E Pψη G E P η G Proof. I can be easily checked ha E P η G is sricly posiive P-a.s. so ha he righ-hand side of 2.35 is well-defined. By our assumpion, he random variable ξ = ψη is P-inegrable, herefore i is enough o show ha E P ξ G=E Q ψ GE P η G. Since he righ-hand side of he las formula defines a G-measurable random variable, we need o verify ha for any se A G, we have ψη dp = E Q ψ GE P η G dp. A Bu for every A G, we ge ψη dp = ψdq = E Q ψ G dq = E Q ψ GηdP A A A A = E P E Q ψ Gη G dp = E Q ψ G E P η G dp Maringale Measure A A Le us reurn o he muliplicaive binomial laice modelling he sock price. In he presen framework, he process S is deermined by he iniial sock price S and he sequence ξ j, j =1,...,T, of independen random variables given by More precisely, S,=,...,T, is defined on he probabiliy space Ω, F, P by means of 2.4, or equivalenly, by he relaion A S +1 = ξ +1 S, <T, 2.36 wih S R +. Le us inroduce he process S of he discouned sock price by seing S = S /B = S /ˆr, T Le D S be he family of decomposiions 5 of Ω generaed by random variables S u,u ; ha is, D S = DS,...,S for every T. I is clear ha he family D S, T, of decomposiions is an increasing family of σ-fields, meaning ha D S D+1 S for every T 1. Noice ha he family D S is also generaed by he family ξ 1,...,ξT of random variables, more precisely D S = Dξ,ξ 1,...,ξ, T, where by convenion ξ =1. The family D S, T, models a discree-ime flow of informaion generaed by he observaions of sock prices. In financial inerpreaion, he decomposiion D S represens he marke informaion available o all invesors a ime. Le us denoe F S = σd S for every T, where σd S is he σ-field generaed by he decomposiion D S. I is clear ha F S = σs,s 1,...,S, T. Finally, we wrie F S =F S T o denoe he family of naural σ-fields of he process S, or briefly, he naural filraion of he process S. 5 We say ha a finie collecion D = {D 1,...,D k } of non-empy evens is a decomposiion of Ω if he evens D 1,...,D k are pairwise disjoin; ha is,if D i D j = for every i j, and he equaliy D 1 D 2... D k =Ω holds.

39 38 CHAPTER 2. THE COX-ROSS-RUBINSTEIN MODEL The nex definiion inroduces a probabiliy measure, P, which is equivalen o any probabiliy measure P from P, such ha he discouned sock price S behaves under P like a fair game wih respec o is naural filraion. We will assume, in addiion, ha P belongs o he class P. In view of he definiion of he class P, any probabiliy measure from P depends on he choice of he underlying parameers u, d and r hrough he value of p only. Furhermore, no generaliy is los by his assumpion; i is no hard o check ha here is no probabiliy measure P ouside P which would be equivalen o any probabiliy measure from he class P, and such ha he discouned sock price S in he CRR model would follow a P -maringale. Therefore, we may adop, wihou loss of generaliy, he following definiion of he maringale measure. Definiion A probabiliy measure P P is called a maringale measure for he discouned sock price process S if E P S+1 F S =S, T 1; 2.38 ha is, if he process S follows a maringale under P wih respec o he filraion F S. In his case, we say ha he discouned sock price S is a P, F S -maringale, or briefly, a P -maringale, if no confusion may arise. In some circumsances, we shall assume ha he sock price is given on an underlying filered probabiliy space Ω, F, P, where he underlying filraion F is sricly greaer han he naural filraion F S generaed by he sock price process. In he presen seing, however, i is convenien o ake F = F S, as is implici in he definiion above. Lemma A maringale measure P P for he discouned sock price S exiss if and only if d<1+r =ˆr<u. In his case, he maringale measure P for S is he unique elemen from he class P ha corresponds o p = p =1+r d/u d. Proof. Using , we may re-express equaliy 2.38 in he following way E P ˆr +1 ξ +1 S F S =ˆr S, T 1, 2.39 or equivalenly ˆr +1 S E P ξ +1 F S =ˆr S, T 1. Since he random variable ξ +1 is independen of he σ-field F S, for he las equaliy o hold i is necessary and sufficien ha E P ξ +1 =ˆr, or explicily, ha he equaliy up + d1 p =ˆr is saisfied. By solving he las equaion for p, we find he unique value of p for which 2.39 is valid. Noice ha he sock price follows under he unique maringale measure P, an exponenial random walk cf. formula 2.5, wih he probabiliy of upward movemen equal o p. This feaure explains why i was possible, wih no loss of generaliy, o resric aenion o he special class of probabiliy measures on he underlying canonical space Ω. I should be sressed, however, ha he maringale probabiliy measure P does no model he observed real-world flucuaions of sock prices. On he conrary, i is merely a echnical ool which proves o be very useful in he arbirage valuaion. Remarks. The noion of a maringale measure or risk-neural probabiliy depends essenially on he choice of a numeraire asse. I can be checked ha he unique maringale measure for he relaive bond price B = B/S is he unique elemen P from he class P ha corresponds o he following value of p see formula p = p = d 1ˆr ud u d.

40 2.2. PROBABILISTIC APPROACH Risk-neural Valuaion Formula For our furher purposes, we find i convenien o re-examine he problem of arbirage opion pricing wihin he framework of he CRR binomial model by means of probabilisic mehods. Observe ha in m periods o he opion expiry dae T, he discouned payoff of a call opion equals ˆr m S T K +, hence i depends on he erminal sock price S T which, of course, is no known a ime T m. We shall show ha he CRR opion valuaion formula esablished in Proposiion may be alernaively derived by he direc evaluaion of he condiional expecaion, under he maringale measure P, of he discouned opion s payoff. Proposiion Consider a European call opion, wih expiry dae T and srike price K, wrien on one share of a common sock whose price S is assumed o follow he CRR muliplicaive binomial process 2.4. Then he arbirage price C T m, given by formula 2.13, coincides wih he condiional expecaion CT m, which equals C T m = E P ˆr m S T K + F S T m, m T. 2.4 Proof. I is sufficien o find explicily he condiional expecaion 2.4. Recall ha S T = S T m ξ T m+1...ξ T = S T m η m, where he sock price S T m is a FT S m-measurable random variable, and he random variable η m = ξ T m+1...ξ T is independen of he σ-field FT S m. By applying he well-known propery of condiional expecaions see Lemma o he random variables ψ = S T m,η= η m and o he funcion hx, y =ˆr m xy K +, one finds ha where he funcion H : R R equals C T m = E P ˆr m S T K + F S T m = HST m, Hx =E P hx, ηm = E P ˆr m xη m K +, x R. Since he random variables ξ T m+1,...,ξ T are muually independen and idenically disribued under P, wih P {ξ j = u} = p =1 P {ξ j = d}, i is also clear ha Hx =ˆr m m j= m p j j 1 p m j xu j d m j K +. Using equaliies p = p u/ˆr and 1 p =1 p d/ˆr, we conclude ha C T m = m j=a m S T m p j 1 p m j Kˆr m p j j 1 p m j, where a = min { j N S T m u j d m j >K}. We may rewrie 2.4 in he following way C = B E P B 1 T S T K + F S = B E P B 1 T X F S, 2.41 where X =S T K +. One migh wonder if he valuaion formula 2.41 remains in force for a larger class of financial models and European coningen claims X. Generally speaking, he answer o his quesion is posiive, even if he ineres rae is assumed o follow a sochasic process. Remarks. I is ineresing o noice ha he CRR valuaion formula 2.13 may be rewrien as follows for simpliciy, we focus on he case = C = S P {ST >K} Kˆr T P {S T >K}, 2.42 where P and P are maringale measures corresponding o he choice of he sock price and he bond price as a numeraire, respecively.

41 4 CHAPTER 2. THE COX-ROSS-RUBINSTEIN MODEL 2.3 Valuaion of American Opions Le us firs consider he case of he American call opion ha is, he opion o buy a specified number of shares, which may be exercised a any ime before he opion expiry dae T, or on ha dae. The exercise policy of he opion holder is necessarily based on he informaion accumulaed o dae and no on he fuure prices of he sock. As in he previous chaper, we will wrie C a o denoe he arbirage price a ime of an American call opion wrien on one share of a sock. By arbirage price of he American call we mean such price process C a, T, ha an exended financial marke model ha is, a marke wih rading in riskless bonds, socks and American call opions remains arbirage-free. Our firs goal is o show ha he price of an American call opion in he CRR arbirage-free marke model coincides wih he arbirage price of a European call opion wih he same expiry dae and srike price. For his purpose, i is sufficien o show ha he American call opion should never be exercised before mauriy, since oherwise he opion wrier would be able o make riskless profi. The argumen hinges on he following simple inequaliy C S K +, T, 2.43 which can be jusified in several ways. An inuiive way of deriving 2.43 is based on no-arbirage argumens. Noice ha since he opion s price C is always non-negaive, i is sufficien o consider he case when he curren sock price is greaer han he exercise price ha is, when S K>. Suppose, on he conrary, ha C <S K for some, i.e., S C >K. Then i would be possible, wih zero ne iniial invesmen, o buy a ime a call opion, shor a sock, and inves he sum S C in a savings accoun. By holding his porfolio unchanged up o he mauriy dae T, we would be able o lock in a riskless profi. Indeed, he value of our porfolio a ime T would saisfy recall ha r ˆr T S C +C T S T > ˆr T K +S T K + S T. We conclude once again ha inequaliy 2.43 is necessary for he absence of arbirage opporuniies. Taking 2.43 for graned, we may deduce he propery C a = C by simple no-arbirage argumens. Suppose, on he conrary, ha he wrier of an American call is able o sell he opion a ime a he price C a >C i is eviden ha, a any ime, an American opion is worh a leas as much as a European opion wih he same conracual feaures; in paricular, C a C. In order o profi from his ransacion, he opion wrier esablishes a dynamic porfolio which replicaes he value process of he European call, and invess he remaining funds in riskless bonds. Suppose ha he holder of he opion decides o exercise i a insan before he expiry dae T. Then he opion s wrier locks in a riskless profi, since he value of porfolio saisfies C S K + +ˆr T C a C >, T. The above reasoning implies ha he European and American call opions are equivalen from he poin of view of arbirage pricing heory; ha is, boh opions have he same price, and an American call should never be exercised by is holder before expiry. The las saemen means also ha a riskneural invesor who is long an American call should be indifferen beween selling i before, and holding i o, he opion s expiry dae provided ha he marke is efficien ha is, opions are neiher underpriced nor overpriced American Pu Opion Since he early exercise feaure of American pu opions was examined in Sec. 1.5, we will focus on he jusificaion of he valuaion formula. Le us denoe by T he class of all sopping imes defined on he filered probabiliy space Ω, F, P, where F = F S for every =,...,T. By a sopping ime we mean an arbirary funcion τ :Ω {,...,T } such ha for any =,...,T, a random even {ω Ω τω =} belongs o he σ-field F. Inuiively, his propery means ha he decision wheher o sop a process a ime ha is, wheher o exercise an opion a ime or no depends

42 2.3. VALUATION OF AMERICAN OPTIONS 41 on he sock price flucuaions up o ime only. Also, le T [,T ] sand for he subclass of hose sopping imes τ which saisfy τ T. Corollary and he preceding discussion sugges he following resul. Proposiion The arbirage price P a of an American pu opion equals P a = max τ T [,T ] E P ˆr τ K S τ + F, T Moreover, he sopping ime τ which realizes he maximum in 2.44 is given by he expression τ = min {u K S u + P a u }, T Proof. The problem of arbirage pricing of American coningen claims wihin a coninuous-ime seup is examined in deail in Chap. 5 below. In paricular, a coninuous-ime counerpar of formula 2.44 is proved in Theorem The verificaion of 2.44 is, of course, much simpler, bu i is based on similar argumens. For hese reasons, he proof is lef o he reader. The sopping ime τ will be referred o as he raional exercise ime of an American pu opion ha is sill alive a ime. I should be poined ou ha τ does no solve he opimal sopping ime for any individual, bu only for hose invesors who are risk-neural. A sraighforward applicaion of he classic Bellman principle 6 reduces he opimal sopping problem 2.44 o an explici recursive procedure which allows us o find he value funcion V p. These observaions lead o he following corollary o Proposiion Corollary Le he non-negaive adaped process V p, T, be defined recursively by seing V p T =K S T +, and V p { = max K S, E P ˆr 1 V p +1 F }, T Then he arbirage price P a of an American pu opion a ime equals V p. Moreover, he raional exercise ime afer ime equals τ = min {u K S u Vu p }. Remarks. I is also possible o go he oher way around ha is, o firs show direcly ha he price P a needs necessarily o saisfy he recursive relaionship { = max K S, E P ˆr 1 P+1 a } F, T 1, 2.47 P a subjec o he erminal condiion PT a =K S T +, and subsequenly derive he equivalen represenaion In he case of he CRR model indeed, in he case of any discree-ime securiy pricing model, he laer approach appears o be he simples way o value American opions. The main reason for his is ha an apparenly difficul valuaion problem is hus reduced o he simple one-period case. To show his we shall argue, as usual, by conradicion. To sar wih, we assume ha 2.47 fails o hold for = T 1. If his is he case, by reasoning along he same lines as in Sec. 1.5, one may easily consruc a ime T 1 a porfolio which produces riskless profi a ime T. Hence, we conclude ha necessarily { PT a 1 = max K S T 1, E P ˆr 1 K S T + } F T. The nex sep is o consider he ime period [T 2,T 1], wih T 1 now playing he role of he erminal dae, and PT a 1 being he erminal payoff. This procedure may be repeaed as many imes as needed. Summarizing, in he case of he CRR model, he arbirage pricing of an American pu reduces o he following simple recursive recipe { P a = max K S, ˆr 1 p P+1 au +1 p P+1 } ad, T 1, I should be observed ha he process S is Markovian under P. This is an immediae consequence of he independence of random variables ξ 1,...,ξ T.

43 42 CHAPTER 2. THE COX-ROSS-RUBINSTEIN MODEL wih P a T =K S T +. Noice ha P au +1 and P ad +1 represen he value of he American pu in he nex sep corresponding o he upward and downward movemen of he sock price saring from a given node on he laice. The above resuls may be easily generalized o he case of an arbirary coningen claim of American syle. 2.4 Opions on a Dividend-paying Sock So far we have assumed ha a sock pays no dividend during an opion s lifeime. Suppose now ha he sock pays dividends, and he dividend policy is of he following specific form: he sock mainains a consan yield, κ, on each ex-dividend dae. We shall resric ourselves o he las period before he opion s expiry. However, he analysis we presen below may be easily exended o he case of muli-period rading. We assume ha he opion s expiry dae T is an ex-dividend dae. This means ha he shareholder will receive a ha ime a dividend paymen d T which amouns o κus T 1 or κds T 1, according o he acual sock price flucuaion. On he oher hand, we posulae ha he ex-dividend sock price a he end of he period will be eiher u1 κs T 1 or d1 κs T 1. This corresponds o he radiional assumpion ha he sock price declines on he ex-dividend dae by exacly he dividend amoun. Therefore, he opion s payoff CT κ is eiher CT u = u1 κs T 1 K + or C d T = d1 κs T 1 K +, depending on he sock price flucuaion during he las period. If someone is long a sock, he or she receives he dividend a he end of he period; a pary in a shor posiion has o make resiuion for he dividend o he pary from whom he sock was borrowed. Under hese assumpions, he replicaing sraegy of a call opion is deermined by he following sysem of equaions independenly of he sign of α T 1 ; ha is, wheher he posiion is long or shor { α T 1 us T 1 + β T 1ˆr = u1 κs T 1 K +, α T 1 ds T 1 + β T 1ˆr = d1 κs T 1 K +. Noe ha, in conras o he opion payoff, he erminal value of he porfolio α T 1,β T 1 is no influenced by he fac ha T is he ex-dividend dae. This nice feaure of porfolio s wealh depends essenially on our assumpion ha he ex-dividend drop of he sock price coincides wih he dividend paymen. Solving he above equaions for α T 1 and β T 1, we find and α T 1 = u κs T 1 K + d κ S T 1 K + S T 1 u d β T 1 = ud κs T 1 K + du κ S T 1 K + ˆru d = Cu T Cd T S u T Sd T = ucd T dcu T ˆru d 2.49, 2.5 where u κ =1 κu and d κ =1 κd. By sandard argumens, we conclude ha he price C κ T 1 of he opion a he beginning of he period equals C κ T 1 = α T 1 S T 1 + β T 1 =ˆr 1 p C u T +1 p C d T, or explicily C κ T 1 =ˆr 1 p u κ S T 1 K + +1 p d κ S T 1 K +, 2.51 where p =ˆr d/u d. Working backwards in ime from he expiry dae, one finds he general formula for he arbirage price of a European call opion, provided ha he ex-dividend daes and he dividend raio κ, 1 are known in advance. If we price a pu opion, he corresponding hedging porfolio a ime T 1 saisfies { α T 1 us T 1 + β T 1ˆr = K u κ S T 1 +, α T 1 ds T 1 + β T 1ˆr = K d κ S T 1 +.

44 Chaper 3 The Black-Scholes Model The opion pricing model developed by Black and Scholes 1973, formalized and exended by Meron 1973, enjoys grea populariy. I is compuaionally simple and, like all arbirage-based pricing models, does no require he knowledge of an invesor s risk preferences. Opion valuaion wihin he Black-Scholes framework is based on he already familiar concep of perfec replicaion of coningen claims. More specifically, we will show ha an invesor can replicae an opion s reurn sream by coninuously rebalancing a self-financing porfolio involving socks and risk-free bonds. For insance, a replicaing porfolio for a call opion involves, a any dae before he opion s expiry dae, a long posiion in sock, and a shor posiion in risk-free bonds. By definiion, he wealh a ime of a replicaing porfolio equals he arbirage price of an opion. Our main goal is o derive closed-form expressions for boh he opion s price and he replicaing sraegy in he Black-Scholes seing. To do his in a formal way, we need firs o consruc an arbirage-free marke model wih coninuous rading. This can be done relaively easily if we ake for graned cerain resuls from he heory of sochasic processes, more precisely, from he Iô sochasic calculus. The heory of Iô sochasic inegraion is presened in several monographs; o cie a few, Ellio 1982, Karazas and Shreve 1988, Proer 199, Revuz and Yor 1991, and Durre In financial lieraure, i is no uncommon o derive he Black-Scholes formula by inroducing a coninuously rebalanced risk-free porfolio conaining an opion and underlying socks. In he absence of arbirage, he reurn from such a porfolio needs o mach he reurns on risk-free bonds. This propery leads o he Black-Scholes parial differenial equaion saisfied by he arbirage price of an opion. I appears, however, ha he risk-free porfolio does no saisfy he formal definiion of a self-financing sraegy. We make hroughou he following basic assumpions concerning marke aciviies: rading akes place coninuously in ime, and unresriced borrowing and lending of funds is possible a he same consan ineres rae. Furhermore, he marke is fricionless, meaning ha here are no ransacion coss or axes, and no discriminaion agains he shor sale. Finally, unless explicily saed oherwise, we will assume ha a sock which underlies an opion does no pay dividends a leas during he opion s lifeime. 3.1 Iô Sochasic Calculus This secion provides a very brief accoun of he Iô sochasic inegraion heory. For more deails we refer he reader, for insance, o monographs Durre 1996, Karazas and Shreve 1988, Chap. 2 3 or Revuz and Yor 1991, Chap. 4 5 and 8 9. Le us consider a probabiliy space Ω, F, P, equipped wih a filraion F = F [,T ] a filraion is an increasing family of σ-fields. Definiion A sample-pahs coninuous, F-adaped process W, wih W =, defined on a filered probabiliy space Ω, F, P, is called a one-dimensional sandard Brownian moion wih 43

45 44 CHAPTER 3. THE BLACK-SCHOLES MODEL respec o he filraion F if, for every u T, he incremen W W u is independen of he σ-field F u, and he probabiliy disribuion of W W u is Gaussian, wih mean and variance u. We shall describe only he mos imporan properies of a Brownian moion. Firsly, i can easily be seen ha a Brownian moion is a coninuous maringale wih respec o he underlying filraion F, since E P W < and E P W F u =E P W W u F u +E P W u F u =W u, 3.1 if u T. I is well known ha almos all sample pahs of a Brownian moion have infinie variaion on every open inerval, so classical Lebesgue-Sieljes inegraion heory canno be applied o define an inegral of a sochasic process wih respec o a Brownian moion. Finally, W is a process of finie quadraic variaion, as he following resul shows. Proposiion For every u< T and an arbirary sequence {T n } of finie pariions T n = { n = u< n 1 <...< n n = } of he inerval [u, ] saisfying lim n δt n =, where we have δt n def = max k=,...,n 1 n k+1 n k, n 1 lim n k= where he convergence in 3.2 is in L 2 Ω, F T, P. W n W n2 = u, 3.2 k+1 k Le W be a sandard one-dimensional Brownian moion defined on a filered probabiliy space Ω, F, P. For simpliciy, he horizon dae T>will be fixed hroughou. We shall inroduce he Iô sochasic inegral as an isomery I from a cerain space L 2 P W of sochasic processes ino he space L 2 = L 2 Ω, F T, P of square-inegrable, F T -measurable random variables. To sar wih, le us denoe by L 2 P W he class of hose progressively measurable for he definiion of a progressively measurable process we refer o, e.g., Karazas and Shreve 1988 processes γ defined on Ω, F, P for which T γ 2 def W = E P γu 2 du <. 3.3 Also, le K sand for he space of elemenary processes, ha is, processes of he form m 1 γ =γ 1 I + γ j I j, j+1], [,T], 3.4 j= where =< 1 <...< m = T, he random variables γ j,j=,...,m 1, are uniformly bounded and F j -measurable, and, finally, he random variable γ 1 is F -measurable. For any process γ K, he Iô sochasic inegral Î T γ wih respec o W over he ime inerval [,T] is defined by he formula Î T γ = T γ u dw u def = m 1 j= γ j W j+1 W j. 3.5 Similarly, he Iô sochasic inegral of γ wih respec o W over any subinerval [,], where T, is defined by seing m 1 def Î γ = γ u dw u = ÎT γ I [,] = γ j Wj+1 W j, 3.6 where x y = min {x, y}. I is easily seen ha for any process γ K, he Iô inegral I γ, [,T], follows a coninuous maringale on he space Ω, F, P; ha is, E P I γ F u =I u γ for u T. j=

46 3.1. ITÔ STOCHASTIC CALCULUS 45 Lemma The class K is a subse of L 2 P W, and T 2 E P γ u dw u = IT γ 2 L 2= γ 2 W 3.7 for any process γ from K. The space L 2 P W of progressively measurable sochasic processes, equipped wih he norm W, is a complee normed linear space ha is, a Banach space. Moreover, he class K of elemenary sochasic processes is a dense linear subspace of L 2 P W. By virue of Lemma 3.1.1, he isomery ÎT :K, W L 2 Ω, F T, P can be exended o an isomery I T :L 2 P W, W L 2 Ω, F T, P. This leads o he following definiion. Definiion For any process γ L 2 P W, he random variable I T γ is called he Iô sochasic inegral of γ wih respec o W over [,T], and i is denoed by T γ u dw u. More generally, for every γ L 2 P W and every [,T], we se I γ = γ u dw u def = I T γi [,], 3.8 so ha he Iô sochasic inegral I γ is a well-defined sochasic process. The nex resul summarizes he mos imporan properies of his process. By Iγ we denoe he sochasic process given by he formula def Iγ = γu 2 du, [,T]. 3.9 Proposiion For any process γ L 2 P W, he Iô sochasic inegral I γ follows a squareinegrable coninuous maringale on Ω, F, P. Moreover, he process is a coninuous maringale on Ω, F, P. I γ 2 Iγ, [,T], 3.1 In a more general framework, if M is a coninuous local maringale, hen we denoe by M he unique, coninuous, increasing adaped process vanishing a zero such ha M 2 M is a local maringale. The process M is referred o as he quadraic variaion of M. In view of Lemma 3.1.2, i is clear ha formula 3.9 is consisen wih his more general definiion. By applying he opional sopping echnique known also as a localizaion, i is possible o exend he definiion of Iô sochasic inegral o he class of all progressively measurable processes γ for which { T } P γu 2 du < = In his case, he Iô inegral Iγ is known o follow a coninuous local maringale on Ω, F, P, in general. Recall ha a process M is said o be a local maringale if here exiss an increasing sequence τ n of sopping imes such ha τ n ends o T a.s., and for every n he process M n, given by he formula { M n M τnωω if τ = n ω >, if τ n ω =, follows a uniformly inegrable maringale. Remarks. A random variable τ :Ω [,T] is called a sopping ime wih respec o he filraion F if, for every [,T], he even {τ } belongs o he σ-field F. For any progressively measurable process γ, and any sopping ime τ, he sopped process γ τ, which is defined by γ τ = γ τ, is also progressively measurable.

47 46 CHAPTER 3. THE BLACK-SCHOLES MODEL Le us denoe by L P W he class of all progressively measurable processes γ saisfying he inegrabiliy condiion I is clear ha his space of sochasic processes is invarian wih respec o he equivalen change of probabiliy measure; ha is, L P W =L P W whenever P and P are muually equivalen probabiliy measures on Ω, F T, and processes W and W are Brownian moions under P and under P, respecively. Since we resric ourselves o equivalen changes of probabiliy measures, given a fixed underlying probabiliy space Ω, F T, P, we shall wrie shorly LW insead of L P W in wha follows. Thus, a process γ is called inegrable wih respec o W if i belongs o he class LW Iô s Lemma In his secion, we shall deal wih he following problem: does he process gx follow a semimaringale if X is a semimaringale and g is a sufficienly regular funcion? I urns ou ha he class of coninuous semimaringales is invarian wih respec o composiions wih C 2 -funcions of course, much more general resuls are also available. We sar by inroducing a paricular class of coninuous semimaringales, referred o as Iô processes. Definiion An adaped coninuous process X is called an Iô process if i admis a represenaion X = X + α u du + β u dw u, [,T], 3.12 for some adaped processes α, β defined on Ω, F, P, which are inegrable in a suiable sense. For he sake of noaional simpliciy, i is cusomary o use a more condensed differenial noaion in which 3.12 akes he following form dx = α d + β dw. A coninuous adaped process X is called a coninuous semimaringale if i admis a decomposiion X = X +M +A, where X is a F -measurable random variable, M is a coninuous local maringale, and A is a coninuous process whose sample pahs are almos all of finie variaion on [,T]. I is clear ha an Iô process follows a coninuous semimaringale and 3.12 gives is canonical decomposiion. We denoe by S c P he class of all real-valued coninuous semimaringales on he probabiliy space Ω, F, P. One-dimensional case. Le us consider a funcion g = gx,, where x R is he space variable, and [,T] is he ime variable. I is eviden ha if X is a coninuous semimaringale and g : R [,T] R is a joinly coninuous funcion, hen he process Y = gx,isf -adaped and has almos all sample pahs coninuous. The nex resul, which is a special case of Iô s lemma, saes ha Y follows a semimaringale, provided ha he funcion g is sufficienly smooh. Theorem Suppose ha g : R [,T] R is a funcion of class C 2,1 R [,T], R. Then for any Iô process X, he process Y = gx,, [,T], follows an Iô process. Moreover, is canonical decomposiion is given by he Iô formula dy = g X, d + g x X,α d + g x X,β dw g xxx,β 2 d. More generally, if X = X + M + A is a real-valued coninuous semimaringale, and g is a funcion of class C 2,1 R [,T], R, hen Y = gx, follows a coninuous semimaringale wih he following canonical decomposiion dy = g X, d + g x X, dx g xxx, d M Mulidimensional case. Le us sar by defining a d-dimensional Brownian moion. A R d - valued sochasic process W =W 1,...,W d defined on a filered probabiliy space Ω, F, P is

48 3.1. ITÔ STOCHASTIC CALCULUS 47 called a d-dimensional sandard Brownian moion if W 1,W 2,...,W d are muually independen onedimensional sandard Brownian moions. In his paragraph, W denoes a d-dimensional sandard Brownian moion. Le γ be an adaped R d -valued process saisfying he following condiion { T } P γ u 2 du < =1, 3.14 where sands for he Euclidean norm in R d. Then he Iô sochasic inegral of γ wih respec o W equals d I γ = γ u dw u = γu i dwu, i [,T] Le X =X 1,X 2,...,X k beak-dimensional process such ha X i = X i + i=1 α i u du + β i u dw u, 3.16 where α i are real-valued adaped processes, and β i are R d -valued processes for i =1, 2,...,k, inegrable in a suiable sense. Le g = gx, be a funcion g : R k [,T] R. Before saing he nex resul, i will be convenien o inroduce he noion of he cross-variaion or quadraic covariaion of wo coninuous semimaringales. If X i = X i + M i + A i are in S c P for i =1, 2, hen X 1,X 2 def = M 1,M 2, where in urn M 1,M 2 def = 1 4 M 1 + M 2,M 1 + M 2 M 1,M 1 M 2,M 2. For insance, if X 1 and X 2 are he Iô processes given by 3.16, hen i is easily seen ha X 1,X 2 = β 1 u β 2 u du, [,T]. Proposiion Suppose ha g is a funcion of class C 2 R k, R. Then he following form of Iô s formula is valid k dgx = g xi X α i d + i=1 i=1 k g xi X β i dw k g xix j X β i β j d. i,j=1 More generally, if processes X i are in S c P for i =1, 2,...,k, hen gx =gx + k i=1 g xi X u dx i u k i,j=1 g xix j X u d X i,x j u. A special case of he Iô formula, known as he inegraion by pars formula, is obained by aking he funcion gx 1,x 2 =x 1 x 2. Corollary Suppose ha X 1,X 2 are real-valued coninuous semimaringales. Then he following inegraion by pars formula is valid X 1 X 2 = X 1 X 2 + X 1 u dx 2 u + X 2 u dx 1 u + X 1,X

49 48 CHAPTER 3. THE BLACK-SCHOLES MODEL Predicable Represenaion Propery In his secion, we shall assume ha he filraion F = F W is he sandard augmenaion 1 of he naural filraion σ{w u u } of he Brownian moion W. In oher words, we assume here ha he underlying probabiliy space is Ω, F W, P, where W is a one-dimensional Brownian moion. Theorem For any random variable X L 2 Ω, F W, P, here exiss a unique predicable process γ from he class L P W such ha T E P γu 2 du < 3.18 and he following equaliy is valid T X = E P X+ γ u dw u Basically, i can be deduced from Proposiion ha any local maringale on he filered probabiliy space Ω, F W, P admis a modificaion wih coninuous sample pahs Girsanov s heorem Le W be a d-dimensional sandard Brownian moion defined on a filered probabiliy space Ω, F, P. For an adaped R d -valued process γ LW, we define he process U by seing U = I γ = γ u dw u, [,T]. 3.2 The process U defined in his way follows, of course, a coninuous local maringale under P. One may check, using Iô s formula, ha he Doléans exponenial of U ha is, he unique soluion EU of he sochasic differenial equaion de U =E U γ dw = E U du, 3.21 wih he iniial condiion E U = 1 is given by he formula E U =E γ u dw u = exp γ u dw u 1 γ u 2 du, 2 i.e., E U = expu U /2. Noe ha EU follows a sricly posiive coninuous local maringale under P. For any probabiliy measure P on Ω, F T equivalen o P, we define he densiy process η by seing def d P η = E P dp F, [,T] I is clear ha η follows a sricly posiive, uniformly inegrable maringale under P; in paricular, η = E P η T F for every [,T]. Observe ha an adaped process X follows a maringale under P if and only if he produc ηx follows a maringale under P. We are in a posiion o sae a classical version of Girsanov s heorem. Theorem Le W be a sandard d-dimensional Brownian moion on a filered probabiliy space Ω, F, P. Suppose ha γ is an adaped real-valued process such ha { } E P E T γ u dw u = If a filraion F is no P-complee,is P-compleion runs as follows. Firs,we pu F = σf N, where N is he class of all P-negligible ses from F T. Second,for any we define ˆF = F +, where F + = F ɛ> +ɛ. Filraion ˆF is hen P-complee and righ-coninuous; i is referred o as he P-augmenaion of F.

50 3.2. THE BLACK-SCHOLES OPTION VALUATION FORMULA 49 Define a probabiliy measure P on Ω, F T equivalen o P by means of he Radon-Nikodým derivaive d P dp = E T γ u dw u, P-a.s Then he process W, which is given by he formula W = W γ u du, [,T], 3.25 follows a sandard d-dimensional Brownian moion on he space Ω, F, P. Obviously, we always have F W F, [,T]. The filraions generaed by W and of W do no coincide, in general. In paricular, if he underlying filraion F is he P-augmenaion of he naural filraion of W, hen we obain F W F W, [,T]. The nex well-known resul shows ha, if he underlying filraion is he naural filraion of a Brownian moion, he densiy process of any probabiliy measure equivalen o P has an exponenial form. Proposiion Assume ha he filraion F is he usual augmenaion of he naural filraion of W ; ha is, F = F W. Then for any probabiliy measure P on Ω, F T equivalen o P, here exiss a d-dimensional process γ, adaped o he filraion F W, and such ha he Radon-Nikodým derivaive of P wih respec o P equals d P = E γ u dw u, P-a.s dp F 3.2 The Black-Scholes Opion Valuaion Formula In Black and Scholes 1973, wo alernaive jusificaions of he opion valuaion formula are provided. The firs relies on he fac ha he risk-free reurn can be replicaed by holding a coninuously revised posiion in he underlying sock and an opion. In oher words, if an opion is no priced according o he Black-Scholes formula, here is a sure profi o be made by some combinaion of eiher shor or long sales of he opion and he underlying asse. The second mehod of derivaion is based on equilibrium argumens which require, in paricular, ha he opion earns an expeced rae of reurn commensurae wih he risk involved in holding he opion as an asse. The firs approach is usually referred o as he risk-free porfolio mehod, while he second is known as he equilibrium derivaion of he Black-Scholes formula. The replicaion approach presened below is based on he observaion ha in he Black-Scholes seing he opion value can be mimicked by holding a coninuously revised posiion in he underlying sock and risk-free bonds. I should be sressed ha, unless oherwise saed, we assume hroughou ha he financial marke we are dealing wih is perfec parially, his was already implici in he definiion of a self-financing rading sraegy Sock Price Le us firs describe sochasic processes which model he prices of primary securiies, a common sock and a risk-free bond. Following Samuelson 1965 and, of course, Black and Scholes 1973, we ake a geomeric or exponenial Brownian moion as a sochasic process which models he sock price. More specifically, he evoluion of he sock price process S is assumed o be described by he following linear sochasic differenial equaion SDE ds = µs d + σs dw, 3.27 where µ R is a consan appreciaion rae of he sock price, σ > is a consan volailiy coefficien, and S R + is he iniial sock price. we wrie R + o denoe he se of all sricly posiive real

51 5 CHAPTER 3. THE BLACK-SCHOLES MODEL numbers. Finally, W, [,T], sands for a one-dimensional sandard Brownian moion defined on a filered probabiliy space Ω, F, P. Le us emphasise ha 3.27 is merely a shorhand noaion for he following Iô inegral equaion S = S + µs u du + σs u dw u, [,T ]. Acually, he sample pahs of a Brownian moion are known o be almos everywhere non-differeniable funcions, wih probabiliy 1, so ha 3.27 can no be seen as a family of ordinary differenial equaions for each fixed elemenary even ω Ω. In he presen conex, i is convenien o assume ha he underlying filraion F =F [,T ] is he sandard augmenaion of he naural filraion F W of he underlying Brownian moion, i.e., ha he equaliy F = F W holds for every [,T ]. Remarks. This assumpion is no essenial if our aim is o value European opions wrien on a sock S. If his condiion were no saisfied, he uniqueness of he maringale measure, and hus also he compleeness of he marke, would fail o hold, in general. This would no affec he arbirage valuaion of sandard opions wrien on a sock S, however. I is elemenary o check, using Iô s formula, ha he process S which equals S = S exp σw +µ 1 2 σ2, [,T ], 3.28 is indeed a soluion of 3.27, saring from S a ime. The uniqueness of a soluion is an immediae consequence of a general resul due o Iô, which saes ha a SDE wih Lipschiz coninuous coefficiens has a unique soluion. I is apparen from 3.28 ha he sock reurns are lognormal, meaning ha he random variable lns /S u has under P a Gaussian probabiliy disribuion for any choice of daes u T. Since for any fixed, he random variable S = fw for some inverible funcion f : R R +, i is clear ha we have F W = σ{w u u } = σ{s u u } = F S. Therefore, he filraion generaed by he sock price coincides wih he naural filraion of he underlying noise process W, and hus F S = F W = F. This means ha he informaion srucure of he model is based on observaions of he sock price process only. Moreover, i is worhwhile o observe ha he sock price S follows a ime-homogeneous Markov process under P wih respec o he filraion F. In paricular, we have E P S u F =E P S u F S =E P S u S =S e µu for every u T. This follows from he fac ha S u = S exp σw u W +µ 1 2 σ2 u, 3.29 and he incremen W u W of he Brownian moion W is independen of he σ-field F, wih he Gaussian law N, u. The second securiy, whose price process is denoed by B in wha follows, represens in our model an accumulaion facor corresponding o a savings accoun known also as a money marke accoun. We assume hroughou ha he so-called shor-erm ineres rae r is consan over he rading inerval [,T ]. The risk-free securiy is assumed o coninuously compound in value a he rae r; ha is, db = rb d, or equivalenly B = e r, [,T ], 3.3 as, by convenion, we ake B =1. Le us observe ha we could have assumed insead ha he ineres rae r, and he sock price volailiy σ, are deerminisic funcions of ime. Also, he appreciaion rae µ could be an adaped sochasic process, saisfying mild regulariy condiions. Exensions of mos resuls presened in his chaper o such a case are raher sraighforward. I should be made clear ha we are no concerned here wih he quesion of wheher he mainained

52 3.2. THE BLACK-SCHOLES OPTION VALUATION FORMULA 51 model is he correc model of asse price flucuaions. Le us agree ha 3.27 can hardly be seen as a realisic model of he real-world behavior of a sock price. On he oher hand, he opion prices obained wihin he Black-Scholes framework are reasonably close o hose observed on he opion exchanges a leas for shor-mauriy opions on liquid insrumens. I is a puzzling quesion as o wha exen his coincidence is a consequence of he nooriey of he Black-Scholes formula among marke praciioners Self-financing Sraegies By a rading sraegy we mean a pair φ =φ 1,φ 2 ofprogressively measurable sochasic processes on he underlying probabiliy space Ω, F, P. The concep of a self-financing rading sraegy in he Black-Scholes marke is formally based on he noion of he Iô inegral. Inuiively, such a choice of sochasic inegral is suppored by he fac ha, in he case of he Iô inegral as opposed o he Fisk-Sraonovich inegral, he underlying process is inegraed in a predicable way, meaning ha we ake is values on he lef-hand end of each infiniesimal ime inerval. Formally, we say ha a rading sraegy φ =φ 1,φ 2 over he ime inerval [,T]isself-financing if is wealh process V φ, which is se o equal V φ =φ 1 S + φ 2 B, [,T], 3.31 saisfies he following condiion V φ =V φ+ φ 1 u ds u + φ 2 u db u, [,T], 3.32 where he firs inegral is undersood in he Iô sense. I is, of course, implicily assumed ha boh inegrals on he righ-hand side of 3.32 are well-defined. I is well known ha a sufficien condiion for his is ha 2 { T } { T } P φ 1 u 2 du < = 1 and P φ 2 u du < = We denoe by Φ he class of all self-financing rading sraegies. I is well known ha arbirage opporuniies are no excluded a priori from he class of self-financing rading sraegies Maringale Measure for he Spo Marke We find i convenien o inroduce he concep of he admissibiliy of a rading sraegy direcly in erms of a maringale measure see Chap. 8 for an alernaive approach. Le us denoe S = S/B. By definiion, a probabiliy measure Q on Ω, F T, equivalen o P, is called a maringale measure for he process S if S is a local maringale under Q. Similarly, a probabiliy measure P is said o be a maringale measure for he spo marke or briefly, a spo maringale measure if he discouned wealh of any self-financing rading sraegy follows a local maringale under P. The following resul shows ha boh hese noions coincide. Lemma A probabiliy measure is a spo maringale measure if and only if i is a maringale measure for he discouned sock price S. Proof. The proof relies on he following equaliy, which easily follows from he Iô formula V φ =V φ+ φ 1 u ds u, [,T ], where V φ =V φ/b and φ is a self-financing sraegy up o ime T. I is now sufficien o make use of he local maringale propery of he Iô sochasic inegral. In he Black-Scholes seing, he maringale measure for he discouned sock price process is unique, and is explicily known, as he following resul shows. 2 Noe ha condiion 3.33 is invarian wih respec o an equivalen change of a probabiliy measure.

53 52 CHAPTER 3. THE BLACK-SCHOLES MODEL Lemma The unique maringale measure Q for he discouned sock price process S is given by he Radon-Nikodým derivaive dq r µ dp = exp W T 1 r µ 2 σ 2 σ 2 T, P-a.s The dynamics of he discouned sock price S under he maringale measure Q are ds = σs dw, 3.35 and he process W which equals W = W r µ, [,T ], σ follows a sandard Brownian moion on a probabiliy space Ω, F, Q. Proof. Essenially, all saemens are direc consequences of Girsanov s heorem cf. Theorem and Proposiion Combining he wo lemmas, we conclude ha he unique spo maringale measure P is given on Ω, F T by means of he Radon-Nikodým derivaive dp r µ dp = exp σ W T 1 r µ 2 2 σ 2 T, P-a.s The discouned sock price S follows under P a sricly posiive maringale, since clearly S = S S = S exp σw 1 2 σ for every [,T ]. Noice also ha in view of 3.35, he dynamics of he sock price S under P are ds = rs d + σs dw, S >, 3.38 and hus he sock price a ime equals S = S exp σw +r 1 2 σ Finally, i is useful o observe ha all filraions involved in he model coincide; ha is, F = F W = F W = F S = F S. We are in a posiion o inroduce he class of admissible rading sraegies. An unconsrained Black-Scholes marke model would involve arbirage opporuniies, so ha reliable valuaion of derivaive insrumens would no be possible. Definiion A rading sraegy φ Φ is called P -admissible if he discouned wealh process V φ =B 1 V φ, [,T], 3.4 follows a maringale under P. We wrie ΦP o denoe he class of all P -admissible rading sraegies. The riple M BS =S, B, ΦP is called he arbirage-free Black-Scholes model of a financial marke, or briefly, he Black-Scholes marke. I is no hard o check ha by resricing our aenion o he class of P -admissible sraegies, we have guaraneed he absence of arbirage opporuniies in he Black-Scholes marke. Consequenly, given a coningen claim X which seles a ime T T and is aainable i.e., can be replicaed by means of a P -admissible sraegy we can uniquely define is arbirage price, π X, as he wealh V φ a ime of any P -admissible rading sraegy φ which replicaes X ha is, saisfies V T φ =X. If no replicaing P -admissible sraegy exiss, 3 he arbirage price of such a claim is no defined. Conforming wih he definiion of an arbirage price, o value a derivaive securiy we will usually search firs for is replicaing sraegy. Anoher approach o he valuaion problem is also possible, as he following simple resul shows. Since all saemens are immediae consequences of definiions above, he proof is lef o he reader. 3 One can show ha his happens only if a claim is no inegrable under he maringale measure P.

54 3.2. THE BLACK-SCHOLES OPTION VALUATION FORMULA 53 Corollary Le X be a P -aainable European coningen claim which seles a ime T. Then he arbirage price π X a ime [,T] in M BS is given by he risk-neural valuaion formula π X =B E P B 1 T X F, [,T] In paricular, he price of X a ime equals π X =E P B 1 T X. For he sake of concreeness, we shall firs consider a European call opion wrien on a sock S, wih expiry dae T and srike price K. Le he funcion c : R + [,T] R be given by he formula cs, =sn d 1 s, Ke r N d 2 s,, 3.42 where and d 1 s, = lns/k+r σ2 σ 3.43 d 2 s, =d 1 s, σ Furhermore, N sands for he sandard Gaussian cumulaive disribuion funcion Nx = 1 x e z2 /2 dz, x R. 2π We adop he following noaional convenion d 1,2 s, = lns/k+r ± 1 2 σ2 σ. Le us denoe by C he arbirage price of a European call opion a ime in he Black-Scholes marke. We are in a posiion o sae he main resul of his chaper. Theorem The arbirage price a ime [,T] of he European call opion wih expiry dae T and srike price K in he Black-Scholes marke is given by he formula C = cs,t, [,T], 3.45 where he funcion c : R + [,T] R is given by Moreover, he unique P -admissible replicaing sraegy φ of he call opion saisfies φ 1 = c s S,T, φ 2 = e r cs,t φ 1 S 3.46 for every [,T]. Proof. We provide wo alernaive proofs of he Black-Scholes resul. The firs relies on he calculaion of he replicaing sraegy. I hus gives no only he valuaion formula his, however, requires solving he Black-Scholes PDE 3.52, bu also explici formulae for he replicaing sraegy. The second mehod makes direc use of he risk-neural valuaion formula 3.41 of Corollary I focuses on he explici compuaion of he arbirage price of he opion, raher han on he derivaion of he hedging sraegy. Firs mehod. We sar by assuming ha he opion price, C, saisfies he equaliy C = vs, for some funcion v : R + [,T] R. We may hus assume ha he replicaing sraegy φ we are looking for has he following form φ =φ 1,φ 2 =gs,,hs, 3.47

55 54 CHAPTER 3. THE BLACK-SCHOLES MODEL for [,T], where g, h : R + [,T] R are unknown funcions. self-financing, he wealh process V φ, which equals Since φ is assumed o be V φ =gs,s + hs,b = vs,, 3.48 needs o saisfy he following: dv φ =gs, ds + hs, db. Under he presen assumpions, he las equaliy can be given he following form dv φ =µ rs gs, d + σs gs, dw + rvs,d, 3.49 since from he second equaliy in 3.48 we obain φ 2 = hs,=b 1 vs, gs,s. We shall search for he wealh funcion v in he class of smooh funcions on he open domain D =, +,T; more exacly, we assume ha v C 2,1 D. An applicaion of Iô s formula yields 4 dvs,= v S,+µS v s S,+ 1 2 σ2 S 2 v ss S, d + σs v s S, dw. Combining he expression above wih 3.49, we arrive a he following expression for he Iô differenial of he process Y, which equals Y = vs, V φ dy = v S,+µS v s S,+ 1 2 σ2 S 2 v ss S, d + σs v s S, dw +r µs gs, d σs gs, dw rvs, d. On he oher hand, in view of 3.48, Y vanishes idenically, hus dy =. By virue of he uniqueness of canonical decomposiion of coninuous semimaringales, he diffusion erm in he above decomposiion of Y vanishes. In our case, his means ha for every [,T]wehave or equivalenly T σs u gsu,u v s S u,u dw u =, S 2 u gsu,u v s S u,u 2 du =. 3.5 For 3.5 o hold, i is sufficien and necessary ha he funcion g saisfies gs, =v s s,, s, R + [,T] We shall assume from now on ha 3.51 holds. Then, using 3.51, we ge sill anoher represenaion for Y, namely { } Y = v S u,u+ 1 2 σ2 Su 2 v ss S u,u+rs u v s S u,u rvs u,u du. I is hus apparen ha Y vanishes whenever v saisfies he following parial differenial equaion PDE, referred o as he Black-Scholes PDE v s, σ2 s 2 v ss s, +rsv s s, rvs, = Since C T = vs T,T=S T K +, we need o impose also he erminal condiion vs, T =s K + for s R +. I is no hard o check by direc calculaion ha he funcion vs, =cs, T, where c 4 Subscrips on v denoe parial derivaives wih respec o he corresponding variables.

56 3.2. THE BLACK-SCHOLES OPTION VALUATION FORMULA 55 is given by , acually solves his problem. 5 I hus remains o check ha he replicaing sraegy φ, which equals φ 1 = gs,=v s S,, φ 2 = hs,=b 1 vs, gs,s, is P -admissible. Le us firs check ha φ is indeed self-financing. Though his propery is here almos rivial by he consrucion of φ, i is neverheless always preferable o check direcly he self-financing propery of a given sraegy. We need o check ha dv φ =φ 1 ds + φ 2 db. Since V φ =φ 1 S + φ 2 B = vs,, by applying Iô s formula, we ge dv φ =v s S, ds σ2 S 2 v ss S, d + v S, d. In view of 3.52, he las equaliy can also be given he following form dv φ =v s S, ds + rvs, d rs v s S, d, and hus dv φ =φ 1 vs, S φ 1 ds + rb d = φ 1 ds + φ 2 db. B This ends he verificaion of he self-financing propery. In view of our definiion of admissibiliy of rading sraegies, we need o verify ha he discouned wealh process V φ, which saisfies V φ =V φ+ v s S u,u ds u, 3.53 follows a maringale under he maringale measure P. By direc calculaion we obain v s s, = N d 1 s, T for every s, R + [,T], and hus, using also 3.35, we find ha V φ =V φ+ σs u N d 1 S u,t u dw u = V φ+ ζ u dw u, where ζ u = σs u N d 1 S u,t u. The exisence of he sochasic inegral is an immediae consequence of he sample pah coninuiy of he process η. From he general properies of he Iô sochasic inegral, i is hus clear ha he discouned wealh V φ follows a local maringale under P. To show ha V φ is a genuine maringale even a square-inegrable maringale, i is enough o observe ha T 2 T T E P ζ u dwu = E P ζu 2 du σ 2 E P Su 2 du <, where he second inequaliy follows easily from he exisence of he exponenial momens of a Gaussian random variable. Second mehod. The second mehod of he proof pus more emphasis on he explici calculaion of he price funcion c. The form of he replicaing sraegy will no be examined here. Since we wish o apply Corollary 3.2.1, we need o check firs ha he coningen claim X =S T K + is aainable in he Black-Scholes marke model, however. This follows easily from he general resuls more specifically, from he predicable represenaion propery see Theorem combined wih he square-inegrabiliy of he random variable X = B 1 T S T K + under he maringale measure P. We conclude ha here exiss a predicable process θ such ha he sochasic inegral V = V + θ u dw u, [,T], More precisely,he funcion v solves he final value problem for he backward PDE 3.52,while he funcion c solves he associaed iniial value problem for he forward PDE. To ge he forward PDE,i is enough o subsiue v s, wih v s, in Boh of hese PDEs are of parabolic ype on D.

57 56 CHAPTER 3. THE BLACK-SCHOLES MODEL follows a square-inegrable coninuous maringale under P, and T T X = B 1 T S T K + = E P X + θ u dwu = E P X + h u dsu, 3.55 where we have pu h = θ /σs. Le us consider a rading sraegy φ ha is given by φ 1 = h, φ 2 = V h S = B 1 V h S, 3.56 where V = B V. Le us check firs ha he sraegy φ is self-financing. Observe ha he wealh process V φ coincides wih V, and hus dv φ = db V = B dv + rv B d = B h ds + rv d = B h B 1 ds rb 1 S d+rv d. This in urn yields dv φ =h ds + rv h S d = ψ 1 ds + ψ 2 db, as expeced. Finally, i is clear ha V T φ =V T =S T K +, so ha φ is in fac a P -admissible replicaing sraegy for X. So far, we have shown ha he call opion is represened by a coningen claim ha is aainable in he Black-Scholes marke M BS. Our goal is now o evaluae he arbirage price of X using he risk-neural valuaion formula. Since F W = F S for every [,T], he riskneural valuaion formula 3.41 can be rewrien as follows C = B E P ST K + B 1 T F S = cs,t 3.57 for some funcion c : R + [,T] R. The second equaliy in 3.57 can be inferred, for insance, from he Markovian propery of S i is easily seen ha S follows a ime-homogeneous Markov process under P. Alernaively, we can make use of equaliy The incremen WT W of he Brownian moion is independen of he σ-field F W = F W ; on he oher hand, he sock price S is manifesly F W -measurable. By virue of he well-known properies of condiional expecaion see Lemma 2.2.1, we ge E P ST K + F S = HS,T, 3.58 where he funcion Hs, T is defined as follows { } + Hs, T =E P se σw T W +r σ2 /2T K for s, R + [,T]. Therefore, i is enough o find he uncondiional expecaion E P ST K + B 1 T = E P ST B 1 T I D E P KB 1 T I D = J1 J 2, where D sands for he se {S T >K}. For J 2 we have J 2 = e rt K P {S T >K} = e rt K P { S exp σwt +r 1 2 σ2 T } >K = e rt K P { } σwt < lns /K+r 1 2 σ2 T = e rt K P { ξ< lns /K+r 1 2 σ2 } σ T = e rt KNd 2 S,T, since he random variable ξ = W T / T has a sandard Gaussian law N, 1 under he maringale measure P. For he second inegral, noe firs ha J 1 = E P S T B 1 T I D=E P S T I D. 3.59

58 3.2. THE BLACK-SCHOLES OPTION VALUATION FORMULA 57 I is convenien o inroduce an auxiliary probabiliy measure P on Ω, F T by seing d P dp = exp σw T 1 2 σ2 T, P -a.s. By virue of Proposiion 3.1.3, he process W = W σ follows a sandard Brownian moion on he space Ω, F, P. Moreover, using 3.37 we obain Combining 3.59 wih 3.6, we find ha S T = S exp σ W T σ2 T. 3.6 J 1 = S P {S T >KB 1 T } = S P {S exp σ W T + 12 σ2 T } >Ke rt { = S P σ W } T < lns /K+r σ2 T. Using similar argumens as for J 2, we find ha J 1 = S Nd 1 S,T. Summarizing, we have shown ha he price a ime of a call opion equals C = cs,t=s N d 1 S,T Ke rt N d 2 S,T, where d 1,2 S,T= lns /K+r ± 1 2 σ2 T σ. T This ends he proof for he special case of =. The valuaion formula for >can be easily deduced from I can be checked ha he probabiliy measure P is he maringale measure corresponding o he choice of he sock price as a numeraire asse, ha is, he unique probabiliy measure, equivalen o P, under which he process B = B/S follows a maringale. Noice ha we have shown ha cf. formula 2.42 C = S P {ST >K} e rt KP {S T >K} Undoubedly, he mos sriking feaure of he Black-Scholes resul is he fac ha he appreciaion rae µ does no ener he valuaion formula. This is no surprising, however, as expression 3.39, which describes he dynamics of he sock price under he maringale measure P, does no involve he sock appreciaion rae µ. More generally, we could have assumed ha he appreciaion rae is no consan, bu is varying in ime, or even follows a sochasic process adaped o he underlying filraion. Assume, for insance, ha he sock price process is deermined by he sochasic differenial equaion i is implicily assumed ha SDE 3.62 admis a unique srong soluion S, which follows a coninuous, sricly posiive process ds = µ, S S d + σs dw, S >, 3.62 where µ :[,T ] R R is a deerminisic funcion saisfying cerain regulariy condiions, and σ :[,T ] R is also deerminisic, wih σ >ɛ>for some consan ɛ. We inroduce he accumulaion facor B by seing B = exp ru du, [,T], 3.63 for a deerminisic funcion r :[,T ] R +. In view of 3.63, we have db = rb d, B =1,

59 58 CHAPTER 3. THE BLACK-SCHOLES MODEL so ha i is clear ha r represens he insananeous, coninuously compounded ineres rae prevailing a he marke a ime. I is easily seen ha, under he presen hypoheses, he maringale measure P is unique, and he risk-neural valuaion formula 3.41 is valid. In paricular, he price of a European call opion equals C = e T ru du E P ST K + F for every [,T]. Noice ha under he maringale measure P, he dynamics of S are 3.64 ds = rs d + σs dw If r and σ are, for insance, coninuous funcions, he unique soluion o 3.65 is known o be S = S exp σu dwu + ru 1 2 σ2 u du. I is now an easy ask o derive a suiable generalizaion of he Black-Scholes formula using Indeed, i appears ha i is enough o subsiue he quaniies rt and σ 2 T in he sandard Black-Scholes formula by T T ru du and σ 2 u du, respecively. The funcion obained in such a way solves he Black-Scholes PDE wih ime-dependen coefficiens. Remarks. Le us sress ha we have worked wihin a fully coninuous-ime seup ha is, wih coninuously rebalanced porfolios. For obvious reasons, such an assumpion is no jusified from he pracical viewpoin. I is hus ineresing o noe ha he Black-Scholes resul can be derived in a discree-ime seup, by making use of he general equilibrium argumens see Rubinsein 1976, Brennan 1979 or Huang and Lizenberger The Pu-Call Pariy for Spo Opions If here are o be no arbirage opporuniies, oherwise idenical pus and calls mus a all imes during heir lives obey, a leas heoreically, he pu-call pariy relaionship. A poin worh sressing is ha equaliy 3.66 does no rely on specific assumpions imposed on he sock price model. Indeed, i is saisfied in any arbirage-free, coninuous-ime model of a securiy marke, provided ha he savings accoun B is modelled by 3.3. Proposiion The arbirage prices of European call and pu opions wih he same expiry dae T and srike price K saisfy he pu-call pariy relaionship for every [,T]. C P = S Ke rt 3.66 Proof. I is sufficien o observe ha he payoffs of he call and pu opions a expiry saisfy he equaliy S T K + K S T + = S T K. Relaionship 3.66 now follows from he risk-neural valuaion formula. Alernaively, one may derive 3.66 using simple no-arbirage argumens. The pu-call pariy can be used o derive a closed-form expression for he arbirage price of a European pu opion. Le us denoe by p : R + [,T] R he funcion ps, =Ke r N d 2 s, sn d 1 s,, 3.67 wih d 1 s, and d 2 s, given by The following resul is an immediae consequence of Proposiion combined wih Theorem

60 3.2. THE BLACK-SCHOLES OPTION VALUATION FORMULA 59 Corollary The Black-Scholes price a ime [,T] of a European pu opion wih srike price K equals P = ps,t, where he funcion p : R + [,T] R is given by In paricular, he price a ime of a European pu opion equals P = Ke rt N d 2 S,T S N d 1 S,T. Since in ypical siuaions i is no difficul o find a proper form of he call-pu pariy, we shall usually resric our aenion o he case of a call opion. In some circumsances, i will be convenien o explicily accoun for he dependence of he opion s price on is srike price K, as well as on he parameers r and σ of he model. For his reason, we shall someimes wrie C = cs,t, K, r, σ and P = ps,t, K, r, σ in wha follows The Black-Scholes PDE Suppose we are given a Borel-measurable funcion g : R R. Then we have he following resul, which generalizes Theorem Le us observe ha he problem of aainabiliy of any P -inegrable European coningen claim can be resolved by invoking he predicable represenaion propery compleeness of he mulidimensional Black-Scholes model is examined laer in his chaper. Corollary Le g : R R be a Borel-measurable funcion, such ha he random variable X = gs T is inegrable under P. Then he arbirage price in M BS of he claim X which seles a ime T is given by he equaliy π X =vs,, where he funcion v : R + [,T] R solves he Black-Scholes parial differenial equaion v σ2 s 2 2 v s 2 + rs v rv =, s s,,,t, 3.68 subjec o he erminal condiion vs, T =gs Sensiiviy Analysis We will now examine he basic feaures of rading porfolios involving opions. Le us sar by inroducing he erminology widely used in relaion o opion conracs. We say ha a a given insan before or a expiry, a call opion is in-he-money and ou-of-he-money if S >Kand S <K,respecively. Similarly, a pu opion is said o be in-he-money and ou-of-he-money a ime when S <Kand S >K,respecively. Finally, when S = K boh opions are said o be a-he-money. The inrinsic values of a call and a pu opions are defined by he formulae respecively, and he ime values equal I C =S K +, I P =K S +, 3.69 J C = C S K +, J P = P K S +, 3.7 for [,T]. I is hus eviden ha an opion is in-he-money if and only if is inrinsic value is sricly posiive. A shor posiion in a call opion is referred o as a covered call if he wrier of he opion hedges his or her risk exposure by holding he underlying sock; in he opposie case, he posiion is known as a naked call. When an invesor who holds a sock also purchases a pu opion on his sock as a proecion agains sock price decline, he posiion is referred o as a proecive pu. While wriing covered calls runcaes, roughly speaking, he righ-hand side of he reurn disribuion and simulaneously shifs i o he righ, buying proecive pus runcaes he lef-hand side of he reurn disribuion and a he same ime shifs he disribuion o he lef. The las effec is due o he fac ha he cos of a pu increases he iniial invesmen of a

61 6 CHAPTER 3. THE BLACK-SCHOLES MODEL porfolio. Noe ha he radiional mean-variance analysis pioneered in Markowiz is no an appropriae performance measure for porfolios conaining opions because of he skewness ha may be inroduced ino porfolio reurns. For more deails on he effeciveness of opion porfolio managemen, he ineresed reader may consul Leland 198, Booksaber 1981, and Booksaber and Clarke 1984, To measure quaniaively he influence of an opion s posiion on a given porfolio of financial asses, we will now examine he dependence of is price on he flucuaions of he curren sock price, ime o expiry, srike price, and oher relevan parameers. For a fixed expiry dae T and arbirary T, we denoe by τ he ime o opion expiry ha is, we pu τ = T. We wrie ps,τ,k,r,σ and cs,τ,k,r,σ o denoe he price of a call and a pu opion, respecively. The funcions c and p are hus given by he formulae cs, τ, K, r, σ =snd 1 Ke rτ Nd and ps, τ, K, r, σ =Ke rτ N d 2 sn d 1, 3.72 where d 1,2 = d 1,2 s, τ, K, σ, r = lns/k+r ± 1 2 σ2 τ σ. τ Recall ha a any ime [,T], he replicaing porfolio of a call opion involves α shares of sock and β unis of borrowed funds, where α = c s S,τ=N d 1 S,τ, β = cs,τ α S The sricly posiive number α, which deermines he number of shares in he replicaing porfolio, is commonly referred o as he hedge raio or, briefly, he dela of he opion. I is no hard o verify by sraighforward calculaions ha c s = Nd 1 =δ>, c ss = nd 1 sσ τ = γ>, c τ = sσ 2 τ nd 1+Kre rτ Nd 2 =θ>, c σ = s τnd 1 =λ>, c r = τke rτ Nd 2 =ρ>, c K = e rτ Nd 2 <, where n sands for he sandard Gaussian probabiliy densiy funcion ha is nx = 1 e x2 /2, x R. 2π Similarly, in he case of a pu opion we ge p s = Nd 1 1= N d 1 =δ<, p ss = nd 1 sσ τ = γ>, p τ = sσ 2 τ nd 1+Kre rτ Nd 2 1 = θ, p σ = s τnd 1 =λ>, p r = τke rτ Nd 2 1 = ρ<, p K = e rτ 1 Nd 2 >. 6 The more recen lieraure include Markowiz 1987,Huang and Lizenberger 1988,and Elon and Gruber 1995.

62 3.2. THE BLACK-SCHOLES OPTION VALUATION FORMULA 61 Consequenly, he dela of a long posiion in a pu opion is a sricly negaive number equivalenly, he price of a pu opion is a sricly decreasing funcion of a sock price. Generally speaking, he price of a pu moves in he same direcion as a shor posiion in he asse. In paricular, in order o hedge a wrien pu opion, an invesor needs o shor a cerain number of shares of he underlying sock. Anoher useful coefficien which measures he relaive change of an opion s price as he sock price moves is he elasiciy. For any dae T, he elasiciy of a call opion is given by he equaliy η c = c s S,τS /C = N d 1 S,τ S /C, and for a pu opion i equals η p = p s S,τS /P = N d 1 S,τ S /P. Le us check ha he elasiciy of a call opion price is always greaer han 1. Indeed, for every [,T], we have η c =1+e rτ KC 1 N d 2 S,τ > 1. This implies also ha C c s S,τS <, so ha he replicaing porfolio of a call opion always involves he borrowing of funds. Similarly, he elasiciy of a pu opion saisfies η p =1 Ke rτ P 1 N d 2 S,τ < 1. This in urn implies ha P S p s S,τ > his inequaliy is obvious anyway and hus he replicaing porfolio of a shor pu opion generaes funds which are invesed in risk-free bonds. These properies of replicaing porfolios have special consequences when he assumpion ha he borrowing and lending raes coincide is relaxed. I is insrucive o deermine he dynamics of he opion price C. Using Iô s formula, one finds easily ha under he maringale measure P we have dc = rc d + σc η c dw. This shows ha he appreciaion rae of he opion price in a risk-neural economy equals he riskfree rae r; however, he volailiy coefficien equals ση c, so ha, in conras o he sock price volailiy, he volailiy of he opion price follows a sochasic process. The posiion dela is obained by muliplying he face value 7 of he opion posiion by is dela. Clearly, he posiion dela of a long call opion or a shor pu opion is posiive; on he conrary, he posiion dela of a shor call opion and of a long pu opion is a negaive number. The posiion dela of a porfolio is obained by summing up he posiion delas of is componens. In his conex, le us make he rivial observaion ha he posiion dela of a long sock equals 1, and ha of a shor sock is 1. I should be sressed ha he opion s or opion porfolio s posiion dela measures only he marke exposure a he curren price levels of underlying asses. More precisely, i gives he firs order approximaion of he change in opion price, which is sufficienly accurae only for a small move in he underlying asse price. To measure he change in he opion dela as he underlying asse price moves, one should use he second derivaive wih respec o s of he opion s price ha is, he opion s gamma. The gamma effec means ha posiion delas also move as asse prices flucuae, so ha predicions of revaluaion profi and loss based on posiion delas are no sufficienly accurae, excep for small moves. I is easily seen ha bough opions have posiive gammas, while sold opions have negaive gammas. A porfolio s gamma is he weighed sum of is opions gammas, and he resuling gamma is deermined by he dominan opions in he porfolio. In his regard, opions close o he money wih a shor ime o expiry have a dominan influence on he porfolio s gamma. Generally speaking, a porfolio wih a posiive gamma is more aracive han a negaive gamma porfolio. Recall ha by hea we have denoed he derivaive of he opion price wih respec o ime o expiry. Generally, a porfolio dominaed by bough opions will have a negaive hea, meaning ha he porfolio will lose value as ime passes oher variables 7 The face value equals he number of underlying asses,e.g.,he face value of an opion on a lo of 1 shares of sock equals 1.

63 62 CHAPTER 3. THE BLACK-SCHOLES MODEL held consan. In conras, shor opions generally have posiive heas. The derivaive of he opion price wih respec o volailiy is known as he vega of an opion. A posiive vega posiion will resul in profis from increases in volailiy; similarly, a negaive vega means a sraegy will profi from falling volailiy. Example Consider a call opion on a sock S, wih srike price $3and wih 3 monhs o expiry. Suppose, in addiion, ha he curren sock price equals $31, he sock price volailiy is σ = 1% per annum, and he risk-free ineres rae is r = 5% per annum wih coninuous compounding. We may assume, wihou loss of generaliy, ha = and T =.25. Using 3.43, we obain approximaely d 1 S,T=.93, and hus d 2 S,T=d 1 S,T σ T =.88. Consequenly, using formula 3.42 and he following values of he sandard Gaussian probabiliy disribuion funcion: N.93=.8238 and N.88=.816, we find ha approximaely C =1.52, φ 1 =.82 and φ 2 = This means ha o hedge a shor posiion in he call opion, which was sold a he arbirage price C =$1.52, an invesor needs o purchase a ime he number δ =.82 shares of sock his ransacion requires an addiional borrowing of 23.9 unis of cash. The elasiciy a ime of he call opion price wih respec o he sock price equals η c = N d 1 S,T S C = Suppose ha he sock price rises immediaely from $31 o $31.2, yielding a reurn rae of.65% fla. Then he opion price will move by approximaely 16.5 cens from $1.52 o $1.685, giving a reurn rae of 1.86% fla. Roughly speaking, he opion has nearly 17 imes he reurn rae of he sock; of course, his also means ha i will drop 17 imes as fas. If an invesor s porfolio involves 5 long call opions each on a round lo of 1 shares of sock, he posiion dela equals 5.82 = 41, so ha i is he same as for a porfolio involving 41shares of he underlying sock. Le us now assume ha an opion is a pu. The price of a pu opion a ime equals alernaively, P can be found from he pu-call pariy 3.66 P =3e.5/4 N N.93 =.15. The hedge raio corresponding o a shor posiion in he pu opion equals approximaely δ =.18 since N.93 =.18, herefore o hedge he exposure, using he Black-Scholes recipe, an invesor needs o shor.18 shares of sock for one pu opion. The proceeds from he opion and shareselling ransacions, which amoun o $5.73, should be invesed in risk-free bonds. Noice ha he elasiciy of he pu opion is several imes larger han he elasiciy of he call opion. If he sock price rises immediaely from $31 o $31.2, he price of he pu opion will drop o less han 12 cens Opion on a Dividend-paying Sock We consider he case when he dividend yield, raher han he dividend payoffs, is assumed o be known. More specifically, we assume ha he sock S coninuously pays dividends a some fixed rae κ. Following Samuelson 1965, we assume ha he effecive dividend rae is proporional o he level of he sock price. Alhough his is raher impracical as a realisic dividend policy associaed wih a paricular sock, Samuelson s model fis he case of a sock index opion reasonably well. The dividend paymens should be used in full, eiher o purchase addiional shares of sock, or o inves in risk-free bonds however, ineremporal consumpion or infusion of funds is no allowed. Consequenly, a rading sraegy φ =φ 1,φ 2 is said o be self-financing when is wealh process V φ, which equals, as usual, V φ =φ 1 S + φ 2 B, saisfies dv φ =φ 1 ds + κφ 1 S d + φ 2 db, or equivalenly dv φ =φ 1 µ + κs d + φ 1 σs dw + φ 2 db.

64 3.2. THE BLACK-SCHOLES OPTION VALUATION FORMULA 63 We find i convenien o inroduce an auxiliary process S = e κ S, whose dynamics are given by he sochasic differenial equaion d S = µ κ S d + σ S dw, where µ κ = µ + κ. In erms of his process we have and V φ =φ 1 e κ S + φ 2 B dv φ =φ 1 e κ d S + φ 2 db. Also, i is no difficul o check ha he discouned wealh V φ saisfies dv φ =φ 1 e κ d S, where S = S B 1. Pu anoher way, we have dv φ =σφ 1 S dw + σ 1 µ κ r d. In view of he las equaliy, he unique maringale measure Q for our model is given by 3.34, bu wih µ replaced by µ κ. The dynamics of V φ under Q are given by he expression while hose of S are dv φ =σφ 1 S d W, d S = σ S d W, 3.74 and he process W = W r µ κ /σ follows a sandard Brownian moion on he probabiliy space Ω, F, Q. I is hus possible o consruc, by defining in a sandard way he class of admissible rading sraegies, an arbirage-free marke in which a risk-free bond and a dividend-paying sock are primary securiies. Assuming ha his is done, he valuaion of sock-dependen coningen claims is now sandard. In paricular, we have he following resul. Proposiion The arbirage price a ime T of a call opion on a sock which pays dividends a a consan rae κ during he opion s lifeime is given by he risk-neural formula C κ = B E Q B 1 T S T K + F, [,T], 3.75 or explicily C κ = S N d 1 S,T Ke rt N d 2 S,T, 3.76 where S = S e κt, and d 1,d 2 are given by Equivalenly, C κ = e κt S N ˆd1 S,T Ke r κt N ˆd2 S,T, where Proof. ˆd 1,2 s, = lns/k+r κ ± 1 2 σ2 σ The firs equaliy is obvious. For he second, noe firs ha we may rewrie 3.75 as follows C κ = e rt E Q ST K + F = e κt e rt E Q ST e κt K + F. Using 3.74, and proceeding along he same lines as in he proof of Theorem 3.2.1, we find ha C κ = e κt c S,T, e κt K,

65 64 CHAPTER 3. THE BLACK-SCHOLES MODEL where c is he sandard Black-Scholes call opion valuaion funcion. Pu anoher way, C κ = c κ S,T, where c κ s, =se κ N ˆd1 s, Ke r N ˆd2 s, and ˆd 1, ˆd 2 are given by Alernaively, o derive he valuaion formula for a call opion or for any European claim of he form X = gs T, we may firs show ha is arbirage price equals v, S, where v solves he following backward PDE v σ2 s 2 2 v +r κs v rv = 3.78 s2 s on,,t, subjec o he sandard erminal condiion vs, T =gs. Under he assumpions of Proposiion 3.2.2, one can show also ha he value a ime of he forward conrac wih expiry dae T and delivery price K is given by he equaliy V K =e κt S e rt K. Consequenly, he forward price a ime T of he sock S, for selemen a dae T, equals F κ S, T =e r κt S I is no difficul o check ha he following version of he pu-call pariy relaionship is valid c κ S,T p κ S,T =e κt S e rt K, 3.8 where p κ S,T sands for he arbirage price a ime of he European pu opion wih mauriy dae T and srike price K. In paricular, if he exercise price equals he forward price of he underlying sock, hen c κ S,T p κ S,T =. As already menioned, formula 3.76 is commonly used by marke praciioners when valuing sock index opions. For his purpose, one needs o assume ha he sock index follows a lognormal process acually, a geomeric Brownian moion. 8 The dividend yield κ, which can be esimaed from he hisorical daa, slowly varies on a monhly or quarerly basis. Therefore, for opions wih a relaively shor mauriy, i is reasonable o assume ha he dividend yield is consan Hisorical Volailiy All poenial pracical applicaions of he Black-Scholes formula hinge on knowledge of he volailiy parameer of he reurn of sock prices. Indeed, of he five variables necessary o specify he model, all are direcly observable excep for he sock price volailiy. The mos naural approach uses an esimae of he sandard deviaion based upon an ex-pos series of reurns from he underlying sock. In he firs empirical ess of he Black-Scholes model, performed by Black and Scholes 1972, he auhors used over-he-couner daa covering he period. The sock volailiies were esimaed from daily daa over he year preceding each opion price observaion. They concluded ha he model overpriced underpriced, respecively opions on socks wih high low, respecively hisorical volailiies. More generally, hey suggesed ha he usefulness of he model depends o a grea exen upon invesors abiliies o make good forecass of he volailiy. In subsequen years, he model was esed by several auhors on exchange-raded opions see, e.g., Galai 1977 confirming he bias in heoreical opion prices observed originally by Black and Scholes. Alhough he esimaion of sock price volailiy from hisorical daa is a fairly sraighforward procedure, some imporan poins should be menioned. Firsly, o reduce he esimaion risk arising from he sampling error, i seems naural o increase he sample size, e.g., by using a longer series of hisorical 8 This does no follow from he assumpion ha each underlying sock follows a lognormal process,unless he sock index is calculaed on he basis of he geomeric average,as opposed o he more commonly used arihmeic average.

66 3.2. THE BLACK-SCHOLES OPTION VALUATION FORMULA 65 observaions or by increasing he frequency of observaions. Unforunaely, here is evidence o sugges ha he variance is non-saionary, so ha exending he observaion period may make maers even worse. Furhermore, in many cases only daily daa are available, so ha here is a limi on he number of observaions available wihin a given period. Finally, since he opion pricing formula is non-linear in he sandard deviaion, an unbiased esimae of he sandard deviaion does no produce an unbiased esimae of he opion price. To summarize, since he volailiy is usually unsable hrough ime, hisorical preceden is a poor guide for esimaing fuure volailiy. Moreover, esimaes of opion prices based on hisorical volailiies are sysemaically biased Implied Volailiy Alernaively, one can infer he invesmen communiy s consensus oulook as o he volailiy of a given asse by examining he prices a which opions on ha asse rade. I was observed in he research of Black and Scholes 1972 ha he acual sandard deviaion ha would resul over he life of an opion would be a beer inpu ino he model if i were known in advance. Since an opion price appears o be an increasing funcion of he underlying sock volailiy, and all oher facors deermining he opion price are known wih cerainy, one can infer he volailiy ha is implied in he observed marke price of an opion. More specifically, he implied volailiy, σ imp say, is derived from he non-linear equaion 9 C = sn d 1 s, T Ke rt N d 2 s, T, where he only unknown parameer is σ, since for C we ake he curren marke price of he call opion. In oher words, he implied volailiy is he value of he sandard deviaion of sock reurns ha, when pu in he Black-Scholes formula, resuls in a model price equal o he curren marke price. The acual value of he implied volailiy σ imp deermined in his way depends, in general, on an opion s conracual feaures ha is, on he value K of he srike price, as well as on he ime T o mauriy. A properly weighed average of hese implied sandard deviaions is used as a measure of he marke forecass of reurn variabiliy. The implied volailiy, considered as a funcion of he opion s srike price, someimes exhibis a specific U-shape. One of he long-sanding problems has been how o reconcile his peculiar feaure of empirical opion prices, referred o as he smile effec, wih he Black-Scholes model. A ypical soluion o his problem relies on a judicious choice of a discree- or coninuous-ime model for sock price reurns see, e.g., Rubinsein 1994 or Derman e al Le us menion in his conex he growing ineres in financial modelling based on sable and hyperbolic disribuions. Brenner and Subrahmanyam 1988 and Corrado and Miller 1996 provide explici approximae formulas for he implied volailiy in he Black-Scholes seing. To derive he Corrado-Miller formulas, we sar wih he expansion of he Gaussian probabiliy disribuion funcion see Suar and Ord 1987, p.184 Nx = x 2 x3 π 6 + x By subsiuing his expansion ino he Black-Scholes call price, we obain he following approximae formula for he call price noe ha cubic and higher order erms are ignored 1 C S 2 + d 1S,T 2 Ke rt 1 π 2 + d 2S,T 2. π Afer sandard manipulaions, we arrive a he following quadraic equaion in he quaniy σ = σ T σ 2 S + K σ 8π C 1 2 S K +2S KlnS / K =, 9 To solve his equaion explicily,one needs o make use of numerical mehods,such as,e.g.,he Newon-Raphson mehod. See Manaser and Koehler 1982 for he reasonable choice of a saring value for he firs ieraion.

67 66 CHAPTER 3. THE BLACK-SCHOLES MODEL where we wrie K = Ke rt. The larges roo of he las equaion is s = C 1 2 2π S K S + K + C 1 2π 2 S K 2 S + K 2S K lns / K S + K. In paricular, when he curren sock price S equals he discouned exercise price, his reduces o he original Brenner-Subrahmanyam formula. A furher gain in accuracy can be obained using lns / K 2S K/S + K, and by subsiuing he number 4 wih he parameer α. In his way, we arrive a he following approximae formula which sill reduces o he Brenner-Subrahmanyam formula if S = Ke rt s C 1 2 2π S K S + K + C 1 2π 2 S K 2 S S + K α K 2 S + K. The clever choice of α allows us o improve he accuracy of wihou affecing a-he-money accuracy. I appears see Corrado and Miller 1996 for he deails ha α = 2 is a reasonable choice; we hus have K+ 2π C s S + K C 1 2 S 1 2 S K 2 1 π S K 2. For he discussion of he accuracy of he las formula, we refer o he original paper by Corrado and Miller Numerical Mehods Since a closed-form expression for he arbirage price of a claim is no always available, an imporan issue is he sudy of numerical mehods which give approximaions of arbirage prices and hedging sraegies. In he Black-Scholes framework, hese mehods include: he use of mulinomial laices rees o approximae coninuous-ime models of securiy prices; procedures based on he Mone Carlo simulaion of random variables; and finie-difference mehods of solving he associaed parial differenial equaions. Le us survey briefly he relevan lieraure. Binomial or, more generally, mulinomial models were sudied by, among ohers, Cox e al and Boyle 1986, who proposed an approximaion of he sock price by means of he rinomial ree. The CRR binomial approximaion of he exponenial Brownian moion was examined in Sec I should be poined ou ha if a coninuous-ime framework is aken as a benchmark, an increase in accuracy is gained by assuming ha he firs wo momens of an approximaing binomial process coincide wih he corresponding momens of he exponenial Brownian moion used o model he sock price. The corresponding modificaion of he CRR model of Chap. 2 runs as follows. For a fixed T > and arbirary n N, we denoe n = T/n. Noice ha for every j =,...,n 1, we have S j+1 n = S j n exp σ Wj+1 n Wj n + r 1 2 σ2 n under he maringale measure P. Therefore, he expeced value of he raio S j+1 n /S j n equals m 1 n =E P S j+1 n /S j n = expr n for every j, and he second momen of his raio is m 2 n =E P S j+1 n /S j n 2 = exp2r n + σ 2 n.

68 3.3. FUTURES MARKET 67 The corresponding values of he parameers u n,d n and p n can be found by solving he following equaions { pn u n +1 p n d n = m 1 n, p n u 2 n +1 p n d 2 n = m 2 n, wih d n = u 1 n. I is worhwhile o poin ou ha he convergence resul of Sec remains valid for he above modificaion of he CRR binomial model. More accurae approximaion resuls were obained by considering a rinomial ree, as proposed in Boyle In a rinomial ree, here are hree possible fuure saes for every node. For a fixed j, le us denoe by p jk i he ransiion probabiliy from he sae s j i a ime j n o he sae s j+1 k a ime j + 1 n. The ransiion probabiliies p jk i and he sae values s j i mus be chosen in such a way ha he laice accuraely approximaes he behavior of he sock price S under he maringale probabiliy. Once he laice is already consruced, he valuaion of European and American coningen claims is done by he sandard backward inducion mehod. In order o evaluae he prices of opions which depend on wo underlying asses, Boyle 1988 exends his echnique o he case of a bivariae mulinomial model. For furher developmens of he binomial approach, which ake ino accoun he presence of he volailiy smile effec, we refer o Rubinsein 1994 and Derman e al The efficien valuaion of derivaive securiies using he Mone Carlo simulaion was sudied by, among ohers, Boyle 1977, Johnson and Shanno 1987, and Duffie and Glynn Hull and Whie 1988 apply he conrol variae echnique in order o improve he efficiency of he finie-difference mehod when valuing American opions on a dividend-paying sock. Finie-difference mehods of solving PDEs are examined in he conex of opion valuaion in Schwarz 1977, Brennan and Schwarz 1978, and Couradon Since a presenaion of hese mehods is beyond he scope of his book, le us menion only ha one can use boh explici and implici finie-difference schemes, as well as he Crank-Nicolson mehod. For a deailed analysis of finie-difference mehods in opion valuaion, we refer he reader o he monographs by Wilmo e al. 1993, 1995, in which he auhors successfully apply he PDE approach o all kinds of sandard and exoic opions. 3.3 Fuures Marke Le us denoe by f S, T, [,T ], he fuures price of a cerain sock S for he dae T. Our aim is o sudy he arbirage pricing of a European coningen claim which seles a ime T, wih T T. The dynamics of fuures prices f = f S, T are given by he familiar expression df = µ f f d + σ f f dw, f >, 3.81 where µ f and σ f > are real numbers, and W, [,T ], sands for a one-dimensional sandard Brownian moion, defined on a probabiliy space Ω, F, P, where F = F W. The unique soluion of SDE 3.81 is given by he formula cf f = f exp σ f W +µ f 1 2 σ2 f, [,T] The price of he second securiy, a risk-free bond, is given as before by 3.3. In he Black-Scholes seing, he fuures price dynamics of a sock S can be found by combining 3.27 wih he following chain of equaliies f = f S, T =F S, T =S e rt, [,T ], 3.83 where, as usual, we wrie F S, T o denoe he forward price of he sock for he selemen dae T. The las equaliy in 3.83 can be easily derived from he absence of arbirage in he spo/forward marke; he second is a consequence of he assumpion ha he ineres rae is deerminisic. If he dynamics of he sock price S are given by he SDE 3.27, hen Iô s formula yields df =µ rf d + σf dw,

69 68 CHAPTER 3. THE BLACK-SCHOLES MODEL wih f = S e rt, so ha f saisfies 3.81 wih µ f = µ r and σ f = σ. Since fuures conracs are no necessarily associaed wih a physical underlying securiy, such as a sock or a bond, we prefer o sudy he case of fuures opions in an absrac way. This means ha we consider 3.81 as he exogenously given dynamics of he fuures price f. However, for he sake of noaional simpliciy, we shall wrie µ = µ f and σ = σ f in wha follows. I follows from 3.83 ha bu also f S, T =F S, T =E P S T F, [,T ], 3.84 f S, T =F S, T =S /B, T, [,T ], 3.85 where B, T sands for he price a ime of he zero-coupon bond ha maures a T. I appears ha under uncerainy of ineres raes, he righ-hand sides of 3.84 and 3.85 characerize he fuures and he forward price of S, respecively see Chap Self-financing Sraegies Le us fix a ime horizon T T. We consider a European coningen claim X which seles a ime T. By a fuures sraegy we mean a pair φ =φ 1,φ 2 of real-valued adaped sochasic processes, defined on he probabiliy space Ω, F, P. Since i coss nohing o ake a long or shor posiion in a fuures conrac, he wealh process V f φ of a fuures sraegy φ equals V f φ =φ 2 B, [,T] We say ha a fuures sraegy φ =φ 1,φ 2 isself-financing if is wealh process V f φ saisfies for every [,T] V f φ =V f φ+ φ 1 u df u + φ 2 u db u We wrie Φ f o denoe he class of all self-financing fuures sraegies Maringale Measure for he Fuures Marke A probabiliy measure P equivalen o P is called he fuures maringale measure if he discouned wealh Ṽ f φ of any sraegy φ Φ f, which equals Ṽ f φ =V f φ/b, follows a local maringale under P. Lemma Le P be a probabiliy measure on Ω, F T equivalen o P. Then P is a fuures maringale measure if and only if he fuures price f follows a local maringale under P. Proof. The discouned wealh Ṽ f for any rading sraegy φ Φ f saisfies dṽ f φ =B 1 φ 1 df + φ 2 db rb 1 V f φ d = φ 1 B 1 df, as 3.86 yields he equaliy B 1 φ 2 db rv f φ d = B 1 B 1 V f φ db rv f φ d =. The saemen of he lemma now easily follows. The nex resul is an immediae consequence of Girsanov s heorem. Proposiion The unique maringale measure P for he process f is given by he Radon- Nikodým derivaive d P dp = exp µ σ W T 1 µ 2 2 σ 2 T, P -a.s. 3.88

70 3.3. FUTURES MARKET 69 The dynamics of he fuures price f under P are df = σf d W, 3.89 and he process W = W + µ/σ follows a sandard Brownian moion on he probabiliy space Ω, F, P. I is clear from 3.89 ha f = f exp σ W 1 2 σ2, [,T ], 3.9 so ha f follows a sricly posiive maringale under P. As expeced, we say ha a fuures sraegy φ Φ f is P-admissible if he discouned wealh Ṽ f φ follows a maringale under P. We shall sudy an arbirage-free fuures marke M f =f,b,φ f P, where Φ f P is he class of all P-admissible fuures rading sraegies. The fuures marke model M f is referred o as he Black fuures marke in wha follows. The noion of an arbirage price is defined in a similar way o he case of he Black-Scholes marke The Black Fuures Opion Formula We shall now derive he valuaion formula for fuures opions, due o Black Le he funcion c f : R + [,T] R be given by Black s fuures formula c f f, =e r fn d1 f, KN d2 f,, 3.91 where d 1,2 f, = lnf/k ± 1 2 σ2 σ 3.92 and N denoes he sandard Gaussian cumulaive disribuion funcion. Before we formulae he main resul of his secion, le us consider once again he fuures conrac wrien on a sock whose dynamics are given by If T = T, he fuures opion valuaion rt resul can be found direcly from he Black-Scholes formula by seing S = f e his applies also o he replicaing sraegy. Inuiively, his follows from he simple observaion ha in his case we have f T = S T a he opion s expiry, and hus he payoffs from boh opions coincide. In pracice, he expiry dae of a fuures opion usually precedes he selemen dae of he underlying fuures conrac ha is, T<T. In such a case we have C f T = f S T,T K + = e rt T S T Ke rt T +, and we may sill value he fuures opion as if i were he spo opion. Such consideraions rely on he equaliy f S, T =F S, T, which in urn hinges on he assumpion ha he ineres rae is a deerminisic funcion. They hus canno be easily exended o he case of sochasic ineres raes. For his reason, we prefer o give below a sraighforward derivaion of he Black fuures formula. Theorem The arbirage price C f in he arbirage-free fuures marke M f of a European fuures call opion, wih expiry dae T and srike price K, is given by he equaliy C f = c f f,t. The fuures sraegy φ Φ f P ha replicaes a European fuures call opion is given by for every [,T]. φ 1 = cf f f,t, φ 2 = e r c f f,t, 3.93

71 7 CHAPTER 3. THE BLACK-SCHOLES MODEL Proof. We shall follow raher closely he proof of Theorem Therefore, we shall focus mainly on he derivaion of Some echnical deails, such as inegrabiliy of random variables or admissibiliy of rading porfolios, are lef aside. Firs mehod. Assume ha he price process C f is of he form C f = vf, for some funcion v : R + [,T] R, and consider a fuures sraegy φ Φ f of he form φ =gf,,hf, for some funcions g, h : R + [,T] R. Since he replicaing porfolio φ is assumed o be self-financing, he wealh process V f φ, which equals V f φ =hs,b = vf,, 3.94 saisfies dv f φ =gf, df + hf, db, or more explicily dv f φ =f µgf, d + f σgf, dw + rvf,d On he oher hand, assuming ha he funcion v is sufficienly smooh, we find ha dvf,= v f,+µf v f f,+ 1 2 σ2 f 2 v ff f, d + σf v f f, dw. Combining he las equaliy wih 3.95, we ge he following expression for he Iô differenial of he process Y = vf, V f φ dy = v f,+µf v f f,+ 1 2 σ2 f 2 v ff f, d + σf v f f, dw µf gf, d σf gf, dw rvf, d =. Arguing along he same lines as in he proof of Theorem 3.2.1, we infer ha and hus also Y = gf, =v f f,, f, R + [,T], 3.96 { } v f u,u+ 1 2 σ2 fuv 2 ff f u,u rvf u,u du =, 3.97 where he las equaliy follows from he definiion of Y. To guaranee he las equaliy we assume ha v saisfies he following parial differenial equaion referred o as he Black PDE v σ2 f 2 v ff rv = 3.98 on,,t, wih he erminal condiion vf,t =f K +. Since he funcion vf, = c f f,t, where c f is given by , is easily seen o solve his problem, o complee he proof i is sufficien o noe ha, by virue of 3.96 and 3.94, he unique P-admissible sraegy φ ha replicaes he opion saisfies φ 1 = gf,=v f f,, φ 2 = hf,=b 1 v f f,. Deails are lef o he reader. Second mehod. Since he random variable X = B 1 T S T K + is inegrable wih respec o he maringale measure P, i is enough o evaluae he condiional expecaion C f = B E P ft K + B 1 T F f = B E P ft K + B 1 T f. This means ha, in paricular for =, we need o find he expecaion ft K + B 1 T = E Pf T B 1 T I 1 D E PKBT I D=I 1 I 2, E P

72 3.3. FUTURES MARKET 71 where D denoes he se {f T >K}. For I 2, we have I 2 = e rt K P{f T >K} = e rt K P { f exp σ W T 1 2 σ2 T } >K, and hus I 2 = e rt K P { σ W } T < lnf /K 1 2 σ2 T = e rt K P { ξ< lnf /K 1 2 σ2 } σ T = e rt KN d2 f,t, since he random variable ξ = W T / T has under P he sandard Gaussian law. To evaluae I 1, we define an auxiliary probabiliy measure ˆP on Ω, F T by seing d ˆP d P = exp σ W T 1 2 σ2 T, P-a.s., and hus cf. 3.9 I 1 = P{f T B 1 T I D} = e rt f ˆP{fT >K}. Moreover, he process Ŵ = W σ follows a sandard Brownian moion on he filered probabiliy space Ω, F, ˆP, and f T = f exp σŵt σ2 T. Consequenly, I 1 = e rt f ˆP {ft >K} { = e rt f ˆP f exp σŵt σ2 T } >K { } = e rt f ˆP σŵt < lnf /K+ 1 2 σ2 T = e rt f N d1 f,t. The general valuaion resul for any dae is a consequence of he Markov propery of f. The mehod of arbirage pricing in he Black fuures marke can be easily exended o any pahindependen claim coningen on he fuures price. In fac, he following corollary follows easily from he firs proof of Theorem As already menioned, in financial lieraure, he parial differenial equaion 3.99 is commonly referred o as he Black PDE. Corollary The arbirage price in M f of any aainable coningen claim X = gf T which seles a ime T is given by π f X =vf,, where he funcion v : R + [,T] R is a soluion of he following parial differenial equaion v σ2 f 2 2 v rv =, f,,,t, 3.99 f2 subjec o he erminal condiion vf,t =gf. Le us denoe by P f = p f f,t he price of a pu fuures opion wih srike price K and T o is expiry dae, provided ha he curren fuures price is f. To find he price of a fuures pu opion, we can use he following resul, whose easy proof is lef o he reader.

73 72 CHAPTER 3. THE BLACK-SCHOLES MODEL Corollary The following relaionship, known as he pu-call pariy for fuures opions, holds for every [,T] Consequenly, C f P f = c f f,t p f f,t =e rt f K. 3.1 p f f,t =e rt KN d 2 f,t f N d 1 f,t, where d 1 f, and d 2 f, are given by Example Suppose ha he call opion considered in Example is a fuures opion. This means, in paricular, ha he price is now inerpreed as he fuures price. Using 3.91, one finds ha he arbirage price of a fuures call opion equals approximaely C f =1.22. Moreover, he porfolio ha replicaes he opion is composed a ime of φ 1 fuures conracs and φ 2 invesed in risk-free bonds, where φ 1 =.75 and φ 2 =1.22. Since he number φ 1 is posiive, i is clear ha an invesor who assumes a shor opion posiion needs o ener φ 1 long fuures conracs. Such a posiion, commonly referred o as he long hedge, is also a generally acceped pracical sraegy for a pary who expecs o purchase a given asse a some fuure dae. To find he arbirage price of he corresponding pu fuures opion, we make use of he pu-call pariy relaionship 3.1. We find ha P f =.23; moreover, for he replicaing porfolio of he pu opion we have φ1 =.25 and φ 2 =.23. Since now φ 1 <, we deal here wih he shor hedge a sraegy ypical for an invesor who expecs o sell a given asse a some fuure dae Opions on Forward Conracs We adop he classic Black-Scholes framework of Sec We will consider a forward conrac wih delivery dae T > wrien on a non-dividend-paying sock S. Recall ha he forward price a ime of a sock S for he selemen dae T equals F S, T =S e rt, [,T ]. This means ha he forward conrac, esablished a ime, in which he delivery price is se o be equal o F S, T is worhless a ime. I should be sressed ha he value of such a conrac a ime u [, T ] is no longer zero, in general. I is inuiively clear ha he value V F, u, T of such a conrac a ime u equals he discouned value of he difference beween he curren forward price of S a ime u and is value a ime, ha is V F, u, T =e rt u S u e rt u S e rt = S u S e ru for every u [, T ]. The las equaliy can also be derived by applying direcly he risk-neural valuaion formula o he claim X = S T F S, T, which seles a ime T. Indeed, we have V F, u, T = B u E P B 1 T S T B 1 T S e rt F u = B u E P ST F u S e rt e rt u = S u S e ru = S u F S, u, since he random variable S is F u -measurable. I is worhwhile o observe ha V F, u, T isin fac independen of he selemen dae T, herefore we may and do wrie V F, u, T =V F, u in wha follows. By definiion, 1 a call opion wrien a ime on a forward conrac wih he expiry 1 Since opions on forward conracs are no raded on exchanges,he definiion of an opion wrien on a forward conrac is largely a maer of convenion.

74 3.3. FUTURES MARKET 73 dae <T <T is simply a call opion, wih zero srike price, which is wrien on he value of he underlying forward conrac. The erminal opion s payoff hus equals C F T = V F, T + = ST S e rt +. I is clear ha he call opion on he forward conrac purchased a ime gives he righ o ener a ime T ino he forward conrac on he sock S wih delivery dae T and delivery price F S, T. If he forward price a ime T is less han i was a ime, he opion is abandoned. In he opposie case, he holder exercises he opion, and eiher eners, a no addiional cos, ino a forward conrac under more favorable condiions han hose prevailing a ime T, or simply akes he payoff of he opion. Assume now ha he opion was wrien a ime, so ha C F T =V F,T + =S T S e rt +. To value such an opion a ime T, we can make use of he Black-Scholes formula wih he fixed srike price K = S e rt. Afer simple manipulaions, we find ha he opion s value a ime is C F = S N d 1 S, S e r N d 2 S,, 3.11 where d 1,2 S,= ln S lns e r ± 1 2 σ2 T σ. T Alernaively, we can make use of Black s fuures formula. Since he fuures price f S T,T coincides wih S T, we have CT F =f S T,T S e rt +. An applicaion of Black s formula yields where f = f S, T, and C F = e rt f N d1 f, S e rt N d2 f,, 3.12 d 1,2 f,= ln f lns e rt ± 1 2 σ2 T σ T Since in he Black-Scholes seing he relaionship f S, T =S e rt is saisfied, i is apparen ha expressions 3.11 and 3.12 are equivalen..

75 74 CHAPTER 3. THE BLACK-SCHOLES MODEL

76 Chaper 4 Foreign Marke Derivaives In his chaper, an arbirage-free model of he domesic securiy marke is exended by assuming ha rading in foreign asses, such as foreign risk-free bonds and foreign socks and heir derivaives, is allowed. We shall work wihin he classic Black-Scholes framework. More specifically, boh domesic and foreign risk-free ineres raes are assumed hroughou o be non-negaive consans, and he foreign sock price and he exchange rae are modelled by means of geomeric Brownian moions. This implies ha he foreign sock price, as well as he price in domesic currency of one uni of foreign currency i.e., he exchange rae will have lognormal probabiliy disribuions a fuure imes. Noice, however, ha in order o avoid perfec correlaion beween hese wo processes, he underlying noise process should be modelled by means of a mulidimensional, raher han a onedimensional, Brownian moion. Our main goal is o esablish explici valuaion formulae for various kinds of currency and foreign equiy opions. Also, we will provide some indicaions concerning he form of he corresponding hedging sraegies. I is clear ha foreign marke conracs of cerain kinds should be hedged boh agains exchange rae movemens and agains he flucuaions of relevan foreign equiies. 4.1 Cross-currency Marke Model All processes considered in wha follows are defined on a common filered probabiliy space Ω, F, P, where he filraion F is assumed o be he P-augmenaion of he naural filraion generaed by a d-dimensional Brownian moion W =W 1,...,W d. The domesic and foreign ineres raes, r d and r f, are assumed o be given real numbers. Consequenly, he domesic and foreign savings accouns saisfy B d = expr d, B f = expr f, [,T ], 4.1 where B d and B f are denominaed in unis of domesic and foreign currency, respecively. The exchange rae process Q, which is used o conver foreign payoffs ino domesic currency, is modelled by he following sochasic differenial equaion dq = Q µq d + σ Q dw, Q >, 4.2 where µ Q R is a consan drif coefficien and σ Q R d denoes a consan volailiy vecor. As usual, he do sands for he Euclidean inner produc in R d, for insance σ Q dw = d σq i dw i. Also, we wrie o denoe he Euclidean norm in R d. Using his noaion, we can make a clear disincion beween models which are based on a one-dimensional Brownian moion, and hose i=1 75

77 76 CHAPTER 4. FOREIGN MARKET DERIVATIVES models in which he mulidimensional characer of he underlying noise process is essenial. Le us make clear ha we adop here he convenion ha he exchange rae process Q is denominaed in unis of domesic currency per uni of foreign currency; ha is, Q represens he domesic price a ime of one uni of he foreign currency. I should be sressed, however, ha he exchange rae process Q canno be reaed on an equal basis wih he price processes of domesic asses; pu anoher way, he foreign currency canno be seen as jus an addiional raded securiy in he domesic marke model, unless he impac of he foreign ineres rae is aken ino accoun. The process Q plays an imporan role as a ool which allows he conversion of foreign marke cash flows ino unis of domesic currency. Moreover, i can also play he role of an opion s underlying asse Domesic Maringale Measure In view of 4.2, he exchange rae a ime equals Q = Q exp σ Q W +µ Q 1 2 σ Q Le us inroduce an auxiliary process Q, given by he equaliy Q def = B f Q /B d = e r f r d Q, [,T ]. I is clear ha Q represens he value a ime of he foreign savings accoun, when convered ino he domesic currency, and discouned by he curren value of he domesic savings accoun. Moreover, i is useful o observe ha Q saisfies or equivalenly, ha he dynamics of Q are Q = Q exp σ Q W +µ Q + r f r d 1 2 σ Q 2, dq = Q µq + r f r d d + σ Q dw. 4.4 I is clear ha he process Q follows a maringale under he original probabiliy measure P if and only if he drif coefficien µ Q saisfies µ Q = r d r f. We shall frequenly make use of he process B f, which equals def = B f Q = e r f Q, [,T ]. 4.5 B f Noe ha B f represens he value a ime of a uni invesmen in a foreign savings accoun, expressed in unis of he domesic currency. In order o exclude arbirage beween invesmens in domesic and foreign bonds, we have o assume ha he drif coefficien of he exchange rae process equals r d r f under an equivalen probabiliy measure P, hereafer referred o as he maringale measure of he domesic marke, or briefly, he domesic maringale measure. I is worhwhile o observe ha a maringale measure P is no unique, in general. Indeed, in our framework, he maringale measure P is associaed wih a soluion ˆη R d of he following equaion µ Q + r f r d + σ Q ˆη =, for which he uniqueness of a soluion need no hold, in general. Sill, for any soluion ˆη of his equaion, he probabiliy measure P given by he usual exponenial formula dp dp = expˆη W T 1 2 ˆη 2 T, P-a.s., 4.6 can play he role of a maringale measure associaed wih he domesic marke. In addiion, he process W, which equals W = W ˆη, [,T ], 4.7 follows a d-dimensional Brownian moion under P wih respec o he underlying filraion. The uniqueness of he maringale measure can be gained by inroducing he possibiliy of rading in

78 4.1. CROSS-CURRENCY MARKET MODEL 77 addiional foreign or domesic asses, foreign or domesic socks, say. In oher words, he uniqueness of a maringale measure holds if he number of non-redundan raded asses, including he domesic savings accoun, equals d + 1, where d sands for he dimensionaliy of he underlying Brownian moion. For insance, if no domesic socks are raded and only one foreign sock is considered, o guaranee he uniqueness of a maringale measure P, and hus he compleeness of he marke, i is enough o assume ha W, and hus also W, is a wo-dimensional Brownian moion. In such a case, he marke model involves hree primary securiies he domesic and foreign savings accouns or equivalenly, domesic and foreign bonds and a foreign sock. In any case, he dynamics of he exchange rae Q under he domesic maringale measure P are easily seen o be dq = Q rd r f d + σ Q dw, Q >, 4.8 where W follows a d-dimensional Brownian moion under P. Inuiively, he domesic maringale measure P is a risk-neural probabiliy as seen from he perspecive of a domesic invesor ha is, an invesor who consanly denominaes he prices of all asses in unis of domesic currency. I is clear ha he arbirage price π X, in unis of domesic currency, of any coningen claim X, which seles a ime T and is also denominaed in he domesic currency, equals π X =e r dt E P X F. 4.9 If a ime T claim Y is denominaed in unis of foreign currency, is arbirage price a ime, expressed in unis of domesic currency, is given by he formula π Y =e r dt E P QT Y F. 4.1 Noice ha he arbirage price of such a claim can be alernaively evaluaed using he maringale measure associaed wih he foreign marke, and ulimaely convered ino domesic currency using he curren exchange rae Q. For his purpose, we need o inroduce an arbirage-free probabiliy measure associaed wih he foreign marke, referred o as he foreign maringale measure Foreign Maringale Measure We shall now ake he perspecive of a foreign-based invesor ha is, an invesor who consisenly denominaes her profis and losses in unis of foreign currency. Since Q is he price a ime of one uni of foreign currency in he domesic currency, i is eviden ha he price a ime of one uni of he domesic currency, expressed in unis of foreign currency, equals R =1/Q. From Iô s formula, we have dq 1 = Q 2 dq + Q 3 d Q, Q, or more explicily dr = R rd r f d + σ Q dw + R σ Q 2 d. Therefore, he dynamics of R under P are given by he expression Equivalenly, where we denoe by R he following process dr = R rf r d d σ Q dw σ Q d dr = R σ Q dw σ Q d, 4.12 R def = R e r d r f = e r f R B d, [,T ]. Observe ha R represens he price process of he domesic savings accoun, expressed in unis of foreign currency, and discouned using he foreign risk-free ineres rae. By virue of 4.12, i is

79 78 CHAPTER 4. FOREIGN MARKET DERIVATIVES easily seen ha he process R follows a maringale under a probabiliy measure P, equivalen o P, which saisfies d P dp = η T, P -a.s on Ω, F T, where η equals η = exp σ Q W 1 2 σ Q 2, [,T ] Any probabiliy measure P defined in his way is referred o as he maringale measure of he foreign marke. If he uniqueness of a domesic maringale measure P is no valid, he uniqueness of a foreign marke maringale measure P does no hold eiher. However, under any foreign maringale measure P, we have dr = R σ Q d W, 4.15 where he process W = W σ Q follows a d-dimensional Brownian moion under P. I is useful o observe ha he dynamics of R under he maringale measure P are given by he following counerpar of 4.8 dr = R rf r d d σ Q d W In financial inerpreaion, a foreign marke maringale measure P is any probabiliy measure on Ω, F T equivalen o P ha excludes arbirage opporuniies beween risk-free and risky invesmens in boh economies, as seen from he perspecive of a foreign-based invesor. For any aainable coningen claim X, which seles a ime T and is denominaed in unis of domesic currency, he arbirage price a ime in unis of foreign currency is given by he equaliy π X =e r f T E P RT X F, [,T] We are now in a posiion o esablish a relaionship which links a condiional expecaion evaluaed under he foreign marke maringale measure P o is counerpar evaluaed under he domesic maringale measure P. We assume here ha P is associaed wih P hrough Proposiion The following formula is valid for any F T -measurable random variable X provided ha he condiional expecaion is well-defined E PX F =E P X exp σ Q WT W 12 σ Q 2 T F Proof. By virue of he Bayes formula we ge E PX F = E P η T X F E P η T F, and hus observe ha η follows a maringale under P E PX F =η 1 E P η T X F =E P η T η 1 X F, as expeced. Le S f be he foreign currency price a ime of a foreign raded sock which pays no dividends. In order o exclude arbirage, we assume ha he dynamics of he price process S f under he foreign maringale measure P are ds f = S f rf d + σ S f d W, S f >, 4.19 wih a consan volailiy coefficien σ S f R d. This means ha he sock price process S f follows ds f = S f rf σ Q σ S f d + σ S f dw 4.2

80 4.2. CURRENCY FORWARD CONTRACTS AND OPTIONS 79 under he domesic maringale measure P associaed wih P. For he purpose of pricing foreign equiy opions, we will someimes find i useful o conver he price of he underlying foreign sock ino he domesic currency. We wrie S f = Q S f o denoe he price of a foreign sock S f expressed in unis of domesic currency. Using Iô s formula, and he dynamics under P of he exchange rae Q, which are dq = Q rd r f d + σ Q dw, 4.21 one finds ha under he domesic maringale measure P, he process S f saisfies d S f = S f rd d +σ S f + σ Q dw The las equaliy shows ha he price process S f behaves as he price process of a domesic sock in he classic Black-Scholes framework; however, he corresponding volailiy coefficien is equal o he superposiion σ S f + σ Q of wo volailiies he foreign sock price volailiy and he exchange rae volailiy. By defining in he usual way he class of admissible rading sraegies, one may now easily consruc a marke model in which here is no arbirage beween invesmens in foreign and domesic bonds and socks. Since his can be easily done, we leave he deails o he reader. 4.2 Currency Forward Conracs and Opions In his secion, we consider derivaive securiies whose value depends exclusively on he flucuaions of exchange rae Q, as opposed o hose securiies which depend also on some foreign equiies. Currency opions, forward conracs and fuures conracs provide an imporan financial insrumen hrough which o conrol he risk exposure induced by he uncerain fuure exchange rae. The deliverable insrumen in a classic foreign exchange opion is a fixed amoun of underlying foreign currency. The valuaion formula ha provides he arbirage price of foreign exchange European-syle opions was esablished independenly in Biger and Hull 1983 and Garman and Kohlhagen They have shown ha if he domesic and foreign risk-free raes are consan, and he dynamics of he exchange rae are given by 4.2, hen a foreign currency opion may be valued by means of a suiable varian of he Black-Scholes opion valuaion formula. More precisely, one may apply formula 3.76, which gives he arbirage price of a European opion wrien on a sock which pays a consan dividend yield Forward Exchange Rae Le us firs consider a foreign exchange forward conrac, wrien a ime, which seles a he fuure dae T. The asse o be delivered by he pary assuming a shor posiion in he conrac is a prespecified amoun of foreign currency, say 1 uni. The pary who assumes a long posiion in a currency forward conrac is obliged o pay a cerain number of unis of a domesic currency, he delivery price. As usual, he delivery price ha makes he forward conrac worhless a ime T is called he forward price a ime of one uni of he foreign currency o be delivered a he selemen dae T. In he presen conex, i is naural o refer o his forward price as he forward exchange rae. We will wrie F Q, T o denoe he forward exchange rae. Proposiion The forward exchange rae F Q, T a ime for he selemen dae T is given by he following formula F Q, T =e r d r f T Q, [,T] Proof. I is easily seen ha if 4.23 does no hold, risk-free profiable opporuniies arise beween he domesic and he foreign marke. Relaionship 4.23, commonly known as he ineres rae pariy, assers ha he forward exchange premium mus equal, in he marke equilibrium, he ineres rae differenial r d r f. A

81 8 CHAPTER 4. FOREIGN MARKET DERIVATIVES relaively simple version of he ineres rae pariy sill holds even when he domesic and foreign ineres raes are no longer deerminisic consans, bu follow sochasic processes. Under uncerain ineres raes, we need o o inroduce he price processes B d, T and B f, T of he domesic and foreign zero-coupon bonds wih mauriy T. A zero-coupon bond wih a given mauriy T is a financial securiy which pays one uni of he corresponding currency a he fuure dae T. Suppose ha zero-coupon bonds wih mauriy T are raded in boh domesic and foreign markes. Then equaliy 4.23 may be exended o cover he case of sochasic ineres raes. Indeed, i is no hard o show, by means of no-arbirage argumens, ha F Q, T = B f, T B d, T Q, [,T], 4.24 where B d, T and B f, T sand for he respecive ime prices of he domesic and foreign zerocoupon bonds wih mauriy T. Noice ha in 4.24, boh B d, T and B f, T should be seen as he domesic and foreign discoun facors raher han he prices. Indeed, prices should be expressed in unis of he corresponding currencies, while discoun facors are merely he corresponding real numbers. Finally, i follows immediaely from 4.21 ha for any fixed selemen dae T, he forward price dynamics under he maringale measure of he domesic economy P are and F Q T,T=Q T Currency Opion Valuaion Formula df Q, T =F Q, T σ Q dw, 4.25 As a firs example of a currency opion, we consider a sandard European call opion, whose payoff a he expiry dae T equals def = NQ T K +, C Q T where Q T is he spo price of he deliverable currency i.e., he spo exchange rae a he opion s expiry dae, K is he srike price in unis of domesic currency per foreign uni, and N>is he nominal value of he opion, expressed in unis of he underlying foreign currency. I is clear ha payoff from he opion is expressed in he domesic currency; also, here is no loss of generaliy if we assume ha N =1. Summarizing, we consider an opion o buy one uni of a foreign currency a a prespecified price K, which may be exercised a he dae T only. Proposiion The arbirage price, in unis of domesic currency, of a currency European call opion is given by he risk-neural valuaion formula C Q = e r dt E P QT K + F, [,T] Moreover, he price C Q is given by he following expression C Q = Q e r f T N h 1 Q,T Ke r dt N h 2 Q,T, where N is he sandard Gaussian cumulaive disribuion funcion, and h 1,2 q, = lnq/k+r d r f ± 1 2 σ2 Q σ Q. Proof. Le us firs examine a rading sraegy in risk-free domesic and foreign bonds, which we call a currency rading sraegy in wha follows. Formally, by a currency rading sraegy we mean an adaped sochasic process φ =φ 1,φ 2. In financial inerpreaion, φ 1 Bf and φ 2 B d represen he amouns of money invesed a ime in foreign and domesic bonds. I is imporan o noe ha

82 4.2. CURRENCY FORWARD CONTRACTS AND OPTIONS 81 boh amouns are expressed in unis of domesic currency see, in paricular, 4.4. A currency rading sraegy φ is said o be self-financing if is wealh process V φ, which equals V φ =φ 1 B f + φ 2 B d, [,T], where B f = B f Q,B d = e r d, saisfies he following relaionship dv φ =φ 1 d B f + φ 2 db d. For he discouned wealh process V φ =e rd V φ of a self-financing currency rading sraegy, we easily ge dv φ =φ 1 de r d Bf =φ 1 dq. On he oher hand, by virue of 4.21, he dynamics of he process Q, under he domesic maringale measure P, are given by he expression dq = σ Q Q dw. Therefore, he discouned wealh V φ of any self-financing currency rading sraegy φ follows a maringale under P. This jusifies he risk-neural valuaion formula Taking ino accoun he equaliy Q T = B f T e r f T, one ges also C Q = e r dt E P QT K + F = e r f T e r dt E P Bf T Ker f T + F = e r f T C B f,t, Ke r f T,r d,σ Q, where C sands for he sandard Black-Scholes call opion price. More explicily, we have C Q = e r f T Bf N d 1 B f,t Ke r f T e r dt N d 2 B f,t = Q e r f T N d 1 B f,t Ke r dt N d 2 B f,t. This proves he formula we wish o show, since d i B f,t, Ke r f T,r d,σ Q =h i Q,T for i = 1, 2. Finally, one finds immediaely ha he firs componen of he self-financing currency rading sraegy ha replicaes he opion equals φ 1 = e r f T N d 1 B f,t = e r f T N h 1 Q,T. Therefore, o hedge a shor posiion, he wrier of he currency call should inves a ime T he amoun expressed in unis of foreign currency φ 1 B f = e r f T N h 1 Q,T in foreign marke risk-free bonds or equivalenly, in he foreign savings accoun. On he oher hand, she should also inves he amoun denominaed in domesic currency C Q Q e r f T N h 1 Q,T in he domesic savings accoun. Remarks. a As menioned earlier, a comparison of he currency opion valuaion formula esablished in Proposiion wih expression 3.76 shows ha he exchange rae Q can be formally seen as he price of a ficiious domesic sock. Under such a convenion, he foreign ineres rae r f can be inerpreed as a dividend yield ha is coninuously paid by his ficiious sock.

83 82 CHAPTER 4. FOREIGN MARKET DERIVATIVES b I is easy o derive he pu-call relaionship for currency opions. Indeed, he payoff in domesic currency of a porfolio composed of one long call opion and one shor pu opion is C Q T P Q T =Q T K + K Q T + = Q T K, where we assume, as before, ha he opions are wrien on one uni of foreign currency. Consequenly, for any [,T], we have C Q P Q = e r f T Q e r dt K c We may also rewrie he currency opion valuaion formula of Proposiion in he following way C Q = e r dt F N d1 F,T KN d2 F,T, 4.28 where F = F Q, T and d 1,2 F, = lnf/k ± 1 2 σ2 Q σ Q, F, R +,T]. This shows ha he currency opion valuaion formula can be seen as a varian of he Black fuures formula 3.91 of Sec Furhermore, i is possible o re-express he replicaing sraegy of he opion in erms of domesic bonds and currency forward conracs. Le us menion ha under he presen assumpions of deerminisic domesic and foreign ineres raes, he disincion beween he currency fuures price and forward exchange rae is no essenial. In marke pracice, currency opions are frequenly hedged by aking posiions in forward and fuures conracs, raher han by invesing in foreign risk-free bonds. 4.3 Foreign Equiy Forward Conracs In a global equiy marke, an invesor may link his foreign sock and currency exposures in a large variey of ways. More specifically, he may choose o combine his invesmens in foreign equiies wih differing degrees of proecion agains adverse moves in exchange raes and sock prices, using forward and fuures conracs as well as a variey of opions Forward Price of a Foreign Sock Le us firs consider an ordinary forward conrac wih a foreign sock being he underlying asse o be delivered ha is, an agreemen o buy a sock on a cerain dae a a cerain delivery price in a specified currency. We shall disinguish beween he wo following cases: a when he delivery price K f is denominaed in he foreign currency, and b when i is expressed in he domesic currency; in he laer case he delivery price will be denoed by K d. I should be emphasized ha in boh siuaions, he value of he forward conrac a he selemen dae T is equal o he spread beween he sock price a ime T and he delivery price expressed in foreign currency. The erminal payoff is hen convered ino unis of domesic currency a he exchange rae ha prevails a he selemen dae T. Summarizing, in unis of domesic currency, he erminal payoffs from he long posiions are V d T K f =Q T S f T Kf in he firs case, and if he second case is considered. V d T K d =Q T S f T R 1 T Kd =Q T S f T Kd = S f T Kd

84 4.3. FOREIGN EQUITY FORWARD CONTRACTS 83 Case a. Observe ha he foreign-currency payoff a selemen of he forward conrac equals X T = S f T Kf. Therefore is value a ime, denominaed in he foreign currency, is V f K f =e r f T E P S f T Kf F =S f e r f T K f. Consequenly, when expressed in he domesic currency, he value of he conrac a ime equals V d K f =Q S f e r f T K f. We conclude ha he forward price of he sock S f, expressed in unis of foreign currency, equals F f S f, T =e r f T S f, [,T] Case b. In his case, equaliy 4.9 yields immediaely V d K d =e r dt E P Sf T Kd F ; hence, by virue of 4.22, he domesic-currency value of he forward conrac wih he delivery price K d denominaed in domesic currency equals V d K d =Q S f e r d T K d. This implies ha he forward price of a foreign sock in domesic currency equals F d S f, T =er d T Sf, [,T], 4.3 so ha, somewha surprisingly, i is independen of he foreign risk-free ineres rae r f Quano Forward Conracs In his secion, we shall examine a quano forward conrac on a foreign sock. 1 Such a conrac is also known as a guaraneed exchange rae forward conrac a GER forward conrac for shor. To describe he inuiion ha underpins he concep of a quano forward conrac, le us consider an invesor who expecs a cerain foreign sock o appreciae significanly over he nex period, and who wishes o capure his appreciaion in his porfolio. Buying he sock, or aking a long posiion in i hrough a forward conrac or call opion, leaves he invesor exposed o exchange rae risk. To avoid having his reurn depend on he performance of he foreign currency, he needs a guaranee ha he can close his foreign sock posiion a an exchange rae close o he one ha prevails a presen. This can be done by enering a quano forward or opion conrac in a foreign sock. In his subsecion, we shall sudy he case of quano forward conracs, leaving he analysis of quano opions o he nex secion. We sar by defining precisely wha is mean by a quano forward conrac in a foreign sock S f. As before, he payoff of a guaraneed exchange rae forward conrac on a foreign sock a selemen dae T is he difference beween he sock price a ime T and he delivery price denominaed in he foreign currency, say K f. However, his payoff is convered ino domesic currency a a predeermined exchange rae, denoed by Q in wha follows. More formally, denoing by V d K f, Q he ime value in domesic currency of he quano forward conrac, we have VT d K f, Q = QS f T Kf. We wish o deermine he righ value of such a conrac a ime before he selemen. Noice ha he erminal payoff of a quano forward conrac is independen of he fuure exchange rae flucuaions during he life of a conrac. Sill, as we shall see in wha follows, is value V d K f, Q depends on he volailiy coefficien σ Q of he exchange rae process Q more precisely, on he scalar 1 Generally speaking,a financial asse is ermed o be a quano produc if i is denominaed in a currency oher han ha in which i is usually raded.

85 84 CHAPTER 4. FOREIGN MARKET DERIVATIVES produc σ Q σ S f ha deermines he insananeous covariance beween he logarihmic reurns of he sock price and he exchange rae. By virue of he risk-neural valuaion formula, he value a ime of he quano forward conrac equals in domesic currency V d K f, Q = Qe r dt E P S f T F K f. To find he condiional expecaion E P S f T F, observe ha by virue of 4.2, he process Ŝ = e δ S f follows a maringale under P, provided ha we ake δ = r f σ Q σ S f. Consequenly, we find easily ha E P S f T F =e δt E P ŜT F =e δt Ŝ = e δt S f, and hus V d K f, Q = Qe r dt e r f σ Q σ S f T S f K f This in urn implies ha he forward price a ime associaed wih he quano forward conrac ha seles a ime T equals in unis of foreign currency ˆF f S f, T =e r f σ Q σ S f T S f = E P S f T F I is ineresing o noe ha ˆF f, T is simply he condiional expecaion of he sock price a he S f selemen dae T, as seen a ime from he perspecive of a domesic-based invesor. Furhermore, a leas when κ = σ Q σ S f, i can also be inerpreed as he forward price of a ficiious dividend-paying sock, wih κ = σ Q σ S f playing he role of he dividend yield cf. formula 3.79 in Sec Foreign Equiy Opions In his secion, we shall sudy examples of foreign equiy opions ha is, opions whose erminal payoff in unis of domesic currency depends no only on he fuure behavior of he exchange rae, bu also on he price flucuaions of a cerain foreign sock Opions Sruck in a Foreign Currency Assume firs ha an invesor wans o paricipae in gains in foreign equiy, desires proecion agains losses in ha equiy, bu is unconcerned abou he ranslaion risk arising from he poenial drop in he exchange rae. We denoe by T he expiry dae and by K f he exercise price of an opion. I is essenial o noe ha K f is expressed in unis of foreign currency. The erminal payoff from a foreign equiy call sruck in foreign currency equals C 1 T def = Q T S f T Kf +. This means, in paricular, ha he erminal payoff is assumed o be convered ino domesic currency a he spo exchange rae ha prevails a he expiry dae. By reasoning in much he same way as in he previous secion, one can check ha he arbirage price of a European call opion a ime equals C 1 = e r dt E P QT S f T Kf + F. Using 4.21, we find ha C 1 = e r dt Q E P {S ft Kf + exp σ Q WT W +λt } F, where λ = r d r f 1 2 σ Q 2. Equivalenly, using 4.18, we ge C 1 = e r f T Q E P S f T Kf + F.

86 4.4. FOREIGN EQUITY OPTIONS 85 Since P is he arbirage-free measure of he foreign economy, i is no hard o esablish he following expression C 1 = Q S f N g 1 S f,t K f e r f T N g 2 S f,t, where g 1,2 s, = lns/kf +r f ± 1 2 σ S f 2 σ S f. An inspecion of he valuaion formula above makes clear ha a hedging porfolio involves a any insan he number Ng 1 S f,t shares of he underlying sock; his sock invesmen demands he addiional borrowing of β d = Q K f e r f T N g 2 S f,t unis of he domesic currency, or equivalenly, he borrowing of β f = K f e r f T N g 2 S f,t unis of he foreign currency. Remarks. The valuaion resul esablished above is in fac quie naural. Indeed, seen from he foreign marke perspecive, he foreign equiy opion sruck in foreign currency can be priced direcly by means of he sandard Black-Scholes formula. As menioned, when dealing wih foreign equiy opions, one can eiher do he calculaions wih reference o he domesic economy, or equivalenly, one may work wihin he framework of he foreign economy and hen conver he final resul ino unis of domesic currency. For example, in he case considered above, o complee he calculaions in he domesic economy, on may use he following elemenary lemma, whose proof is lef o he reader. Lemma Le ξ,η be a zero-mean, joinly Gaussian non-degenerae, wo-dimensional random variable on a probabiliy space Ω, F, P. Then for arbirary posiive real numbers a and b, we have E P ae ξ 1 2 Var ξ be η 1 2 Var η + = anh bnh k, 4.33 where h = 1 k lna/b+ 1 2 k and k = Var ξ η Opions Sruck in Domesic Currency Assume now ha an invesor wishes o receive any posiive reurns from he foreign marke, bu wans o be cerain ha hose reurns are meaningful when ranslaed back ino his own currency. In his case he migh be ineresed in a foreign equiy call sruck in domesic currency, wih payoff a expiry CT 2 def = S f T Q T K d + = S f T Kd +, where he srike price K d is expressed in domesic currency. Due o he paricular form of he opion s payoff, i is clear ha i is now convenien o sudy he opion from he domesic perspecive. To find he arbirage price of he opion a ime, i is sufficien o calculae he following condiional expecaion C 2 = e r dt E P Sf T Kd + F, and by virue of , he sock price expressed in unis of domesic currency S f has he following dynamics under P d S f = S f rd d +σ S f + σ Q dw. Therefore, arguing as in he proof of he classic Black-Scholes formula, one finds easily ha he opion s price, expressed in unis of domesic currency, is given by he formula C 2 = S f N l 1 S f,t e r dt K d N l 2 S f,t, where l 1,2 s, = lns/kd +r d ± 1 2 σ S f + σ Q 2 σ S f + σ Q.

87 86 CHAPTER 4. FOREIGN MARKET DERIVATIVES Quano Opions Assume, as before, ha an invesor wishes o capure posiive reurns on his foreign equiy invesmen, bu now desires o eliminae all exchange risk by fixing an advance rae a which he opion s payoff will be convered ino domesic currency. A he inuiive level, such a conrac can be seen as a combinaion of a foreign equiy opion wih a currency forward conrac. By definiion, he payoffofaquano call i.e., a guaraneed exchange rae foreign equiy call opion a expiry is se o be CT 3 def = QS f T Kf +, where Q is he prespecified exchange rae a which he conversion of he opion s payoff is made. Noice ha he quaniy Q is denominaed in he domesic currency per uni of foreign currency, and he srike price K f is expressed in unis of foreign currency. Since he payoff from he quano opion is expressed in unis of domesic currency, is arbirage price equals C 3 = Qe r dt E P S f T Kf + F The proofs of he nex wo proposiions are lef o he reader as exercises. Proposiion The arbirage price a ime of a European quano call opion wih expiry dae T and srike price K f equals in unis of domesic currency C 3 = Qe r dt S f e δt N c 1 S f,t K f N c 2 S f,t, where δ = r f σ Q σ S f and c 1,2 s, = lns/kf +δ ± 1 2 σ S f 2 σ S f Equiy-linked Foreign Exchange Opions Finally, assume ha an invesor desires o hold foreign equiy regardless of wheher he sock price rises or falls ha is, he is indifferen o he foreign equiy exposure, however, wishes o place a floor on he exchange rae risk of his foreign invesmen. An equiy-linked foreign exchange call an Elf-X call, for shor wih payoff a expiry in unis of domesic currency. C 4 T def = Q T K + S f T, where K is a srike exchange rae expressed in domesic currency per uni of foreign currency, is hus a combinaion of a currency opion wih an equiy forward. The arbirage price in unis of domesic currency of a European Elf-X call wih expiry dae T equals C 4 = e r dt E P QT K + S f T F. We shall firs value an equiy-linked foreign exchange call opion using he domesic maringale measure. Proposiion The arbirage price, expressed in domesic currency, of a European equiy-linked foreign exchange call opion, wih srike exchange rae K and expiry dae T, is given by he following formula C 4 = S f Q N w 1 Q,T Ke γt N w 2 Q,T, 4.35 where γ = r d r f + σ Q σ S f and w 1,2 q, = lnq/k+γ ± 1 2 σ Q 2 σ Q.

88 Chaper 5 American Opions In conras o he holder of a European opion, he holder of an American opion is allowed o exercise his righ o buy or sell he underlying asse a any ime before or a he expiry dae. This special feaure makes he arbirage pricing of American opions much more involved han he valuaion of sandard European claims. We know already ha arbirage valuaion of American claims for insance, wihin he framework of he binomial CRR model sudied in Chap. 2 is closely relaed o specific opimal sopping problems. Inuiively, one migh expec ha he holder of an American opion will choose her exercise policy in such a way ha he expeced payoff from he opion will be maximized. Maximizaion of he expeced discouned payoff under subjecive probabiliy would lead, of course, o non-uniqueness of he price. I appears, however, ha for he purpose of arbirage valuaion, he maximizaion of he expeced discouned payoff should be done under he maringale measure ha is, under risk-neural probabiliy. Therefore, he uniqueness of he arbirage price of an American claim holds. One of he earlies works o examine he relaionship beween he early exercise feaure of American opions and opimal sopping problems was he paper in McKean I should be made clear ha he arbirage valuaion of derivaive securiies was no ye discovered a his ime, however. For his reason, he opimal sopping problem associaed wih he opimal exercise of American pu was sudied in McKean 1965 under an acual probabiliy P, raher han under he maringale measure P, as is done nowadays. Basic feaures of American opions, wihin he framework of arbirage valuaion heory, were already examined in van Moerbeke However, mahemaically rigorous valuaion resuls for American claims were firs esablished by means of arbirage argumens in Bensoussan 1984 and Karazas 1988, For an exhausive survey of resuls and echniques relaed o he arbirage pricing of American opions, we refer he reader o Myneni Valuaion of American Claims We place ourselves wihin he classic Black-Scholes seup. Hence, he prices of primary securiies ha is, he sock price, S, and he savings accoun, B are modelled by means of he following differenial equaions ds = µs d + σs dw, S >, 5.1 where µ R and σ>are real numbers, and db = rb d, B =1, 5.2 wih r R, respecively. As usual, we denoe by W he sandard Brownian moion defined on a filered probabiliy space Ω, F, P, where F = F W. For he sake of noaional convenience, we assume here ha he underlying Brownian moion W is one-dimensional. In he conex of arbirage valuaion of American coningen claims, i is convenien o assume ha an individual may wihdraw funds o finance his consumpion needs. For any fixed, we denoe 87

89 88 CHAPTER 5. AMERICAN OPTIONS by A he cumulaive amoun of funds ha are wihdrawn and consumed by an invesor up o ime. The erm consumed refers o he fac ha he wealh is dynamically diminished according o he process A. The process A is assumed o be progressively measurable wih non-decreasing and RCLL sample pahs; also, by convenion, A = A =. We say ha A represens he consumpion sraegy, as opposed o he rading sraegy φ. I is hus naural o call a pair φ, A arading and consumpion sraegy in S,B. In he presen conex, he formal definiion of a self-financing sraegy reads as follows. Definiion A rading and consumpion sraegy φ, A ins, B is self-financing on [, T]if is wealh process V φ, A, which equals saisfies for every [,T] V φ, A =φ 1 S + φ 2 B, [,T], 5.3 V φ, A =V φ, A+ φ 1 u ds u + φ 2 u db u A. 5.4 In view of 5.4, i is clear ha A models he flow of funds ha are no reinvesed in primary securiies, bu raher are pu aside forever. By convenion, we say ha he amoun of funds represened by A is consumed by he holder of he dynamic porfolio φ, A up o ime. Eliminaing he componen φ 2 yields he following equivalen form of 5.4 dv = rv d + φ 1 S µ r d + σdw da, where we wrie V o denoe he wealh process V φ, A. Equivalenly, dv = rv d + ζ µ r d + σζ dw da, 5.5 where ζ = φ 1 S represens he amoun of cash invesed in shares a ime. The unique soluion of he linear SDE 5.5 is given by an explici formula V = B V + µ rζ u B 1 u du B 1 u da u + σζ u B 1 u dw u, which holds for every [,T]. We conclude ha he wealh process of any self-financing rading and consumpion sraegy is uniquely deermined by he following quaniies: he iniial endowmen V, he consumpion process A, and he process ζ represening he amoun of cash invesed in shares. In oher words, given an iniial endowmen V, here is one-o-one correspondence beween self-financing rading and consumpion sraegies φ, A and wo-dimensional processes ζ,a. We will someimes find i convenien o idenify a self-financing rading and consumpion sraegy φ, A wih he corresponding pair ζ,a, where ζ = φ 1 S. Recall ha he unique maringale measure P for he Black-Scholes spo marke saisfies dp r µ dp = exp W T 1 r µ 2 σ 2 σ 2 T, P-a.s. I is easily seen ha he dynamics of he wealh process V under he maringale measure P are given by he following expression dv = rv d + σζ dw da, where W follows he sandard Brownian moion under P. This yields immediaely V = B V B 1 u da u + σζ u Bu 1 dwu.

90 5.1. VALUATION OF AMERICAN CLAIMS 89 Therefore, an auxiliary process Z, which is given by he formula Z def = V + B 1 u da u = V + σζ u B 1 u dw u, where V = V /B, follows a local maringale under P. We say ha a self-financing rading and consumpion sraegy φ, A isadmissible if he condiion E P T ζu 2 du = E P T φ 1 us u 2 du < is saisfied so ha Z is a P -maringale. Similarly o Sec , his assumpion is imposed in order o exclude pahological examples of arbirage opporuniies from he marke model. We are now in a posiion o formally inroduce he concep of a coningen claim of American syle. To his end, we ake an arbirary coninuous reward funcion g : R + [,T] R saisfying he linear growh condiion. An American claim wih he reward funcion g and expiry dae T is a financial securiy which pays o is holder he amoun gs, when exercised a ime. The wrier of an American claim wih he reward funcion g acceps he obligaion o pay he amoun gs, a any ime. I should be emphasized ha he choice of he exercise ime is a discreion of he holder of an American claim ha is, of a pary assuming a long posiion. In order o formalize he concep of an American claim, we need o inroduce firs a suiable class of admissible exercise imes. Since we exclude clairvoyance, he admissible exercise ime τ is assumed o be a sopping ime of filraion F. Le us recall ha a random variable τ :Ω, F T, P [,T]isa sopping ime of filraion F if, for every [,T], he even {τ } belongs o he σ-field F. Since in he Black-Scholes model we have F = F W = F W = F S, any sopping ime of he filraion F is also a sopping ime of he filraion F S generaed by he sock price process S. In inuiive erms, i is assumed hroughou ha he decision o exercise an American claim a ime is based on he observaions of sock price flucuaions up o ime, bu no afer his dae. This inerpreaion is consisen wih our general assumpion ha he σ-field F represens he informaion available o all invesors a ime. Le us denoe by T [,T ] he se of all sopping imes of he filraion F which saisfy τ T wih probabiliy 1. Definiion An American coningen claim X a wih he reward funcion g : R + [,T] R is a financial insrumen consising of: a an expiry dae T ; b he selecion of a sopping ime τ T [,T ] ; and c a payoff X a τ = gs τ,τ on exercise. Typical examples of American claims are American opions wih consan srike price K and expiry dae T. The payoffs of American call and pu opions, when exercised a he random ime τ, are equal o X τ = S τ K + and Y τ = K S τ + respecively. Our aim is o derive he raional price and o deermine he raional exercise ime of an American coningen claim by means of purely arbirage argumens. To his end, we shall firs inroduce a specific class of rading sraegies. For exposiional simpliciy, we shall search for he price of an American claim X a a ime ; he general case can be reaed along he same lines, bu is more cumbersome from he noaional viewpoin. I will be sufficien o consider a very special class of rading sraegies associaed wih he American coningen claim X a, namely he buy-and-hold sraegies. By a buy-and-hold sraegy associaed wih an American claim X a, we mean a pair c, τ, where τ T [,T ] and c is a real number. In financial inerpreaion, a buy-and-hold sraegy c, τ assumes ha c > unis of he American securiy X a are acquired or shored, if c< a ime, and hen held in he porfolio up o he exercise ime τ. Observe ha such a sraegy excludes rading in he American claim afer he iniial dae. In oher words, dynamic rading in he American claim is no considered a his sage. Le us assume ha here exiss a marke price, say U, a which he American claim X a rades in he marke a ime. Our firs ask is o find he righ value of U by means of no-arbirage argumens as menioned above, he argumens which lead o he arbirage valuaion of he claim

91 9 CHAPTER 5. AMERICAN OPTIONS X a a ime >are much he same as in he case of =, herefore he general case is lef o he reader. Definiion By a self-financing rading sraegy in S, B, X a, we mean a collecion φ, A, c, τ, where φ, A is a rading and consumpion sraegy in S, B and c, τ is a buy-and-hold sraegy associaed wih X a. In addiion, we assume ha on he random inerval τ,t] wehave φ 1 =, φ 2 = φ 1 τ S τ B 1 τ + φ 2 τ + cgs τ,τb 1 τ. 5.6 I will soon become apparen ha i is enough o consider he cases of c = 1 and c = 1; ha is, he long and shor posiions in he American claim X a. An analysis of condiion 5.6 shows ha he definiion of a self-financing sraegy φ, A, c, τ implicily assumes ha he American claim is exercised a a random ime τ, exising posiions in shares are closed a ime τ, and all he proceeds are invesed in risk-free bonds. For breviy, we shall someimes wrie ψ o denoe he dynamic porfolio φ, A, c, τ in wha follows. Noe ha he wealh process V ψ of any self-financing sraegy in S, B, X a saisfies he following iniial and erminal condiions V ψ =φ 1 S + φ 2 + cu 5.7 and V T ψ =e rt τ φ 1 τ S τ + cgs τ,τ + e rt φ 2 τ. 5.8 In wha follows, we shall resric our aenion o he class of admissible rading sraegies ψ = φ, A, c, τ ins, B, X a, which are defined in he following way. Definiion A self-financing rading sraegy φ, A, c, τ ins, B, X a is said o be admissible if a rading and consumpion sraegy φ, A is admissible and A T = A τ. The class of all admissible sraegies φ, A, c, τ is denoed by Ψ. Le us inroduce he class Ψ of hose admissible rading sraegies ψ for which he iniial wealh saisfies V ψ <, and he erminal wealh has he non-negaive value; ha is 1 V T ψ =φ 2 T B. In order o precisely define an arbirage opporuniy, we have o ake ino accoun he early exercise feaure of American claims. I is inuiively clear ha i is enough o consider wo cases a long and a shor posiion in one uni of an American claim. This is due o he fac ha we need o exclude he exisence of arbirage opporuniies for boh he seller and he buyer of an American claim. Indeed, he posiion of boh paries involved in a conrac of American syle is no longer symmeric, as i was in he case of European claims. The holder of an American claim can acively choose his exercise policy. The seller of an American claim, on he conrary, should be prepared o mee his obligaions a any random ime. We herefore se down he following definiion of arbirage and an arbirage-free marke model. Definiion There is arbirage in he marke model wih rading in he American claim X a wih iniial price U if eiher a here is long arbirage, i.e., here exiss a sopping ime τ such ha for some rading and consumpion sraegy φ, A he sraegy φ, A, 1,τ belongs o he class Ψ, or b here is shor arbirage, i.e., here exiss a rading and consumpion sraegy φ, A such ha for any sopping ime τ he sraegy φ, A, 1,τ belongs o he class Ψ. In he absence of arbirage in he marke model, we say ha he model is arbirage-free. Definiion can be reformulaed in he following way: here is absence of arbirage in he marke if he following condiions are saisfied: a for any sopping ime τ and any rading and consumpion sraegy φ, A, he sraegy φ, A, 1,τ is no in Ψ ; and b for any rading and consumpion sraegy φ, A, here exiss a sopping ime τ such ha he sraegy φ, A, 1,τis 1 Since he exisence of a sricly posiive savings accoun is assumed,one can alernaively define he class Ψ as he se of hose sraegies ψ from Ψ for which V ψ =,V T ψ =φ 2 T B T, and he laer inequaliy is sric wih posiive probabiliy.

92 5.1. VALUATION OF AMERICAN CLAIMS 91 no in Ψ. Inuiively, under he absence of arbirage in he marke, he holder of an American claim is unable o find an exercise policy τ and a rading and consumpion sraegy φ, A ha would yield a risk-free profi. Also, under he absence of arbirage, i is no possible o make risk-free profi by selling he American claim a ime, provided ha he buyer makes a clever choice of he exercise dae. More precisely, here exiss an exercise policy for he long pary which prevens he shor pary from locking in a risk-free profi. By definiion, he arbirage price a ime of he American claim X a, denoed by π X a, is ha level of he price U which makes he model arbirage-free. Our aim is now o show ha he assumed absence of arbirage in he sense of Definiion leads o a unique value for he arbirage price π X a ofx a as already menioned, i is no hard o exend his reasoning in order o deermine he arbirage price π X a of he American claim X a a any dae [,T]. Also, we shall find he raional exercise policy of he holder ha is, he sopping ime ha excludes he possibiliy of shor arbirage. The following auxiliary resul relaes he value process associaed wih he specific opimal sopping problem o he wealh process of a cerain admissible rading sraegy. For any reward funcion g, we define an adaped process V by seing V = ess sup τ T [,T ] E P e rτ gs τ,τ F 5.9 for every [,T], provided ha he righ-hand side in 5.9 is well-defined. Proposiion Le V be an adaped process defined by formula 5.9 for some reward funcion g. Then here exiss an admissible rading and consumpion sraegy φ, A such ha V = V φ, A for every [,T]. Proof. We shall give he ouline of he proof for echnical deails, we refer o Karazas 1988 and Myneni Le us inroduce he Snell envelope J of he discouned reward process Z = e r gs,. By definiion, he process J is he smalles supermaringale majoran o he process Z. From he general heory of opimal sopping, we know ha J = ess sup τ T [,T ] E P e rτ gs τ,τ F = ess sup E τ T [,T ] P Z τ F for every [,T], so ha V = e r J. Since J is a RCLL regular supermaringale of class DL, 2 i follows from general resuls ha J admis he unique Doob-Meyer decomposiion J = M H, where M is a square-inegrable maringale and H is a coninuous non-decreasing process wih H =. Consequenly, d e r J = re r J d + e r dm e r dh. By virue of he predicable represenaion propery see Theorem we have M = M + ξ u dw u, [,T], for some progressively measurable process ξ wih E P T ξ2 u du <. Hence, upon seing φ 1 = e r ξ σ 1 S 1, φ 2 = J ξ σ 1, A = e ru dh u, 5.1 we conclude ha he process V represens he wealh process of some admissible rading and consumpion sraegy. 2 Basically,one needs o check ha he family {J τ τ T [,T ] } of random variables is uniformly inegrable under P. We refer he reader o Sec. 1.4 in Karazas and Shreve 1998 for he definiion of a regular process and for he concep of he Doob-Meyer decomposiion of a semimaringale.

93 92 CHAPTER 5. AMERICAN OPTIONS By he general heory of opimal sopping, we know also ha he random ime τ ha maximizes he expeced discouned reward afer he dae is he firs insan a which he process J drops o he level of he discouned reward, ha is τ = inf {u [, T ] J u = Z u }, [,T] In oher words, he opimal under P exercise policy of he American claim wih reward funcion g is given by he equaliy τ = inf {u [,T] J u = e ru gs u,u } Observe ha he sopping ime τ is well-defined i.e., he se on he righ-hand side is non-empy wih probabiliy 1, and necessarily V τ = gs τ,τ In addiion, he sopped process J τ is a maringale, so ha he process H is consan on he inerval [,τ ]. This means also ha A = on he random inerval [,τ ], so ha no consumpion is presen before ime τ. We find i convenien o inroduce he following definiion. Definiion An admissible rading and consumpion sraegy φ, A is said o be a perfec hedging agains he American coningen claim X a wih reward funcion g if, wih probabiliy 1, V φ gs,, [,T] We wrie ΦX a o denoe he class of all perfec hedging sraegies agains he American coningen claim X a. From he majorizing propery of he Snell envelope, we infer ha he rading and consumpion sraegy φ, A inroduced in he proof of Proposiion is a perfec hedging agains he American claim wih reward funcion g. Moreover, his sraegy has he special propery of minimal iniial endowmen amongs all admissible perfec hedging sraegies agains he American claim. We shall now explicily deermine π X a by assuming ha rading in he American claim X a would no desroy he arbirage-free feaures of he Black-Scholes model. Theorem There is absence of arbirage in he sense of Definiion in he marke model wih rading in an American claim if and only if he price π X a is given by he formula π X a = sup τ T [,T ] E P e rτ gs τ,τ More generally, he arbirage price a ime of an American claim wih reward funcion g equals π X a = ess sup τ T [,T ] E P e rτ gs τ,τ F. Proof. We shall follow Myneni Le us assume ha he marke price of he opion is U > V. We shall show ha in his case, he American claim is overpriced ha is, a shor arbirage is possible. Le φ, A be he rading and consumpion sraegy considered in he proof of Proposiion see formula 5.1. Suppose ha he opion s buyer selecs an arbirary sopping ime τ T [,T ] as his exercise policy. Le us consider he following sraegy ˆφ, Â, 1,τ observe ha in implemening his sraegy, we do no need o assume ha he exercise ime τ is known in advance ˆφ 1 = φ 1 I [,τ], ˆφ 2 = φ 2 I [,τ] + φ 2 τ + φ 1 τ S τ Bτ 1 gs τ,τbτ 1 I τ,t], and  = A τ. Since φ, A is assumed o be a perfec hedging, we have ˆφ 1 τ S τ + ˆφ 2 τ B τ gs τ,τ, so ha ˆφ 2 T B T, P -a.s. On he oher hand, by consrucion, he iniial wealh of ˆφ, Â, 1,τ

94 5.2. AMERICAN CALL AND PUT OPTIONS 93 saisfies ˆφ 1 S + ˆφ 2 U = V U <. We conclude ha he sraegy ˆφ, Â, 1,τ is a shor arbirage opporuniy ha is, a risk-free profiable sraegy for he seller of he American claim X a. Suppose now ha U <V, so ha he American claim is underpriced. We shall now consruc an arbirage opporuniy for he buyer of his claim. In his case, we may and do assume ha he chooses he sopping ime τ as an exercise ime. In addiion, we assume ha he holds a dynamic porfolio ˆφ, Â. Noice ha he process  vanishes idenically, since τ = τ. This means ha no consumpion is involved in he sraegy chosen by he buyer. Furhermore, he iniial wealh of his porfolio saisfies ˆφ 1 S ˆφ 2 + U = U V < and he erminal wealh is zero, since in view of 5.13 he wealh of he porfolio a he exercise ime τ vanishes. This shows ha by making a clever choice of exercise policy, he buyer of he American claim is able o lock in a risk-free profi. We conclude ha he arbirage price π X a necessarily coincides wih V, since oherwise arbirage opporuniies would exis in he marke model. 5.2 American Call and Pu Opions We shall now focus our aenion on he case of American call and pu opions wih consan srike price K. The reward funcions we shall sudy in wha follows are g c s, =s K + and g p s, = K s +, where he rewards g c and g p correspond o he call and he pu opions, respecively. I will be convenien o inroduce he discouned rewards X =S K + /B and Y =K S + /B. For a coninuous semimaringale Z, and a fixed a R, we denoe by L a Z he righ semimaringale local ime of Z, given explicily by he formula L a Z def = Z a Z a sgn Z u a dz u for every [,T] by convenion we se sgn = 1. I is well known ha he local ime L a Z of a coninuous semimaringale Z is an adaped process whose sample pahs are almos all coninuous, non-decreasing funcions. Moreover, for an arbirary convex funcion f : R R, he following decomposiion, referred o as he Iô-Tanaka-Meyer formula, is valid fz =fz + f Z u dz u + 1 L a Z µda, 2 R where f is he lef-hand side derivaive of f, and µ denoes he second derivaive of f, in he sense of disribuions. An applicaion of he Iô-Tanaka-Meyer formula yields 3 and X = X + Y = Y I {Su>K} Bu 1 σsu dwu + rk du I {Su<K} Bu 1 σsu dwu + rk du Bu 1 dl K u S Bu 1 dl K u S. Since he local ime is known o be an increasing process, i is eviden ha if he srike price is a posiive consan and he ineres rae is non-negaive, hen he discouned reward process X follows a submaringale under he maringale measure P, ha is E P X F u X u, u T. On he oher hand, he discouned reward Y of he American pu opion wih consan exercise price is a submaringale under P, provided ha r. Summarizing, we have he following useful resul. 3 Obviously we have L S K =LK S and L S K =L K S. I is also possible o show ha L K S = L K S.

95 94 CHAPTER 5. AMERICAN OPTIONS Corollary The discouned reward X Y, respecively of he American call opion pu opion, respecively wih consan srike price follows a submaringale under P if r if r, respecively. Le us now examine he raional exercise policy of he holder of an opion of American syle. We shall use hroughou he superscrips c and p o denoe he quaniies associaed wih he reward funcions g c and g p, respecively. In paricular, e rτ S τ K + F V c = ess sup τ T [,T ] E P = e r ess sup τ T [,T ] E P X τ F, and V p = ess sup τ T [,T ] E P e rτ K S τ + F = e r ess sup τ T [,T ] E P Y τ F. We know already ha in he case of a consan srike price and under a non-negaive ineres rae r, he discouned reward X of a call opion follows a P -submaringale. Consequenly, he Snell envelope J c of X equals J c = ess sup τ T [,T ] E P X τ F = E P X T F This in urn implies ha for every dae, he raional exercise ime afer ime of he American call opion wih a consan srike price K is he opion s expiry dae T he same propery holds for he American pu opion, provided ha r. In oher words, under a non-negaive ineres rae, an American call opion wih consan srike price should never be exercised before is expiry dae, and hus is arbirage price coincides wih he Black-Scholes price of a European call. In oher words, in usual circumsances, he American call opion wrien on a non-dividend-paying sock is always worh more alive han dead, hence i is formally equivalen o he European call opion wih he same conracual feaures. This shows ha in he case of consan srike price and non-negaive ineres rae r, only an American pu opion wrien on a non-dividend-paying sock requires furher examinaion. 5.3 Early Exercise Represenaion of an American Pu The following resul is a consequence of a much more general heorem esablished in El Karoui and Karazas For he sake of concreeness, we shall focus on he case of he pu opion. We refer he reader o Myneni 1992 for he proof. Proposiion The Snell envelope J p admis he following decomposiion J p = E P e T rt K S T + F + E P e ru I {τu=u} rk du F. Recall ha V p = e r J p for every. Consequenly, he price P a of an American pu opion saisfies P a = V p = E P e rt K S T + F T + E P e ru I {τu=u} rk du F. Furhermore, in view of he Markov propery of he sock price process S, i is clear ha for any sopping ime τ T [,T ], we have E P e rτ K S τ + F = E P e rτ K S τ + S.

96 5.3. EARLY EXERCISE REPRESENTATION OF AN AMERICAN PUT 95 We conclude ha he price of an American pu equals P a = P a S,T for a cerain funcion P a : R + [,T] R. To be a bi more explici, we define he funcion P a s, by seing P a s, T = sup τ T [,T ] E P e rτ K S τ + S = s, 5.17 where he expecaion on he righ-hand side is condiional on he even {S = s}. I is possible o show ha he funcion P a s, u is decreasing and convex in s, and increasing in u. Le us denoe by C and D he coninuaion region and sopping region, respecively. The sopping region D is defined as ha subse of R + [,T] for which he sopping ime τ saisfies τ = inf {u [, T ] S u,u D} for every [,T]. The coninuaion region C is, of course, he complemen of D in R + [,T]. Noe ha in erms of he funcion P a, we have and D = {s, R + [,T] P a s, T =K s + } C = {s, R + [,T] P a s, T > K s + }. Le us define he funcion b :[,T] R + by seing b T = sup {s R + P a s, T =K s + }. I can be shown ha he graph of b is conained in he sopping region D. This means ha i is only raional o exercise he pu opion a ime if he curren sock price S is a or below he level b T. For his reason, he value b T is commonly referred o as he criical sock price a ime. I is someimes convenien o consider he funcion c :[,T] R + which is given by he equaliy c =b T. For any [,T], he opimal exercise ime τ afer ime saisfies or equivalenly τ = inf {u [, T ] K S u + = P a S u,t u }, τ = inf {u [, T ] S u b T u } = inf {u [, T ] S u c u }. We quoe he following lemma from van Moerbeke Lemma If he srike price is consan, hen c is non-decreasing, infiniely smooh over,t and lim T c = lim T b T =K. By virue of he nex resul, he price of an American pu opion may be represened as he sum of he arbirage price of he corresponding European call opion and he so-called early exercise premium. Corollary The following decomposiion of he price of an American pu opion is valid P a T = P + E P e ru I {Su<b T u}rk du F, where P a = P a S,T is he price of he American pu opion, and P = P S,T is he Black-Scholes price of a European pu wih srike K. Decomposiion of he price provided by Corollary is commonly referred o as he early exercise premium represenaion of an American pu. I was derived independenly, by differen

97 96 CHAPTER 5. AMERICAN OPTIONS means and a various levels of sricness, by Kim 199, Jacka 1991, and Jamshidian 1992 see also Carr e al for oher represenaions of he price of an American pu. For = we have T P a = P + E P e ru I {Su<b T u} rk du, 5.18 where P is he price a ime of he European pu. Observe ha if r =, hen he early exercise premium vanishes, which means ha he American pu is equivalen o he European pu and hus should no be exercised before he expiry dae. Taking ino accoun he dynamics of he sock price under P, we can make represenaion 5.18 more explici, namely T P a S,T=P S,T+rK e ru N lnb T u/s ρu σ u du, where ρ = r σ 2 /2, and N denoes he sandard Gaussian cumulaive disribuion funcion. A similar decomposiion is valid for any insan [,T], provided ha he curren sock price S belongs o he coninuaion region C; ha is, he opion should no be exercised immediaely. We have P a S,T = P S,T T lnb + rk e ru T u/s ρu N σ du, u where P S,T sands for he price of a European pu opion of mauriy T, and S is he curren level of he sock price. A change of variables leads o he following equivalen expression lnb P a s, =P s, +rk e ru u/s ρu N σ du, u which is valid for every s, C. If S = b T, hen we have necessarily P a S,T = K b T, so ha clearly P a b,=k b for every [,T]. This simple observaion leads o he following inegral equaion, which is saisfied by he opimal boundary funcion b lnb K b =P b,+rk e ru u/b ρu N σ du. u Unforunaely, a soluion o his inegral equaion is no explicily known, and hus i needs o be solved numerically. On he oher hand, he following bounds for he price P a s, are easy o derive lnk/s ρu P a s, P s, rk e ru N σ du u and lnb P a s, P s, rk e ru N /s ρu σ du, u where b sands for he opimal exercise boundary of a perpeual pu ha is, an American pu opion wih expiry dae T =. To his end, i is enough o show ha for any mauriy T, he values of he opimal sopping boundary b lie beween he srike price K and he level b, i.e., K b b, [,T]. I should be sressed ha he value of b is known o be see McKean 1965, van Moerbeke 1976 or he nex secion b = 2rK 2r + σ

98 5.4. FREE BOUNDARY PROBLEM Free Boundary Problem The free boundary problem relaed o he opimal sopping problem for an American pu opion was firs examined by McKean 1965 and van Moerbeke For a fixed expiry dae T and consan srike price K, we denoe by P a S,T he price of an American pu a ime [,T]; in paricular, P a S T, =K S T +. I will be convenien o denoe by L he following differenial operaor Lv = 1 2 σ2 s 2 2 v v + rs rv = Av rv, 5.2 s2 s where A sands for he infiniesimal generaor of he one-dimensional diffusion process S considered under he maringale measure P. Also, le L sand for he following differenial operaor L v = v + Lv = v + Av rv. The proof of he following proposiion, which focuses on he properies of he price funcion P a s, of an American pu, can be found in van Moerbeke 1976 see also McKean 1965 and Jacka Recall ha we denoe c =b T. Proposiion The American pu value funcion P a s, is smooh on he coninuaion region C wih 1 P a s s, for all s, C. The opimal sopping boundary b is a coninuous and non-increasing funcion on,t], hence c is a non-decreasing funcion on [,T, and lim T b T = lim c =K. T On C, he funcion P a saisfies P a s, =LP a s, ; ha is, P a s, = 1 2 σ2 s 2 P a sss, +rs P a s s, rp a s,, s, C. Furhermore, we have lim s, s b = K b,,t], lim P a s, = K s +, s R +, lim s, s =,,T], P a s, K s +, s, R +,T]. In order o deermine he opimal sopping boundary, one needs o impose an addiional condiion, known as he smooh fi principle, which reads as follows for he proof of he nex resul, we refer he reader o van Moerbeke Proposiion The parial derivaive Ps a s, is coninuous a.e. across he sopping boundary c ; ha is lim P s c s a s, = for almos every [,T]. We are now in a posiion o sae wihou proof he resul, which characerizes he price of an American pu opion as he soluion o he free boundary problem see van Moerbeke 1976, Jacka Theorem Le G be an open domain in R + [,T wih coninuously differeniable boundary c. Assume ha v : R + [,T] is a coninuous funcion such ha u C 3,1 G, he funcion gs, = ve s, has Tychonov growh see Myneni 1992 and v saisfies L v =on G; ha is v s, σ2 s 2 v ss s, +rs v s s, rvs, = 5.22

99 98 CHAPTER 5. AMERICAN OPTIONS for every s, G, and vs, > K s +, s, G, vs, = K s +, s, G c, vs, T = K s +, s R +, lim ss, s c = 1, [,T. Then he funcion P a s, =vs, T for every s, R + [,T] is he value funcion of he American pu opion wih srike price K and mauriy T. Moreover, he se C = G is he opion s coninuaion region, and he funcion b =ct, [,T], represens he criical sock price. We shall now apply he above heorem o a perpeual pu ha is, an American pu opion which has no expiry dae i.e., wih mauriy T =. Since he ime o expiry of a perpeual pu is always infinie, he criical sock price becomes a real number b K, and he PDE 5.22 becomes he following ordinary differenial equaion ODE for he funcion v s =vs, 1 2 σ2 s 2 d2 v ds 2 s+rs dv ds s rv s =, s b,, 5.23 wih v s =K s for every s [,b ]. Our aim is o show ha equaliy 5.19 is valid. For his purpose, observe ha ODE 5.23 admis a general soluion of he form v s =c 1 s d1 + c 2 s d2, s b,, 5.24 where c 1,c 2,d 1 and d 2 are consans. Using he boundary condiion, which reads v b =K b, and he smooh fi condiion dv lim s = 1, s b ds we find ha b v s =K b s and hus, in paricular, equaliy 5.19 is valid. 2rσ 2, s b,, Approximaions of he American Pu Price Since no closed-form expression for he value of an American pu is available, in order o value American opions one needs o use a numerical procedure. I appears ha he use of he CRR binomial ree, alhough remarkably simple, is far from being he mos efficien way of pricing American opions. Various approximaions of he American pu price on a non-dividend-paying sock were examined in Brennan and Schwarz 1977, Johnson 1983, MacMillan 1986, and Broadie and Deemple Le us commen briefly on he approximae valuaion mehod proposed by Geske and Johnson Basically, he Geske-Johnson approximaion relies on he discreizaion of he ime parameer and he applicaion of backward inducion, as in any oher sandard discree-ime approach. However, in conras o he space-ime discreizaion used in he mulinomial rees approach or in he finie difference mehods, he approach of Geske and Johnson makes use of he exac disribuion of he vecor of sock prices S 1,...,S n, where 1 <... < n = T are he only admissible deerminisic exercise imes. 4 In oher words, he decision o exercise an opion can be made a any of he daes 1,..., n only. Le us sar by considering he special case when n = 2 and 1 = T/2, 2 = T. Noe ha if n =1, he opion can be exercised a 1 = T only, so ha i is 4 An opion which may be exercised early,bu only on predeermined daes,is commonly referred o as Bermudan opion.

100 5.5. APPROXIMATIONS OF THE AMERICAN PUT PRICE 99 equivalen o a European pu. To find an approximae value for an American pu, we shall argue by backward inducion. Suppose ha he opion was no exercised a ime 1. Then he value of he opion a ime 1 is equal o he value of a European pu opion wih mauriy 2 1 = T/2, given he iniial sock price S 1 = S T/2. The price of a European pu is given, of course, by he sandard Black-Scholes formula, denoed by P S T/2,T/2. The criical sock price b 1 a ime 1 = T/2 solves he equaion K S T/2 = P S T/2,T/2, hence i can be found by numerical mehods. Moreover, i is clear ha he value V T/2 of he opion a ime T/2 saisfies V T/2 = { P ST/2,T/2 if S T/2 >b 1, K S T/2 if S T/2 b 1. Noe ha i is opimal o exercise he opion a ime 1 if and only if S T/2 b 1. To find he value of he opion a ime, we need firs o evaluae he expecaion V S =E P e rt/2 V T/2, or equivalenly V S =e rt/2 E P e rt/2 S T K + I {ST/2 >b 1 } +K S T/2 I {ST/2 b 1 }. Noice ha he laer expecaion can be expressed in erms of he probabiliy law of he wodimensional random variable S T/2,S T ; equivalenly, one exploi he join law of W T/2 and W T. A specific, quasi-explici represenaion of V S in erms of wo-dimensional Gaussian cumulaive disribuion funcion is in fac a maer of convenience. The approximae value of an American pu wih wo admissible exercise imes, T/2 and T, equals P2 a S,T=V S. The same ieraive procedure may be applied o an arbirary finie sequence of imes 1 <... < n = T. In his case, he Geske-Johnson approximaion formula involves inegraion wih respec o a n-dimensional Gaussian probabiliy densiy funcion. I appears ha for hree admissible exercise imes, T/3, 2T/3 and T, he approximae quasi-analyical valuaion formula provided by he Geske and Johnson mehod is roughly as accurae as he binomial ree wih 15ime seps. For any naural n, le us denoe by Pn a S,T he Geske-Johnson opion s approximae value associaed wih admissible daes i = T i/n, i =1,...,n. I is possible o show ha he sequence Pn a S,T converges o he opion s exac price P a S,T when he number of seps ends o infiniy, so ha he sep lengh ends o zero. To esimae he limi P a S,T, one can make use of any exrapolaion echnique, for insance Richardson s approximaion scheme. Le us briefly describe he laer echnique. Suppose ha he funcion F saisfies F h =F + c 1 h + c 2 h 2 + oh 2 in he neighborhood of zero, so ha F kh =F + c 1 kh + c 2 k 2 h 2 + oh 2 and F lh =F + c 1 lh + c 2 l 2 h 2 + oh 2 for arbirary l>k>1. Ignoring he erm oh 2, and solving he above sysem of equaions for F, we obain denoes approximae equaliy F F h+ a b F h F kh + F kh F lh, 5.26 c c where a = ll 1 kk 1, b= kk 1 and c = l 2 k 1 lk kk 1. Le us wrie Pn a o denoe Pn a S,T for n =1, 2, 3 in paricular, P1 a S,T is he European pu price P S,T. For n = 3 upon seing k =3/2, l= 3 and P1 a = F lh, P2 a = F kh, P3 a = F h, we ge he following approximae formula P a S,T P3 a P 3 a P2 a 1 2 P 2 a P1 a. Bunch and Johnson 1992 argue ha he Geske-Johnson mehod can be furher improved if he exercise imes are chosen ieraively in such a way ha he opion s approximae value is maximized.

101 1 CHAPTER 5. AMERICAN OPTIONS 5.6 Opion on a Dividend-paying Sock Since mos raded opions on socks are unproeced American call opions wrien on dividendpaying socks, i is worhwhile o commen briefly on he valuaion of hese conracs. A call opion is said o be unproeced if i has no conraced proecion agains he sock price decline ha occurs when a dividend is paid. I is inuiively clear ha an unproeced American call wrien on a dividend-paying sock is no equivalen o he corresponding opion of European syle, in general. Suppose ha a known dividend, D, will be paid o each shareholder wih cerainy a a prespecified dae T D during he opion s lifeime. Furhermore, assume ha he ex-dividend sock price decline equals δd for a given consan δ [, 1]. Le us denoe by S TD and P TD = S TD δd respecively he cum-dividend and ex-dividend sock prices a ime T D. I is clear ha he opion should evenually be exercised jus before he dividend is paid ha is, an insan before T D. Consequenly, as firs noed by Black 1975, he lower bound for he price of such an opion is he price of he European call opion wih expiry dae T D and srike price K. This lower bound is a good esimae of he exac value of he price of he American opion whenever he probabiliy of early exercise is large ha is, when he probabiliy P{C TD <S TD K} is large, where C TD = CP TD,T T D,Kis he Black-Scholes price of he European call opion wih mauriy T T D and exercise price K. Hence, early exercise of he American call is more likely he larger he dividend, he higher he sock price S TD relaive o he srike price K, and he shorer he ime-period T T D beween expiry and dividend paymen daes. An analyic valuaion formula for unproeced American call opions on socks wih known dividends was esablished by Roll However, i seems o us ha Roll s original reasoning, which refers o opions ha expire an insan before he ex-dividend dae, assumes implicily ha he holder of an opion may exercise i before he ex-dividend dae, bu apparenly is no allowed o sell i before he ex-dividend dae. To avoid his discrepancy, we prefer insead o consider European opions which expire on he ex-dividend dae i.e., afer he ex-dividend sock price decline. Before formulaing he nex resul, we need o inroduce some noaion. Le us denoe by b he cum-dividend sock price level above which he original American opion will be exercised a ime T D, so ha Cb δd, T T D,K=b K I is worhwhile o observe ha Cs δd, T T D,K <s K when s b,, and Cs δd, T T D,K >s K for every s,b. Noe ha he firs wo erms on he righ-hand side of equaliy 5.28 below represen he values of European opions, wrien on a sock S, which expire a ime T and on he ex-dividend dae T D, respecively. The las erm, CO T D,b K, represens he price of a so-called compound opion see Sec To be more specific, we deal here wih a European call opion wih srike price b K which expires on he ex-dividend dae T D, and whose underlying asse is he European call opion, wrien on S, wih mauriy T and srike price K. The compound opion will be exercised by is holder a he ex-dividend dae T D if and only if he is prepared o pay b K for he underlying European opion. Since he value of he underlying opion afer he ex-dividend sock price decline equals CP TD,T T D,K, he compound opion is exercised whenever CP TD,T T D,K=CS TD δd, T T D,K >b K, ha is, when he cum-dividend sock price exceeds b his follows from he fac ha he price of a sandard European call opion is an increasing funcion of he sock price, combined wih equaliy Proposiion The arbirage price C a T,K of an unproeced American call opion wih expiry dae T > T D and srike price K, wrien on a sock which pays a known dividend D a ime T D, equals C a T,K= C T,K+C T D,b CO T D,b K 5.28 for [,T D ], where b is he soluion o 5.27.

102 Chaper 6 Exoic Opions In he preceding chapers, we have focused on he wo sandard classes of opions ha is, call and pu opions of European and American syle. The aim of his chaper is o sudy examples of more sophisicaed opion conracs. For convenience, we give he generic name exoic opion o any opion conrac which is no a sandard European or American opion. I should be made clear ha we shall resric our aenion o he case of exoic spo opions. We find i convenien o classify he large family of exoic opions as follows: a packages opions ha are equivalen o a porfolio of sandard European opions, cash and he underlying asse sock, say; b forward-sar opions opions ha are paid for in he presen bu received by holders a a prespecified fuure dae; c chooser opions opion conracs ha are chosen by heir holders o be call or pu a a prescribed fuure dae; d compound opions opion conracs wih oher opions playing he role of he underlying asses; e binary opions conracs whose payoff is defined by means of some binary funcion; f barrier opions opions whose payoff depends on wheher he underlying asse price reaches some barrier during he opion s lifeime; g Asian opions opions whose payoff depends on he average price of he underlying asse during a prespecified period; h baske opions opions wih a payoff depending on he average of prices of several asses; i lookback opions opions whose payoff depends, in paricular, on he minimum or maximum price of he underlying asse during opions lifeimes; 6.1 Packages An arbirary financial conrac whose erminal payoff is a piecewise linear funcion of he erminal price of he underlying asse may be seen as a package opion ha is, a combinaion of sandard opions, cash and he underlying asse. Unless explicily saed oherwise, we shall place ourselves wihin he classic Black-Scholes framework Collars Le K 2 >K 1 > be fixed real numbers. The payoff a expiry dae T from he long posiion in a collar opion equals def CL T = min { } max {S T,K 1 },K 2. I is easily seen ha he payoff CL T can be represened as follows CL T = K 1 +S T K 1 + S T K 2 +, 11

103 12 CHAPTER 6. EXOTIC OPTIONS so ha a collar opion can be seen as a porfolio of cash and wo sandard call opions. This implies ha he arbirage price of a collar opion a any dae before expiry equals CL = K 1 e rt + CS,T, K 1 CS,T, K 2, where Cs, T, K =Cs, T, K, r, σ sands for he Black-Scholes call opion price a ime, where he curren level of he sock price is s, and he exercise price of he opion equals K see formula Break Forwards By a break forward we mean a modificaion of a ypical forward conrac, in which he poenial loss from he long posiion is limied by some prespecified number. More explicily, he payoff from he long break forward is defined by he equaliy BF T def = max {S T,F} K, where F = F S,T=S e rt is he forward price of a sock for selemen a ime T, and K>F is some consan. The delivery price K is se in such a manner ha he break forward conrac is worhless when i is enered ino. Since i is clear ha for every [,T], BF T =S T F + + F K, BF = CS,T, F +F Ke rt. In paricular, he righ level of K, K say, is given by he expression K = e rt S + CS,T,S e rt. Using he Black-Scholes valuaion formula, we end up wih he following equaliy where d 1 and d 2 are given by Range Forwards K = e rt S 1+N d 1 S,T N d 2 S,T, A range forward may be seen as a special case of a collar one wih zero iniial cos. Is payoff a expiry is RF T def = max { min {S T,K 2 },K 1 } F = max { min {ST F, K 2 F },K 1 F }, where K 1 <F <K 2, and as before F = F S,T=S e rt. I appears convenien o decompose he payoff of a range forward in he following way RF T = S T F +K 1 S T + S T K 2 +. Indeed, he above represenaion of he payoff implies direcly ha a range forward may be seen as a porfolio composed of a long forward conrac, a long pu opion wih srike price K 1, and finally a shor call opion wih srike price K 2. Furhermore, is price a equals RF = S S e r + P S,T, K 1 CS,T, K 2. As menioned earlier, he levels K 1 and K 2 should be chosen in such a way ha he iniial value of a range forward equals.

104 6.2. FORWARD-START OPTIONS Forward-sar Opions Le us consider wo daes, say T and T, wih T <T. A forward-sar opion is a conrac in which he holder receives, a ime T a no addiional cos, an opion wih expiry dae T and exercise price K equal o S T. On he oher hand, he holder mus pay a ime an up-fron fee, he price of a forward-sar opion. Le us consider he case of a forward-sar call opion, wih erminal payoff FS T def = S T S T +. To find he price a ime [,T ] of such an opion, i suffices o consider is value a he delivery dae T, ha is FS T = CS T,T T,S T. Since we resric our aenion o he classic Black-Scholes model, i is easily seen ha and hus he opion s value a ime equals CS T,T T,S T =S T C1,T T, 1, FS = S C1,T T, 1 = CS,T T,S. If a sock coninuously pays dividends a a consan rae κ, he above equaliy should be modified as follows FS κ = e κt C κ S,T T,S, where C κ sands for he call opion price derived in Proposiion Similar formulae can be derived for he case of a forward-sar pu opion. 6.3 Chooser Opions A chooser opion is an agreemen in which one pary has he righ o choose a some fuure dae T wheher he opion is o be a call or pu opion wih a common exercise price K and remaining ime o expiry T T. Therefore, he payoff a T of a chooser opion is CH T def = max { CS T,T T,K,PS T,T T,K }, while is erminal payoff is given by he expression CH T =S T K + I A +K S T + I A c, where A sands for he following even, which belongs o he σ-field F T A = { ω Ω CS T,T T,K >PS T,T T,K } and A c is he complemen of A in Ω. The call-pu pariy implies ha and hus P S T,T T,K=CS T,T T,K S T + Ke rt T, CH T = max { CS T,T T,K,CS T,T T,K S T + Ke rt T}, or finally CH T = CS T,T T,K+Ke rt T S T +. The las equaliy implies immediaely ha he sandard chooser opion is equivalen o he porfolio composed of a long call opion and a long pu opion wih differen exercise prices and differen expiry daes, so ha is arbirage price equals CH = CS,T, K+P S,T, Ke rt T

105 14 CHAPTER 6. EXOTIC OPTIONS for every [,T ]. In paricular, using he Black-Scholes formula, we ge for = CH = S Nd1 N d 1 + Ke rt N d 2 Nd 2, where and d 1,2 = lns /K+r ± 1 2 σ2 T σ T d 1,2 = lns /K+rT ± 1 2 σ2 T σ T. 6.4 Compound Opions A compound opion see Geske 1977, 1979a is a sandard opion wih anoher sandard opion being he underlying asse. One can disinguish four basic ypes of compound opions: call on a call, pu on a call, call on a pu, and, finally, pu on a pu. Le us consider, for insance, he case of a call on a call compound opion. For wo fuure daes T and T, wih T <T, and wo exercise prices K and K, consider a call opion wih exercise price K and expiry dae T on a call opion wih srike price K and mauriy T. I is clear ha he payoff of he compound opion a ime T is def = +, CS T,τ,K K CO T where CS T,τ,K sands for he value a ime T of a sandard call opion wih srike price K and expiry dae T = T + τ. In he Black-Scholes framework, we obain he following equaliy Moreover, since under P we have Cs, τ, K =sn d 1 s, τ, K Ke rτ N d 2 s, τ, K. S T = S exp σ T ξ +r 1 2 σ2 T, where ξ has a sandard Gaussian probabiliy law under P, he price of he compound opion a ime equals CO = e rt gxn ˆd1 Ke rτ N ˆd 2 K nx dx, where ˆd i = d i gx,τ,k for i =1, 2, he funcion g : R R equals gx =S exp σ T x +r 1 2 σ2 T x and, finally, he consan x = inf {x R Cgx,τ,K K }. Sraighforward calculaions yield and d 1 gx,τ,k= lns /K+σ T x + rt σ 2 T σ2 T σ T T d 2 gx,τ,k= lns /K+σ T x + rt 1 2 σ2 T σ T T. 6.5 Digial Opions By a digial or binary opion we mean a conrac whose payoff depends in a disconinuous way on he erminal price of he underlying asse. The simples examples of binary opions are cash-ornohing opions and asse-or-nohing opions. The payoffs a expiry of a cash-or-nohing call and pu opions are def def BCC T = X I {ST >K}, BCP T = X I {ST <K},

106 6.6. BARRIER OPTIONS 15 where in boh cases X sands for a prespecified amoun of cash. Similarly, for he asse-or-nohing opion we have def def BAC T = S T I {ST >K}, BAP T = S T I {ST <K} for a call and pu respecively. All opions inroduced above may be easily priced by means of he risk-neural valuaion formula. Somewha more complex binary opions are he so-called gap opions, whose payoff a expiry equals GC T def = S T XI {ST >K} = BAC T BCC T for he call opion, and GP T def = X S T I {ST <K} = BCP T BAP T for he corresponding pu opion. Once again, pricing hese opions involves no difficulies. As a las example of a binary opion, le us menion a supershare, whose payoff is SS T def = S T K 1 I {K1<S T <K 2} for some posiive consans K 1 <K 2. The price of such an opion a ime is easily seen o equal SS = S N h 1 S,T N h 2 S,T, K 1 where h i s, = lns /K i +r σ2 σ. 6.6 Barrier Opions The generic erm barrier opions refers o he class of opions whose payoff depends on wheher or no he underlying prices hi a prespecified barrier during he opions lifeimes. For closed-form expressions for prices of various barrier opions such as knock-in and knock-ou opions we refer o Rubinsein and Reiner 1991, Kuniomo and Ikeda 1992, and Carr 1995 see also Cheuk and Vors 1996 for a numerical approach. Le us also menion he paper by Heynen and Ka 1994, which is devoed o so-called parial barrier opions ha is, barrier opions in which he underlying price is moniored for barrier his only during a prespecified period during an opion s lifeime. To give he flavor of he mahemaical echniques used when dealing wih barrier opions, we shall examine here a specific kind of currency barrier opion, namely he down-and-ou call opion. The payoff a expiry of a down-and-ou call opion equals in unis of domesic currency C 1 T def = Q T K + I {min T Q H}, where K and H are consans. I follows from he formula above ha he down-and-ou opion becomes worhless or is knocked ou if, a any ime prior o he expiry dae T, he curren exchange rae Q falls below a predeermined level H. I is hus eviden ha a down-and-ou opion is less valuable han a sandard currency opion. Our aim is o find an explici formula for he so-called knock-ou discoun. Ou-of-he-money knock-ou opion. Suppose firs ha he inequaliies H<Kand H<Q are saisfied. From he general feaures of a down-and-ou call, i is clear ha he opion is knocked ou when i is ou-of-he-money. Recall ha under he domesic maringale measure P we have cf. 4.8 Q = Q e σ QW +λ = Q e X,

107 16 CHAPTER 6. EXOTIC OPTIONS where X = σ Q W + λ for [,T], and λ = r d r f 1 2 σ2 Q. Therefore, where m T = min T X, and hus where D sands for he se {ω Ω min T Q H} = {ω Ω m T lnh/q }, C 1 T =Q T KI {QT K, min T Q H} = Q e X T I D KI D, D = {ω Ω X T lnk/q,m T lnh/q }. We conclude ha he price a ime of a down-and-ou call opion admis he following represenaion C 1 = e r dt Q E P e X T I D e r d T KP {D}, where P is he maringale measure of he domesic marke. In order o direcly calculae C 1 by means of inegraion, we need o find firs he join probabiliy disribuion of random variables X T and m T. One can show ha for all x, y such ha y and y x, we have x + λt P { X T x, m T y} = N σ T e 2λyσ 2 N x +2y + λt where, for he sake of noaional convenience, we wrie σ in place of σ Q. Consequenly, he probabiliy densiy funcion of X T,m T equals fx, y = 22y x σ 3 T 3/2 e 2λyσ 2 x +2y + λt n σ T for y, y x, where n sands for he sandard Gaussian densiy funcion. From he above i follows, in paricular, ha P lnq /K+λT {D} = N σ 2λσ 2 lnh 2 /Q K+λT H/Q N T σ. T To find he expecaion def I 1 = E P e X T I D = E P e X T I {XT lnk/q,m T lnh/q }, we need o evaluae he double inegral A e x fx, ydxdy, where A = {x, y; x lnk/q,y lnh/q,y, y x}. Sraighforward bu raher cumbersome inegraion leads o he following resul σ T I 1 = e r d r f T N h 1 Q,T H/Q 2λσ 2 +2 N c 1 Q,T,, where and h 1,2 q, = lnq/k+r d r f ± 1 2 σ2 σ c 1,2 q, = lnh2 /qk+r d r f ± 1 2 σ2 σ.

108 6.7. ASIAN OPTIONS 17 By collecing and rearranging he formulae above, we conclude ha he price a iniiaion of he knock-ou opion admis he following represenaion recall ha we wrie σ = σ Q where cf. Proposiion C 1 = C Q J = Sandard Call Price Knockou Discoun, 6.1 C Q = Q e r f T Nh 1 Ke r dt Nh 2 and J = Q e r f T H/Q 2λσ 2 +2 Nc 1 Ke rdt H/Q 2λσ 2 Nc 2, where h 1,2 = h 1,2 Q,T and c 1,2 = c 1,2 Q,T. Noice ha he proof of his formula can be subsanially simplified by an applicaion of Girsanov s heorem. We define an auxiliary probabiliy measure P by seing d P dp = exp σwt 1 2 σ2 T = η T, P -a.s. I follows from he Girsanov heorem ha he process W = W σ follows a sandard Brownian moion under he probabiliy measure P. Moreover, aking ino accoun he definiion of X, we find ha E P e X T I D = e r d r f T E P ηt I D and hus I 1 = e r d r f T P{D} = e r d r f T P{ XT lnk/q,m T lnh/q }. Finally, he semimaringale decomposiion of he process X under P is X = σ W +r d r f σ2, [,T], hence for every y, y x, we have P{D} = N h 1 Q,T H/Q 2λσ 2 +2 N c 1 Q,T. Represenaion 6.1 of he opion s price now follows easily. 6.7 Asian Opions An Asian opion or an average opion is a generic name for he class of opions of European or American syle whose erminal payoff is based on average asse values during some period wihin he opions lifeimes. Due o heir averaging feaure, Asian opions are paricularly suiable for hinly raded asses or commodiies. Acually, in conras o sandard opions, Asian opions are more robus wih respec o manipulaions near heir expiry daes. Typically, hey are also less expensive han sandard opions. Le T be he exercise dae, and le T <T sand for he beginning dae of he averaging period. Then he payoff a expiry of an Asian call opion equals C A T def = A S T,T K +, 6.2 where 1 T A S T,T= S u du 6.3 T T T is he arihmeic average of he asse price over he ime inerval [T,T], Kis he fixed srike price, and he price S of he sock is assumed o follow a geomeric Brownian moion. The main difficuly in pricing and hedging Asian opions is due o he fac ha he random variable A S T,Tdoesno have a lognormal disribuion. This feaure makes he ask of finding an explici formula for he price of an Asian opion surprisingly involved. For his reason, early sudies of Asian opions were based

109 18 CHAPTER 6. EXOTIC OPTIONS eiher on approximaions or on he direc applicaion of he Mone Carlo mehod. The numerical approach o he valuaion of Asian opions, proposed independenly in Ruiens 199 and Vors 1992, is based on he approximaion of he arihmeic average using he geomeric average. Noe firs ha i is naural o subsiue he coninuous-ime average A S T,T wih is discree-ime counerpar A n ST,T= 1 n 1 S Ti, 6.4 n i= where T i = T + it T /n. Furhermore, he arihmeic average A n S T,T can be replaced wih he geomeric average, denoed by G n S T,T in wha follows. Recall ha he random variables S Ti,i=1,...,n, are given explicily by he expression S Ti = S T exp σw T i W T +r 1 2 σ2 T i T, where W is a sandard Brownian moion under he maringale measure P. Therefore, he geomeric average admis he following represenaion G n ST,T= n 1 i= 1/n σ S Ti = cst exp n n 1 i= n i 1WT i+1 WT i for a sricly posiive consan c. In view of he independence of incremens of he Brownian moion, he las formula makes clear ha he geomeric average G n S T,T has a lognormal disribuion under P. The approximae Black-Scholes-like formula for he price of an Asian call opion can hus be easily found by he direc evaluaion of he condiional expecaion C T n = e rt T E P G n S T,T K + FT. I appears, however, ha such an approach significanly underprices Asian call opions. To overcome his deficiency, one may direcly approximae he rue disribuion of he arihmeic average using an approximae disribuion, ypically a lognormal law wih he appropriae parameers 1 see Levy 1992, Turnbull and Wakeman 1991, and Bouaziz e al Anoher approach, iniiaed by Carverhill and Clewlow 199, relies on he use of he fas Fourier ransform o calculae he densiy of he sum of random variables as he convoluion of individual densiies. The second sep in his mehod involves numerical inegraion of he opion s payoff funcion wih respec o his densiy funcion. Kemna and Vors 199 apply he Mone Carlo simulaion wih variance reducion o price Asian opions. They replace A S T,T wih he arihmeic average 6.4, so ha he approximae value of an Asian call opion is given by he formula { n } C T n = e rt T E P S Ti K FT. n Since he random variables S Ti,i=1,...,n, are given by an explici formula, hey can easily be generaed using any sandard procedure. Le us firs consider he special case of an Asian opion which is already known o be in-hemoney, i.e., assume ha <T belongs o he averaging period, and he pas values of sock price are such ha 1 T 1 A S T,T= S u du > S u du K. 6.5 T T T T T T In his case, he value a ime of he Asian opion equals C A = S 1 e rt rt T i= e rt 1 K T T S u du. 6.6 T 1 I should be noed ha explici formulae for all momens of he arihmeic average are available see Geman and Yor 1993.

110 6.8. BASKET OPTIONS 19 Indeed, under 6.5, he price of he opion saisfies or equivalenly C A C A e rt = T T Furhermore, we have = e rt E P E P E P 1 T T T S u du F T T S u du K F, + e rt 1 S u du K. T T T T S u du F = r 1 S e rt 1, since recall ha S = e r S T T T S e rt 1 = e rt ST e r S = de ru Su= rs u du + e ru dsu. For an Asian opion which is no known a ime o be in-he-money a ime T, an explici valuaion formula is no available see, however, Geman and Yor 1992, 1993 for quasi-explici pricing formulas. 6.8 Baske Opions A baske opion, as suggesed by is name, is a kind of opion conrac which serves o hedge agains he risk exposure of a baske of asses ha is, a prespecified porfolio of asses. Generally speaking, a baske opion is more cos-effecive han a porfolio of single opions, as he laer over-hedges he exposure, and coss more han a baske opion. An inuiive explanaion for his feaure is ha he baske opion akes ino accoun he correlaion beween differen risk facors. For insance, in he case of a srong negaive correlaion beween wo or more underlying asses, he oal risk exposure may almos vanish, and his nice feaure is no refleced in payoffs and prices of single opions. Le us observe ha from he analyical viewpoin, here is a close analogy beween baske opions and Asian opions. Le us denoe by S i,i=1,...,k he price processes of k underlying asses, which will be referred o as socks in wha follows. In his case, i seems naural o refer o such a baske opion as he sock index opion in marke pracice, opions on a baske of currencies are also quie common. The payoff a expiry of a baske call opion is defined in he following way C B T def = k + w i ST i K =AT K +, 6.7 i=1 where w i is he weigh of he i h asse, so ha k i=1 w i =1. Noe ha by A T we denoe here he weighed arihmeic average k A T = w i ST i. i=1 We assume ha each sock price S i follows a geomeric Brownian moion. More explicily, under he maringale measure P we have ds i = S i rd+ˆσi dw 6.8 for some non-zero vecors ˆσ i R k, where W =W 1,...,W k sands for a k-dimensional Brownian moion under P, and r is he risk-free ineres rae. Observe ha for any fixed i, we can find a sandard one-dimensional Brownian moion W i such ha ds i = S i rd+ σi d W i 6.9

111 11 CHAPTER 6. EXOTIC OPTIONS and σ i = ˆσ i, where ˆσ i is he Euclidean norm of ˆσ i. Le us denoe by ρ i,j he insananeous correlaion coefficien ρ i,j = ˆσ i ˆσ j σ i σ j = ˆσ i ˆσ j ˆσ i ˆσ j. We may hus alernaively assume ha he dynamics of price processes S i are given by 6.9, where W i,i=1,...,k are one-dimensional Brownian moions, whose cross-variaions saisfy W i, W j = ρ i,j, i, j =1,...,k. Le us reurn o he problem of valuaion of baske opions. For similar reasons as hose applying o Asian opions, baske opions are raher inracable analyically. Rubinsein 1991 developed a simple echnique of pricing baske opions on a bivariae binomial laice, hus generalizing he sandard Cox-Ross-Rubinsein mehodology. Unforunaely, his numerical mehod is very imeconsuming, especially where here are several underlying asses. To overcome his, Genle 1993 proposed valuaion of a baske using an approximaion of he weighed arihmeic mean in he form of is geomeric counerpar his follows he approach of Ruiens 199 and Vors 1992 o Asian opions. For a fixed T, le us denoe by ŵ i he modified weighs ŵ i = w i S i k j=1 w js j = w i F S i, T k j=1 w jf S j, T, 6.1 where F S i, T is he forward price a ime of he i h asse for he selemen dae T. We may rewrie 6.7 as follows 2 k k CT B = w j F S j, T ŵ i Si T K + k = w j F S j, T ÃT K +, j=1 i=1 where S i T = Si T /F S i, T, Ã T = k i=1 ŵi S i T, and K = j=1 K k j=1 w jf S j, T = e rt K k j=1 w js j The arbirage price a ime of a baske call opion hus equals C B = e rt k j=1 w j F S j, T E P ÃT K + F, or equivalenly k C B = w j S j E P ÃT K + F j=1 The nex sep relies on an approximaion of he weighed arihmeic mean k i=1 ŵi S T i using a similarly weighed geomeric mean. More specifically, we approximae he price C B of he baske opion using ĈB, which is given by he formula for he sake of noaional simpliciy, we pu = in wha follows k Ĉ B = w j S j E P GT ˆK +, 6.13 where G T = k i=1 S i T ŵi, and j=1 ˆK = K + E P G T ÃT This represenaion is inroduced because i appears o give a beer approximaion of he price of a baske opion han formula 6.7.

112 6.8. BASKET OPTIONS 111 In view of 6.8, we have recall ha F S i,t=e rt S i S i T = S i T /F S i,t=eˆσi W T σ2 i T/2, and hus he weighed geomeric average G T equals G T = e c1 W T c2t/2 = e η T c 2T/2, 6.15 wih η T = c 1 WT, where c 1 = k i=1 ŵiˆσ i and c 2 = k i=1 ŵiσi 2. We conclude ha he random variable G T is lognormally disribued under P. More precisely, he random variable η T in 6.15 has Gaussian law wih zero mean and he variance 3 { k Var η T = E P = = k i,j,l,m=1 k i,j=1 i,l=1 k ŵ iˆσ il WT l j,m=1 ŵ i ŵ j ˆσ ilˆσ jm E P W l T WT m ŵ i ŵ j ˆσ i ˆσ j T = v 2 T, } ŵ j ˆσ jm WT m where v 2 = k i,j=1 ρ i,j ŵ i ŵ j σ i σ j. Noice also ha he las erm on he righ-hand side of 6.14 equals 1, since k E P ÃT = ŵ j S j E P e rt ST i k = ŵ j =1, j=1 and he expeced value E P G T equals E P G T =e v2 c 2T/2 E P e η T 1 2 Var η T = e v2 c 2T/2 def = c. We conclude ha ˆK = K + c 1. The expecaion in 6.13 can now be evaluaed explicily, using he following simple lemma cf. Lemma Lemma Le ξ be a Gaussian random variable on Ω, F, P wih zero mean and he variance σ 2 >. For any sricly posiive real numbers a and b, we have where h = σ 1 lna/b+ 1 2 σ. j=1 E P ae ξ 1 2 σ2 b + = anh bnh σ, 6.16 We have E P GT ˆK + = E P ce η T 1 2 Var η T K + c 1 +, so ha ξ = η T,a= c and b = K + c 1. In view of Lemma 6.8.1, he following resul is sraighforward. Proposiion The approximae value ĈB of he price C B of a baske call opion wih srike price K and expiry dae T equals k Ĉ B = w j S j cn l 1 T K + c 1N l 2 T, 6.17 j=1 3 We use here,in paricular,he equaliy E P W l T W m T = δlm T, where δ lm sands for Kronecker s dela ha is, δ lm equals 1 if l = m, and zero oherwise.

113 112 CHAPTER 6. EXOTIC OPTIONS where { 1 c = exp 2 k ρ i,j ŵ i ŵ j σ i σ j i,j=1 k } ŵ j σj 2 T, and where he modified weighs ŵ i are given by 6.1, K is given by 6.11, and 6.9 Lookback Opions j=1 l 1,2 = ln c ln K + c 1 ± 1 2 v2 v. Lookback opions are anoher example of pah-dependen opions i.e., opion conracs whose payoff a expiry depends no only on he erminal prices of he underlying asses, bu also on asse price flucuaions during he opions lifeimes. We shall firs examine he wo following cases: ha of a sandard lookback call opion, wih payoff a expiry LC T def = S T m S T + = S T m S T, 6.18 where m S T = min [,T ] S ; and ha of a sandard lookback pu opion, whose erminal payoff equals LP T def = M S T S T + = M S T S T, 6.19 where MT S = max [,T ] S. Noe ha a lookback opion is no a genuine opion conrac since he European lookback opion is always exercised by is holder a is expiry dae. I is clear ha he arbirage prices of a lookback opion are and LC = e rt E P S T F e rt E P m S T F =I 1 I 2 LP = e rt E P M S T F e rt E P S T F =J 1 J 2 for he lookback call and pu, respecively. Sandard lookback opions were firs sudied by Goldman e al So-called limied risk and parial lookback opions boh of European and American syle were examined in Conze and Viswanahan Proposiion The price a ime [,T] of a European lookback call opion equals lns/m+r1 τ LC = sn σ me rτ lns/m+r2 τ N τ σ τ sσ2 lnm/s 2r N r1 τ σ rτ sσ2 m 2rσ 2 lnm/s+r2 τ + e N τ 2r s σ, τ where s = S,m= m S,τ= T, and r 1,2 = r ± 1 2 σ2. Equivalenly, LC = sn d me rτ N d σ τ sσ2 N d 2r rτ sσ2 m 2rσ 2 + e N 2r s d +2rσ 1 τ, where d = lns/m+r 1τ σ. τ In paricular, if s = S = m = m S, hen by seing d = r 1 τ/σ, we ge LC = s Nd e rτ Nd σ τ σ2 σ2 N d+e rτ 2r 2r Nd σ τ.

114 6.9. LOOKBACK OPTIONS 113 The nex resul, which is also saed wihou proof, deals wih he lookback pu opion see Goldman e al Proposiion The price of a European lookback pu opion a ime equals LP = sn lns/m+r 1τ σ + Me rτ N lns/m+r 2τ τ σ τ + sσ2 2r N lns/m+r1 τ σ rτ sσ2 M 2rσ 2 lns/m r2 τ e N τ 2r s σ τ where s = S,M= M S,τ= T, and r 1,2 = r ± 1 2 σ2. Equivalenly, LP = sn ˆd+Me rτ N ˆd + σ τ+s σ2 N ˆd 2r rτ sσ2 M 2rσ 2 e N 2r s ˆd 2rσ 1 τ, where ˆd = lns/m+r σ2 τ σ. τ In paricular, if s = S = M = M S, hen, denoing d = r 1 τ/σ, we obain LP = s N d+e rτ N d + σ τ+ σ2 σ2 Nd e rτ 2r 2r N d + σ τ. Noice ha an American lookback call opion is equivalen o is European counerpar. Indeed, he process Z = e r S m S =S A is a submaringale, since he process A has non-increasing sample pahs wih probabiliy 1. American and European lookback pu opions are no equivalen, however. The following bounds for he price LP a of an American lookback pu opion can be esablished LP LP a e rτ LP + S e rτ 1.,

115 114 CHAPTER 6. EXOTIC OPTIONS

116 Par II Fixed-income Markes 115

117

118 Chaper 7 Ineres Raes and Relaed Conracs By a fixed-income marke we mean ha secor of he global financial marke on which various ineres rae-sensiive insrumens, such as bonds, swaps, swapions, caps, ec. are raded. In realworld pracice, several fixed-income markes operae; as a resul, many conceps of ineres raes have been developed. There is no doub ha managemen of ineres rae risk, by which we mean he conrol of changes in value of a sream of fuure cash flows resuling from changes in ineres raes, or more specifically he pricing and hedging of ineres rae producs, is an imporan and complex issue. I creaes a demand for mahemaical models capable of covering all sors of ineres rae risks. Due o he somewha peculiar way in which fixed-income securiies and heir derivaives are quoed in exising markes, heoreical erm srucure models are ofen easier o formulae and analyse in erms of ineres raes which are differen from he convenional marke raes. In his chaper, we give an overview of various conceps of ineres raes. We also describe he mos imporan financial conracs relaed o ineres raes, and markes a which hey are raded. A more deailed descripion of real-world bond and swap markes can be found in Ray 1993 and Das 1994, respecively. Grabbe 1995 focuses on conracs relaed o inernaional financial markes. 7.1 Zero-coupon Bonds Le T > be a fixed horizon dae for all marke aciviies. By a zero-coupon bond a discoun bond of mauriy T we mean a financial securiy paying o is holder one uni of cash a a prespecified dae T in he fuure. This means ha, by convenion, he bond s principal known also as face value or nominal value is one dollar. We assume hroughou ha bonds are defaul-free; ha is, he possibiliy of defaul by he bond s issuer is excluded. The price of a zero-coupon bond of mauriy T a any insan T will be denoed by B, T ; i is hus obvious ha BT,T = 1 for any mauriy dae T T. Since here are no oher paymens o he holder, in pracice a discoun bond sells for less han he principal before mauriy ha is, a a discoun hence is name. This is because one could carry cash a virually no cos, and hus would have no incenive o inves in a discoun bond cosing more han is face value. We shall usually assume ha, for any fixed mauriy T T, he bond price B,T follows a sricly posiive and adaped process on a filered probabiliy space Ω, F, P. 117

119 118 CHAPTER 7. INTEREST RATES AND RELATED CONTRACTS Term Srucure of Ineres Raes Le us consider a zero-coupon bond wih a fixed mauriy dae T T. The simple rae of reurn from holding he bond over he ime inerval [, T ] equals 1 B, T B, T = 1 B, T 1. The equivalen rae of reurn, wih coninuous compounding, is commonly referred o as a coninuously compounded yield-o-mauriy on a bond. Formally, we have he following definiion. Definiion An adaped process Y, T defined by he formula Y, T = 1 ln B, T, [,T, 7.1 T is called he yield-o-mauriy on a zero-coupon bond mauring a ime T. The erm srucure of ineres raes, known also as he yield curve, is he funcion ha relaes he yield Y, T o mauriy T. I is obvious ha, for arbirary fixed mauriy dae T, here is a one-o-one correspondence beween he bond price process B, T and is yield-o-mauriy process Y, T. Given he yield-o-mauriy process Y, T, he corresponding bond price process B, T is uniquely deermined by he formula 1 B, T =e Y,T T, [,T]. 7.2 The discoun funcion relaes he discoun bond price B, T o mauriy T. A he heoreical level, he iniial erm srucure of ineres raes may be represened eiher by he family of curren bond prices B,T, or by he iniial yield curve Y,T, as B,T=e Y,T T, T [,T ]. 7.3 In pracice, he erm srucure of ineres raes is derived from he prices of several acively raded ineres rae insrumens, such as Treasury bills, Treasury bonds, swaps and fuures. Noe ha he yield curve a any given day is deermined exclusively by marke prices quoed on ha day. The shape of an hisorically observed yield curve varies over ime; he observed yield curve may be upward sloping, fla, descending, or humped. There is also srong empirical evidence ha he movemens of yields of differen mauriies are no perfecly correlaed. Also, he shor-erm ineres raes flucuae more han long-erm raes; his may be parially explained by he ypical shape of he erm srucure of yield volailiies, which is downward sloping. These feaures mean ha he consrucion of a reliable model for sochasic behavior of he erm srucure of ineres raes is a ask of considerable complexiy Forward Ineres Raes Le f, T be he forward ineres rae a dae T for insananeous risk-free borrowing or lending a dae T. Inuiively, f, T should be inerpreed as he ineres rae over he infiniesimal ime inerval [T,T + dt ] as seen from ime. As such, f, T will be referred o as he insananeous, coninuously compounded forward rae, or shorly, insananeous forward rae. In conras o bond prices, he concep of an insananeous forward rae is a mahemaical idealizaion raher han a quaniy observable in pracice. As we shall see in wha follows, however, one of he popular approaches o he bond price modelling is based on he exogenous specificaion of a family f, T, T T, of insananeous forward ineres raes. Given such a family f, T, he bond prices are hen defined by seing T B, T = exp f, u du, [,T] We assume ha he limi of Y, T, as ends o T, exiss.

120 7.1. ZERO-COUPON BONDS 119 On he oher hand, if he family of bond prices B, T is sufficienly smooh wih respec o mauriy T, he implied insananeous forward ineres rae f, T is given by he formula ln B, T f, T = 7.5 T which, indeed, can be seen as he formal definiion of he insananeous forward rae f, T. Alernaively, he insananeous forward rae can be seen as a limi case of a forward rae f, T, U which prevails a ime for riskless borrowing or lending over he fuure ime inerval [T,U]. The rae f, T, U is in urn ied o he zero-coupon bond price by means of he formula B, U B, T = e f,t,uu T, T U, 7.6 or equivalenly ln B, T ln B, U f, T, U =. 7.7 U T Observe ha Y, T =f,, T, as expeced indeed, invesing a ime in T -mauriy bonds is clearly equivalen o lending money over he ime inerval [, T ]. On he oher hand, under suiable echnical assumpions, he convergence f, T = lim U T f, T, U holds for every T. For convenience, we focus on ineres raes ha are subjec o coninuous compounding. In pracice, ineres raes are quoed on an acuarial basis, raher han as coninuously compounded raes. For insance, he acuarial rae or effecive rae a, T a ime for mauriy T i.e., over he ime inerval [, T ] is given by he following relaionship 1 + a, T T = e f,,t T = e Y,T T, T. This means, of course, ha he bond price B, T equals 1 B, T =, T. 1 + a, T T Similarly, he forward acuarial rae a, T, U prevailing a ime over he fuure ime period [T,U] is se o saisfy 1 + a, T, U U T = exp f, T, UU T = B, T /B, U Shor-erm Ineres Rae Mos radiional sochasic ineres rae models are based on he exogenous specificaion of a shorerm rae of ineres. We wrie r o denoe he insananeous ineres rae also referred o as a shor-erm ineres rae, or spo ineres rae for risk-free borrowing or lending prevailing a ime over he infiniesimal ime inerval [, + d]. In a sochasic seup, he shor-erm ineres rae is modelled as an adaped process, say r, defined on a filered probabiliy space Ω, F, P for some T >. We assume hroughou ha r is a sochasic process wih almos all sample pahs inegrable on [,T ] wih respec o he Lebesgue measure. We may hen inroduce an adaped process B of finie variaion and wih coninuous sample pahs, given by he formula B = exp r u du, [,T ]. 7.8 Equivalenly, for almos all ω Ω, he funcion B = B ω solves he differenial equaion db = r B d, wih he convenional iniial condiion B =1. In financial inerpreaion, B represens he price process of a risk-free securiy which coninuously compounds in value a he rae r. The process B is referred o as an accumulaion facor or a savings accoun in wha follows. Inuiively, B represens he amoun of cash accumulaed up o ime by saring wih one uni of cash a ime, and coninually rolling over a bond wih infiniesimal ime o mauriy.

121 12 CHAPTER 7. INTEREST RATES AND RELATED CONTRACTS 7.2 Coupon-bearing Bonds A coupon-bearing bond is a financial securiy which pays o is holder he amouns c 1,...,c m a he daes T 1,...,T m. Unless explicily saed oherwise, we assume ha he ime variable is expressed in years. Obviously he bond price, say B c, a ime can be expressed as a sum of he cash flows c 1,...,c m discouned o ime, namely B c = j=1 m c j B, T j. 7.9 j=1 A real bond ypically pays a fixed coupon c and repays he principal N. Therefore, we have c j = c for j =1,...,m 1 and c m = c+n. The main difficuly in dealing wih bond porfolios is due o he fac ha mos bonds involved are coupon-bearing bonds, raher han zero-coupon bonds. Alhough he coupon paymens and he relevan daes are preassigned in a bond conrac, he fuure cash flows from holding a bond are reinvesed a raes ha are no known in advance. Therefore, he oal reurn on a coupon-bearing bond which is held o mauriy or for a lesser period of ime appears o be uncerain. As a resul, bonds wih differen coupons and cash flow daes may no be easy o compare. The sandard way o circumven his difficuly is o exend he noion of a yield-o-mauriy o coupon-bearing bonds. Consider a bond which pays m idenical yearly coupons c a he daes 1,...,m,and he principal N afer m years. Is yield-o-mauriy a ime, denoed by Ỹc, may be found from he following relaionship m c B c = 1 + Ỹc + N. 7.1 j 1 + Ỹc m Since he coupon paymens are usually deermined by a preassigned ineres rae r c > known as a coupon rae, his may be rewrien as follows B c = m r c N 1 + Ỹc + N j 1 + Ỹc m j=1 I is clear ha in his case he yield does no depend on he face value of he bond. Noice ha he price B c equals he bond s face value N whenever r c = Ỹc; in his case a bond is said o be priced a par. Similarly, we say ha a bond is priced below par i.e., a a discoun when is curren price is lower han is face value: B c <N,or equivalenly, when is yield-o-mauriy exceeds he coupon rae: Ỹ c >r c. Finally, a bond is priced above par i.e., a a premium when B c >N,ha is, when Ỹc <r c. In he case of coninuous compounding, he corresponding yield-o-mauriy Y c saisfies B c = m ce jyc + Ne myc, 7.12 j=1 where B c sands for he curren marke price of he bond. Le us now focus on zero-coupon bonds i.e., c = and N = 1. The iniial price B,m of a zero-coupon bond can easily be found provided is yield-o-mauriy Ỹ,m is known. Indeed, we have B,m= Ỹ,mm. Similarly, in a coninuous-ime framework, we have B,T=e Y,T T, where B,T is he iniial price of a uni zero-coupon bond of mauriy T, and Y,T sands for is yield-o-mauriy.

122 7.3. INTEREST RATE FUTURES 121 Definiion The discreely compounded yield-o-mauriy Ỹci a ime i on a coupon-bearing bond which pays he posiive deerminisic cash flows c 1,...,c m a he daes 1 <...<m T is given implicily by means of he formula B c i = m j=i+1 where B c i sands for he price of a bond a he dae i<m. c j, Ỹci j i In a coninuous-ime framework, we have he following definiion of he bond s yield-o-mauriy. Definiion If a bond pays he posiive cash flows c 1,...,c m a he daes T 1 <...<T m T, hen is coninuously compounded yield-o-mauriy Y c =Y c ; c 1,...,c m,t 1,...,T m is uniquely deermined by he following relaionship B c = T j> c j e YcTj, 7.14 where B c denoes he bond price a ime <T m. Noe ha on he righ-hand side of , respecively, he coupon paymen a ime i a ime, respecively is no aken ino accoun. Consequenly, he price B c i B c, respecively is he price of a bond afer he coupon a ime i a ime, respecively has been paid. We focus mainly on he coninuously compounded yield-o-mauriy Y c. I is common o inerpre he yield-o-mauriy Y c as a proxy for he uncerain reurn on a bond; his means ha i is implicily assumed ha all coupon paymens occurring afer he dae are reinvesed a he rae Y c. Since his canno, of course, be guaraneed, he yield-o-mauriy should be seen as a very rough approximaion of he uncerain reurn on a coupon-bearing bond. On he oher hand, he reurn on a discoun bond is cerain, herefore he yield-o-mauriy deermines exacly he reurn on a discoun bond. I is worhwhile o noe ha for every, an F -adaped random variable Y c is uniquely deermined for any given collecion of posiive cash flows c 1,...,c m and daes T 1,...,T m, provided ha he bond price a ime is known. Le us conclude his secion by observing ha he bond price moves inversely o he bond s yieldo-mauriy. Moreover, i can also be checked ha he moves are asymmeric, so ha a decrease in yields raises bond prices more han he same increase lowers bond prices. This specific feaure of he bond price is referred o as convexiy. Finally, i should be sressed ha he uncerain reurn on a bond comes from boh he ineres paid and from he poenial capial gains or losses caused by he fuure flucuaions of he bond price. Therefore, he erm fixed-income securiy should no be aken lierally, unless we consider a bond which is held o is mauriy. 7.3 Ineres Rae Fuures Unil he inroducion of financial fuures, he fuures marke consised only of conracs for delivery of commodiies. In 1975, he Chicago Board of Trade CBT creaed he firs financial fuures conrac, a fuures conrac for so-called morgage-backed securiies. Morgage-backed securiies are bonds collaerized wih a pool of governmen-guaraneed home morgages. Since hese securiies are issued by he Governmen Naional Morgage Associaion GNMA, he corresponding fuures conrac is commonly referred o as Ginnie Mae fuures Treasury Bond Fuures Treasury bond fuures conracs were inroduced on he Chicago Board of Trade in Nominally, he underlying insrumen of a T-bond fuures conrac is a 15-year T-bond wih an 8% coupon.

123 122 CHAPTER 7. INTEREST RATES AND RELATED CONTRACTS T-bond fuures conracs and T-bond fuures opions rade wih up o one year o mauriy. As usual, he fuures conrac specifies precisely he ime and oher relevan condiions of delivery. Delivery is made on any business day of he delivery monh, wo days afer he delivery noice i.e., he declaraion of inenion o make delivery is passed o he exchange. The invoice price received by he pary wih a shor posiion equals he bond fuures selemen price muliplied by he delivery facor for he bond o be delivered, plus he accrued ineres. The delivery facor, calculaed for each deliverable bond issue, is based on he coupon rae and he ime o he bond s expiry dae. Basically, i equals he price of a uni bond wih he same coupon rae and mauriy, assuming ha he yield-o-mauriy of he bond equals 8%. For insance, for a bond wih m years o mauriy and coupon rae r c, he conversion facor δ equals δ = 2m j=1 r c / j + 1, m so ha δ>1δ<1, respecively whenever r c >. 8r c <.8, respecively. Noe ha he adjusmen facor δ makes he yields of each deliverable bond roughly equal for a pary paying he invoice price. In paricular, if he selemen fuures price 2 is close o 1, his yield is approximaely 8%. A any given ime, here are abou 3bonds ha can be delivered in he T-bond fuures conrac basically, any bond wih a leas 15 years o mauriy. The cheapes-o-deliver bond is ha deliverable issue for which he difference Quoed bond price Selemen fuures price Conversion facor is leas. Pu anoher way, he cheapes-o-deliver bond is he one for which he basis b i = Bc f i δ i is minimal, where Bc i is he curren price of he i h deliverable bond, f is he bond fuures selemen price, and δ i is he conversion facor of he i h bond. Usually, he marke is able o forecas he cheapes-o-deliver bond for a given delivery monh. A change in he shape of he yield curve or a change in he level of yields ofen means a swich in cheapes-o-deliver bond, however. This is because, as yields change, a securiy wih a slighly differen coupon or mauriy may become cheaper for marke makers o deliver. Before he delivery monh, he calculaion of he cheapeso-deliver issue also involves he cos of carry ne financing cos of a given bond; he op delivery choice is he issue wih he lowes afer-carry basis. Due o he change of yield level or new bond issue as ime passes, he op delivery choice also changes. The possibiliy of such an even, which may be seen as an addiional source of risk, makes he valuaion of fuures conracs and heir use for hedging purposes more involved Bond Opions Currenly raded bond-relaed opions spli ino wo caegories: OTC bond opions and T-bond fuures opions. The marke for he firs class of bond opions is made by primary dealers and some acive rading firms. The long i.e., 3-year bond is he mos popular underlying insrumen of OTC bond opions; however, opions on shorer-erm issues are also available o cusomers. Since a large number of differen ypes of OTC bond opions exiss in he marke, he marke is raher illiquid. Mos opions are wrien wih one monh or less o expiry. They usually rade a-he-money. This convenion simplifies quoaion of bond opion prices. Opions wih exercise prices ha are up o wo poins ou-of-he-money are also common. Bond opions are used by raders o immunize heir posiions from he direcion of fuure price changes. For insance, if a dealer buys call opions from a clien, he usually sells cash bonds in he open marke a he same ime. Like all ypical exchange-raded opions, T-bond fuures opions have fixed srike prices and expiry daes. Srike prices come in wo-poin incremens. The opions are wrien on he firs four delivery monhs of a fuures conrac noe ha he delivery of he T-bond fuures conrac occurs only every hree monhs. In addiion, a 1-monh opion is raded unless he nex monh is he 2 I is cusomary o quoe boh he bond price and he bond fuures price for a $1 face value bond.

124 7.3. INTEREST RATE FUTURES 123 delivery monh of he fuures conrac. The opions sop rading a few days before he corresponding delivery monh of he underlying fuures conrac. The T-bond fuures opion marke is highly liquid. An open ineres in one opion conrac may amoun o $5 billion in face value his corresponds o 5, opion conracs. For a deailed descripion of he T-bond marke, we refer o Ray Treasury Bill Fuures The Treasury bill T-bill, for shor is a bill of exchange issued by he U.S. Treasury o raise money for emporary needs. I pays no coupons, and he invesor receives he face value a mauriy. T-bills are issued on a regular schedule in 3-monh, 6-monh and 1-year mauriies. In he T-bill fuures conrac, he underlying asse is a 9-day T-bill. The common marke pracice is o quoe a discoun bond, such as he T-bill, no in erms of he yield-o-mauriy, bu raher in erms of so-called discoun raes. The discoun rae represens he size of he price reducion for a 36-day period for insance, a bill of face value 1which maures in 9days and is sold a a discoun rae 1% is priced a Formally, a discoun rae R b, T known also as bankers discoun yield of a securiy which pays a deerminisic cash flow X T a he fuure dae T, and has he price X a ime <T, equals R b, T = X T X 36 X T T, 7.16 where T is now expressed in days. In paricular, for a discoun bond his gives R b, T = 1 P, T =1 B, T T T, where P, T B, T, respecively sands for he cash price of a bill wih face value 1 wih he uni face value, respecively and T days o mauriy. Conversely, given a discoun rae R b, T of a bill, we find is cash price from he following formula P, T = 1 1 R b, T T 36 For a jus-issued 9-day T-bill, he above formulae can be furher simplified. Indeed, we have The bill yield on a T-bill equals R b, 9 = 41 P, 9/1, P, 9 = R b, 9. Y b, T = 1 P, T P, T 36 T = 1 B, T B, T. 36 T = R b, T B, T, so ha i represens he annualized wih no compounding ineres rae earned by he bill owner. In erms of a bill yield Y b, T, is price B, T equals B, T = 1 1+Y b, T T /36. Le us now examine marke convenions associaed wih T-bill fuures conracs. In he T-bill fuures conrac, he underlying asse is he 9-day Treasury bill. In conras o T-bills, which are quoed in erms of he discoun rae, T-bill fuures are quoed in erms of he price for a 1 face value bill. In paricular, he T-bill fuures quoed price a mauriy equals 1 minus he T-bill quoe in percenage erms. The marking o marke procedure is based, however, on he corresponding cash price of a given fuures conrac, which is calculaed from he formula P b = f b = R f, where f b is he marke quoaion of T-bill fuures a ime, and R f = R f T,T + 9 represens he fuures discoun rae over he fuure ime inerval [T,T + 9] implied by T-bill fuures conrac noe

125 124 CHAPTER 7. INTEREST RATES AND RELATED CONTRACTS ha R f is expressed here in percenage erms. For insance, if he quoaion for T-bill fuures is f b =95.2, he implied fuures discoun rae equals R f =4.98% and hus he corresponding cash fuures price, which is used in marking o he marke, equals P b = = = per $1face value bill, or equivalenly, $987,55per fuures conrac he nominal size of one T-bill fuures conrac which rades on he CME is $1 million Eurodollar Fuures Since convenions associaed wih marke quoaions of he LIBOR rae and Eurodollar fuures conracs are close o hose examined above, we shall describe hem in a raher succinc way. Eurodollar fuures and fuures opions have raded on he CME since 1981 and 1985, respecively. 3 Eurodollar fuures and relaed Eurodollar fuures opions rade wih up o five years o mauriy. A Eurodollar fuures opion is of American syle; one opion covers one fuures conrac and i expires a he selemen dae of he underlying Eurodollar fuures conrac. Formally, he underlying insrumen of a Eurodollar fuures conrac is he 3-monh LIBOR rae. A he selemen dae of a Eurodollar fuures conrac, he CME surveys 12 randomly seleced London banks, which are asked o give heir percepion of he rae a which 3-monh Eurodollar deposi funds are currenly offered by he marke o prime banks. A suiably rounded average of hese quoes serves o calculae he Eurodollar fuures price a selemen cf. Amin and Moron Le us sress ha he LIBOR is defined as he add-on yield ; ha is, he acual ineres paymen on a 3-monh Eurodollar ime deposi equals LIBOR numbers of days for invesmen/36 per uni invesmen. In our framework, he spo LIBOR a ime on a Eurodollar deposi wih a mauriy of τ days is formally defined as or equivalenly l, + τ = 36 τ B, + τ = In paricular, a 3-monh LIBOR equals 1 B, + τ 1, 1 1+l, + ττ/36. l, +9=4B 1, +9 1, l, +9=4 B 1, +9 1, where l, + 9 represens a 3-monh LIBOR expressed in percenage erms. Eurodollar fuures conrac is always seled in cash no physical delivery is possible. A Eurodollar fuures price ft l on he selemen day T is ied o he curren level of a 3-monh LIBOR a ime T and hus o he price of a zero-coupon bond hrough he convenional formula ft l = 1 lt,t +9 = 1 4 B 1 T,T More specifically, he quoed price f T l is given for 1$ of a nominal value, so ha a mauriy dae T we have f T l =1 1 lt,t +9 =1 lt,t + 9 = B 1 T,T Therefore, a 3-monh fuures LIBOR l f =1 f l implici in a Eurodollar fuures conrac converges o a 3-monh LIBOR expressed in percenage erms as he ime argumen ends o he selemen dae T. Le us now focus on he valuaion of a Eurodollar fuures conrac. A marke quoaion of Eurodollar fuures conracs is based on he same rule as he quoaion of T-bill fuures. Explicily, 3 Eurodollar fuures rade also on he LIFFE since 1982 and SIMEX since 1984.

126 7.4. INTEREST RATE SWAPS 125 if f l sands for he marke quoaion of Eurodollar fuures, hen he value of a conrac which is used in marking o he marke equals P l = f l =1 1 4 l f per $1 of nominal value. The nominal value of one Eurodollar fuures conrac is $1 million; one basis poin is hus worh $25 when he conrac is marked o he marke daily. For insance, he quoed Eurodollar fuures price f l =94.47 corresponds o a 3-monh LIBOR fuures rae of lf = 5.53%, and o he price $986,175 of one Eurodollar fuures conrac. If he nex day he quoed price rises o i.e., he 3-monh LIBOR fuures rae declines o 5.52%, he value of one conrac appreciaes by $25 o $986,2. We end his secion by a shor descripion of marke convenions relaed o Eurodollar fuures opions. The owner of a Eurodollar fuures call opion obains a long posiion in he fuures conrac wih a fuures price equal o he opion s exercise price; he call wrier obains a shor fuures posiion. On marking o marke, he call owner receives he cash difference beween he marked-o-marke fuures price and he exercise price. 7.4 Ineres Rae Swaps Generally speaking, a swap is an agreemen beween wo paries o exchange cash flows a some fuure daes according o a prearranged formula. In a classic swap conrac, he value of he swap a he ime i is enered ino, as well as a he end of is life, is zero. In a plain vanilla ineres rae swap, one pary, say A, agrees o pay o he oher pary, say B, amouns deermined by a fixed ineres rae on a noional principal a each of he paymen daes. A he same ime, he pary B agrees o pay o he pary A ineres a a floaing reference rae on he same noional principal for he same period of ime. Thus an ineres rae swap can be used o ransform a floaing-rae loan ino a fixed-rae loan or vice versa. In essence, a swap is a long posiion in a fixed-rae coupon bond combined wih shor posiions in floaing-rae noes alernaively, i can be seen as a porfolio of specific forward conracs. In a payer swap, he fixed rae is paid a he end or, depending on conracual feaures of he swap, a he beginning of each period, and he floaing rae is received herefore, i may also be ermed a fixed-for-floaing swap. Similarly, a receiver swap or a floaing-for-fixed swap is one in which an invesor pays a floaing rae and receives a fixed rae on he same noional principal. In a payer swap seled in arrears, he floaing rae paid a he end of a period is se a he beginning of his period. We say ha a swap is seled in advance if paymens are made a he beginning of each period. Noice ha paymens of a swap which seles in advance are he paymens, discouned o he beginning of each period, of he corresponding swap seled in arrears. However, he discouning convenions vary from counry o counry. In some cases, boh sides of a swap are discouned using he same floaing rae; in ohers, he floaing is discouned using he floaing and he fixed using he fixed. Le us consider an arbirary collecion T = T<T 1 <...<T n of fuure daes. Formally, a forward sar swap or briefly, a forward swap is a swap agreemen enered a he rade dae T wih paymen daes T 1 <...<T n if a swap is seled in arrears or T <...<T n 1 if a swap is seled in advance. For mos swaps, a fee he up-fron cos is negoiaed beween he couner-paries a he rade dae. The forward swap rae is ha value of he fixed rae which makes he value of he forward swap zero. The marke gives quoes on swap raes, i.e., he fixed raes a which financial insiuions offer o heir cliens ineres rae swap conracs of differing mauriies, wih fixed quarerly, semi-annual or annual paymen schedules. The mos ypical opion conrac associaed wih swaps is a swapion ha is, an opion on he value of he underlying swap or, equivalenly, on he swap rae. Le us commen briefly on a more convenional class of conracs, widely used by companies o hedge he ineres rae risk. Consider a company which forecass ha i will need o borrow cash a a fuure dae, say T, for he period [T,U]. The company will be, of course, unhappy if he ineres rae rises by he ime he loan is required. A commonly used conrac, which serves o reduce ineres rae risk exposure by locking ino a rae of ineres, is a forward rae agreemen. A forward rae agreemen an FRA is a conrac in which wo paries a seller of a conrac and

127 126 CHAPTER 7. INTEREST RATES AND RELATED CONTRACTS a buyer agree o exchange, a some fuure dae, ineres paymens on he noional principal of a conrac. I will be convenien o assume ha his paymen is made a he end of he period, say a ime U. The cash flow is deermined by he lengh of he ime-period, say [T,U], and by wo relevan ineres raes: he prespecified rae of ineres and he risk-free rae of ineres prevailing a ime T. The buyer of an FRA hus pays ineres a a preassigned rae and receives ineres a a floaing reference rae which prevails a ime T. Noe ha an FRA may be seen as an example of a forward conrac, he conrac s underlier being an uncerain fuure cash flow ineress o be paid a ime U. A ypical use of a forward rae agreemen is a long posiion in an FRA combined wih a loan aken a ime T over he period [T,U]. A synheic version of such a sraegy is a forwardforward loan ha is, a combinaion of a longer-erm loan and a shorer-erm deposi a company jus akes a loan over [,U] and makes a deposi over [,T]. Assuming a fricionless marke, he rae of ineres a company manages o lock ino on is loan using he above sraegy will coincide wih he prespecified rae of ineres in forward rae agreemens proposed o cusomers by financial insiuions a no addiional charge. Indeed, insead of manufacuring a forward-forward loan, a company may alernaively buy a no charge a forward rae agreemen and ake a ime T a loan on he spo marke boh conracs should refer o he same noional principal. We shall examine firs a forward rae agreemen wrien a ime wih he reference period [T,U]. We may and do assume, wihou loss of generaliy, ha he noional principal of he conrac is 1. Le us firs inroduce some noaion. We denoe by r,t he coninuously compounded ineres rae for risk-free borrowing and lending over he ime-period [, T]. I is clear ha, barring arbirage opporuniies beween bank deposis and he zero-coupon bond marke, he T -mauriy spo rae r,t should saisfy e r,t T = B 1,T. In oher words, he ineres rae r,t coincides wih he coninuously compounded yield on a T -mauriy discoun bond ha is, r,t=y,t for every T. As menioned earlier, he buyer of an FRA receives a ime U a cash flow corresponding o an ineres rae se a ime T, and pays ineres according o a rae preassigned a ime. The level of he prespecified rae, loosely ermed a forward ineres rae, is chosen in such a way ha he conrac is worhless a he dae i is enered ino. Le us denoe by f,t,u his level of ineres rae, corresponding o an FRA wrien a ime and referring o he period [T,U]. The forward rae f,t,u may alernaively be seen as a coninuously compounded ineres rae, prevailing a ime, for risk-free borrowing or lending over he ime period [T,U]. I is no difficul o deermine he righ level of he forward rae f,t,u by sandard no-arbirage argumens. By considering wo alernaive rading sraegies, i is easy o esablish he following relaionship e Ur,U = e Tr,T e f,t,uu T. More generally, he forward rae f, T, U saisfies f, T, U = Ur, U Tr, T U T for every T U, where r, T is he fuure spo rae, as from ime, for risk-free borrowing or lending over he ime period [, T ]. Noe ha f,t,t=f,t, i.e., he rae f,t,t if well-defined coincides wih he insananeous forward ineres rae f,t. For similar reasons, he equaliy f, T, T =f, T is valid. If r, T =Y, T, hen in erms of bond prices we have cf. 7.7 ln B, T ln B, U f, T, U =. U T

128 Chaper 8 Models of he Shor-erm Rae The aim of his chaper is o survey he mos popular models of he shor-erm ineres rae. For convenience, we shall work hroughou wihin a coninuous-ime framework; a deailed presenaion of a discree-ime approach o erm srucure modelling is done in Jarrow We sar his chaper by addressing he exisence and uniqueness of an arbirage-free family of bond prices relaed o a given shor-erm rae process. To obain more explici resuls, we hen assume ha he shorerm ineres rae is modelled eiher as an Iô process or, even more specifically, as a one-dimensional diffusion process. 8.1 Arbirage-free Family of Bond Prices Recall ha, by convenion, a zero-coupon bond pays one uni of cash a a prescribed dae T in he fuure. The price a any insan T of a zero-coupon bond of mauriy T is denoed by B, T ; i is hus clear ha, necessarily, BT,T = 1 for any mauriy dae T T. Furhermore, since here are no inervening ineres paymens, in marke pracice he bond sells for less han he principal before he mauriy dae. Essenially, his follows from he fac ha i is possible o inves money in a risk-free savings accoun yielding a non-negaive ineres rae or a leas o carry cash a virually no cos. We assume hroughou ha for any mauriy T T, he price process B, T, [,T], follows a sricly posiive and adaped process on a filered probabiliy space Ω, F, P, where he filraion F is generaed by he underlying d-dimensional sandard Brownian moion W. Suppose ha an adaped process r, given on Ω, F, P, models he shor-erm ineres rae, meaning ha he savings accoun process B saisfies 7.8. Definiion A family B, T, T T, of adaped processes is called an arbirage-free family of bond prices relaive o r if: a BT,T = 1 for every T [,T ]; and b here exiss a probabiliy measure P equivalen o P such ha for any T [,T ], he relaive bond price follows a maringale under P. Z, T =B, T /B, [,T], 8.1 Any probabiliy measure P of Definiion is called a maringale measure for he family B, T relaive o r, or briefly, a maringale measure for he family B, T if no confusion may arise. 1 The reader migh wonder why i is assumed ha he relaive price Z follows a maringale, and no merely a local maringale, under P. The main reason is ha under such an assumpion we have rivially Z, T =E P Z T,T F for T, so ha he bond price saisfies B, T =B E P B 1 T F, [,T] In wha follows,we shall disinguish beween spo and forward maringale measures. In his conex,he maringale measure of Definiion should be seen as a spo maringale measure for he family B, T. 127

129 128 CHAPTER 8. MODELS OF THE SHORT-TERM RATE In oher words, for any maringale measure P of an arbirage-free family of bond prices, we have B, T =E P e T r u du F, [,T]. 8.3 Conversely, given any non-negaive shor-erm ineres rae r defined on a probabiliy space Ω, F, P, and any probabiliy measure P on Ω, F T equivalen o P, he family B, T given by 8.3 is easily seen o be an arbirage-free family of bond prices relaive o r. Le us observe ha if a family B, T saisfies Definiion 8.1.1, hen necessarily he bond price process B,TisaP - semimaringale, as a produc of a maringale and a process of finie variaion ha is, a produc of wo P -semimaringales. Therefore, i is also a P-semimaringale, since he probabiliy measures P and P are assumed o be muually equivalen see Theorem III.2in Proer Expecaions Hypoheses Suppose ha equaliy 8.3 is saisfied under he acual probabiliy measure P, ha is B, T =E P e T r u du F, [,T]. 8.4 Equaliy 8.4 is radiionally referred o as he local expecaions hypohesis L-EH for shor, or a risk-neural expecaions hypohesis. The erm local expecaions refers o he fac ha under 8.4, he curren bond price equals he expeced value, under he acual probabiliy, of he bond price in he nex infiniesimal period, discouned a he curren shor-erm rae. This propery can be made more explici in a discree-ime seing see Ingersoll 1987 or Jarrow In our framework, given an arbirage-free family of bond prices relaive o a shor-erm rae r, i is eviden ha 8.3 holds, by definiion, under any maringale measure P. This does no mean, however, ha he local expecaions hypohesis, or any oher radiional form of expecaions hypohesis, is saisfied under he acual probabiliy P. The reurn-o-mauriy expecaions hypohesis RTM-EH assumes ha he reurn from holding any discoun bond o mauriy is equal o he reurn expeced from rolling over a series of a single-period bonds. Is coninuous-ime counerpar reads as follows 1 B, T = E P e T r u du F, [,T], for every T T. Finally, he yield-o-mauriy expecaions hypohesis YTM-EH assers ha he yield from holding any bond is equal o he yield expeced from rolling over a series of a single-period bonds. In a coninuous-ime framework, his means ha for any mauriy dae T T, we have B, T = exp { E T P r u du F }, [,T]. The las formula may also be given he following equivalen form Y, T = 1 T T E P r u du F, or finally f, T =E P r T F, [,T]. 8.5 In view of 8.5, under he yield-o-mauriy expecaions hypohesis, he forward ineres rae f, T is an unbiased esimae, under he acual probabiliy P, of he fuure shor-erm ineres rae r T. For his reason, he YTM-EH is also frequenly referred o as he unbiased expecaions hypohesis. We shall see in wha follows ha condiion 8.5 is always saisfied no under he acual probabiliy, however, bu under he so-called forward maringale measure for he given dae T. Noe ha if he shor-erm rae r is a deerminisic funcion, hen all expecaions hypoheses coincide, and follow easily from he absence of arbirage.

130 8.2. CASE OF ITÔ PROCESSES Case of Iô Processes In a coninuous-ime framework, i is cusomary o model he shor-erm rae of ineres by means of an Iô process, or more specifically, as a one-dimensional diffusion 2 process. We shall firs examine he general case of a shor-erm ineres rae which follows an Iô process. We hus assume ha he dynamics of r are given in a differenial form dr = µ d + σ dw, r >, 8.6 where µ and σ are adaped sochasic processes wih values in R and R d, respecively. Recall ha 8.6 is a shorhand form of he following inegral represenaion r = r + µ u du + σ u dw u, [,T]. I is hus implicily assumed ha µ and σ saisfy he suiable inegrabiliy condiions, so ha he process r is well-defined. In financial inerpreaion, he underlying probabiliy measure P is believed o reflec a subjecive assessmen of he marke upon he fuure behavior of he shorerm ineres rae. In oher words, he underlying probabiliy P is he acual probabiliy, as opposed o a maringale probabiliy measure for he bond marke, which we are now going o consruc. Le us recall, for he reader s convenience, a few basic facs concerning he noion of equivalence of probabiliy measures on a filraion of a Brownian moion. Firsly, i is well known ha any probabiliy measure Q equivalen o P on Ω, F T is given by he Radon-Nikodým derivaive dq dp = E T λ u dw u def = η T, P-a.s. 8.7 for some predicable R d -valued process λ. The member on he righ-hand side of 8.7 is he Doléans exponenial, which is given by he following expression def η = E λ u dw u = exp λ u dw u 1 λ u 2 du. 2 Given an adaped process λ, we wrie P λ o denoe he probabiliy measure whose Radon-Nikodým derivaive wih respec o P is given by he righ-hand side of 8.7. In view of Girsanov s heorem, he process = W λ u du, [,T ], 8.8 W λ follows a d-dimensional sandard Brownian moion under P λ. I should be sressed ha he naural filraions of he Brownian moions W and W λ do no coincide, in general. The following resul deals wih he behavior of he shor-erm ineres rae r and he bond price B, T under a probabiliy measure P λ equivalen o P more specifically, under a probabiliy measure P λ which is a maringale measure in he sense of Definiion see Arzner and Delbaen 1989 for relaed resuls. Proposiion Assume he shor-erm ineres rae r follows an Iô process under he acual probabiliy P, as specified by 8.6. Le B, T be an arbirage-free family of bond prices relaive o r. For any maringale measure P = P λ of Definiion 8.1.1, he following holds. i The process r saisfies under P λ dr =µ + σ λ d + σ dw λ. 8.9 ii There exiss an adaped R d -valued process b λ, T such ha db, T =B, T r d + b λ, T dw λ Generally speaking,a diffusion process is an arbirary srong Markov process wih coninuous sample pahs. In our framework,a diffusion process is given as a srong soluion of a sochasic differenial equaion SDE driven by he underlying Brownian moion W.

131 13 CHAPTER 8. MODELS OF THE SHORT-TERM RATE Consequenly, for any fixed mauriy T,T ], we have B, T =B,T B exp b λ u, T dwu λ 1 2 b λ u, T 2 du. Proof. To show i, i is enough o combine 8.6 wih 8.8. For ii, i is sufficien o observe ha he process M = Z η follows a local maringale under P. In view of Theorem 3.1.2, we have M = Z, T η = Z,T+ γ u dw u, [,T], for some F-adaped process γ. Applying Iô s formula, we obain dz, T =dm η 1 =η 1 γ M λ dw λ u, 8.11 where we have used dη 1 = η 1 λ dw λ = η 1 λ dw λ d. Equaliy 8.1 now follows easily from 8.11, once again by Iô s formula. The las assered formula is also eviden. Le us commen on he consequences of he resuls above. Suppose ha he shor-erm ineres rae r saisfies 8.6 under a probabiliy measure P. Le P = P λ be an arbirary probabiliy measure equivalen o P. Then we may define a bond price B, T by seing B, T =E P e T r u du F W λ, [,T] I follows from 8.1 ha he bond price B, T saisfies, under he acual probabiliy P, r db, T =B, T λ b λ, T d + b λ, T dw This means ha he insananeous reurns from holding he bond differ, in general, from he shorerm ineres rae r. In financial lieraure, he addiional erm is commonly referred o as he risk premium or he marke price for risk. I is usually argued ha due o he riskiness of a zero-coupon bond, i is reasonable o expec ha he insananeous reurn from holding he bond will exceed ha of a risk-free securiy i.e, of a savings accoun in a marke equilibrium.unforunaely, since our argumens refer only o he absence of arbirage beween primary securiies and derivaives ha is, we place ourselves in a parial equilibrium framework, we are unable o idenify he risk premium. Summarizing, we have a cerain degree of freedom: if he shor-erm rae r is given by 8.6, hen any probabiliy measure P equivalen o P can formally be used o consruc an arbirage-free family of bond prices hrough formula Noice, however, ha if he acual probabiliy measure P is used o define he bond price hrough 8.12, he marke prices for risk vanish. We end his secion by examining he problem of maching he iniial yield curve. Given a shor-erm ineres rae process r and a probabiliy measure P, he iniial erm srucure B,T is uniquely deermined by he formula B,T=E P e T ru du, T [,T ] This feaure of bond price models based on a specified shor-erm ineres rae process makes he problem of maching he curren yield curve much more cumbersome han in he case of models which incorporae he iniial erm srucure as an inpu of he model.

132 8.3. SINGLE-FACTOR MODELS Single-facor Models In his secion, we survey he mos widely acceped single-facor models of he shor-erm rae. I is assumed hroughou ha he dynamics of r are specified under he maringale probabiliy measure P i.e., he risk premium vanishes idenically. The underlying Brownian moion W is assumed o be one-dimensional. In his sense, he models are based on a single source of uncerainy, i.e., hey belong o he class of single-facor models. Vasicek s model. The model analysed in Vasicek 1977 is one of he earlies models of erm srucure see also Richard 1978 and Dohan The diffusion process proposed by Vasicek is a mean-revering version of he Ornsein-Uhlenbeck process. The shor-erm ineres rae r is defined as he unique srong soluion of he SDE dr = a br d + σdw, 8.15 where a, b and σ are sricly posiive consans. I is well known ha he soluion of 8.15 is a Markov process wih coninuous sample pahs and Gaussian incremens. 3 I is eviden ha Vasicek s model, as any Gaussian model, allows for negaive values of nominal ineres raes. This propery is manifesly incompaible wih no-arbirage in he presence of cash in he economy. Le us consider any securiy whose payoff depends on he shor-erm rae r as he only sae variable. More specifically, we assume ha his securiy is of European syle, pays dividends coninuously a a rae hr,, and yields a erminal payoff G T = gr T a ime T. Using he well-known relaionship beween diffusion processes and he PDEs, one can show ha he price process G of such a securiy admis he represenaion G = vr,, where he funcion v : R [,T ] R solves he following valuaion PDE v r, +1 2 σ2 2 v v r, +a br r, rvr, +hr, =, r2 r subjec o he erminal condiion vr, T = gr. Solving his equaion wih h = and gr = 1, Vasicek showed ha he price of a zero-coupon bond is B, T =vr,,t=e m,t n,t r, 8.16 where n, T = 1 b 1 e bt 8.17 and m, T = σ2 2 T T n 2 u, T du a nu, T du To esablish his resul, i is enough o assume ha he bond price is given by 8.16, wih he funcions m and n saisfying mt,t=nt,t=, and o make use of he fundamenal PDE. By separaing erms which do no depend on r, and hose ha are linear in r, we arrive a he following sysem of differenial equaions m, T =an, T 1 2 σ2 n 2, T, n, T =bn, T 1, 8.19 wih mt, T = nt, T =, which in urn yields easily he expressions above. One may check ha we have db, T =B, T r d + σn, T dw, 8.2 so ha he bond price volailiy equals b, T =σn, T, wih n, T given by If he bond price admis represenaion 8.16, hen obviously Y, T = n, T r m, T, T 3 The soluion o 8.15 is known o admi a saionary disribuion,namely,a Gaussian disribuion wih he mean a/b and he variance σ 2 /2b.

133 132 CHAPTER 8. MODELS OF THE SHORT-TERM RATE and hus he bond s yield, Y, T, is an affine funcion of he shor-erm rae r. For his reason, models of he shor-erm rae in which he bond price saisfies 8.16 for some funcions m and n are ermed affine models of he erm srucure. Jamshidian 1989a obained closed-form soluions for he prices of a European opion wrien on a zero-coupon and on a coupon-bearing bond for Vasicek s model. He showed ha he arbirage price a ime of a call opion on a U-mauriy zero-coupon bond, wih srike price K and expiry T U, equals le us menion ha Jamshidian implicily used he forward measure echnique, which is presened in Sec. 9.2 C = B, T E Q ξη K + F, where η = B, U/B, T, and Q sands for some probabiliy measure equivalen o P. The random variable ξ is independen of he σ-field F under Q, and has under Q a lognormal law such ha he variance Var Q ln ξ equals v U, T, where T vu 2, T = b, T b, U 2 du = σ2 1 e 2bT 2b 3 1 e bu 2. The bond opion valuaion formula esablished in Jamshidian 1989a reads as follows where for every T U C = B, UN h 1, T KB, T N h 2, T, 8.21 h 1,2, T = ln B, U/B, T ln K ± 1 2 v2 U, T v U, T I is imporan o observe ha he coefficien a does no ener he bond opion valuaion formula. This suggess ha he acual value of he risk premium has no impac whasoever on he bond opion price a leas if i is deerminisic; he only relevan quaniies are in fac he bond price volailiies b, T and b, U. To accoun for he risk premium, i is enough o make an equivalen change of he probabiliy measure in 8.2. Since he volailiy of he bond price is invarian wih respec o such a ransformaion of he underlying probabiliy measure, he bond opion price is independen of he risk premium, provided ha he bond price volailiy is deerminisic. Cox-Ingersoll-Ross model. The general equilibrium approach o erm srucure modelling developed by Cox e al. 1985b CIR, for shor leads o he following modificaion of he mean-revering diffusion of Vasicek, known as he square-roo process dr = a br d + σ r dw, 8.23 where a, b and σ are sricly posiive consans. Due o he presence of he square-roo in he diffusion coefficien, he CIR diffusion akes only posiive values; i can reach zero, bu i never becomes negaive. In a way similar o he previous case, he price process G = vr,ofany sandard European ineres rae derivaive, which seles a ime T, can be found, in principle, by solving he valuaion PDE v r, +1 2 σ2 r 2 v v r, +a br r, rvr, +hr, =, r2 r subjec o he erminal condiion vr, T = gr. Cox e al. 1985b found closed-form soluions for he price of a zero-coupon bond. If we assume ha he bond price B, T saisfies 8.16, hen using he valuaion PDE above, we find ha he funcion n solves, for each fixed mauriy dae T, he Riccai equaion n, T 1 2 σ2 n 2, T bn, T +1=, nt,t=, 8.24 and m saisfies m, T =an, T, mt,t=. 8.25

134 8.3. SINGLE-FACTOR MODELS 133 Solving his, we obain m, T = 2a { σ 2 ln γe bτ/2 } γ cosh γτ + 1, n, T = 2b sinh γτ sinh γτ γ cosh γτ + 1, 2b sinh γτ where τ = T and 2γ =b 2 +2σ 2 1/2. Closed-form expressions for he price of an opion on a zero-coupon bond and an opion on a coupon-bearing bond in he CIR framework were derived in Cox e al. 1985b and in Longsaff 1993, respecively. Since hey are raher involved, and will no be used in wha follows, we refer he ineresed reader o he original papers for deails. Le us only menion ha hey involve he cumulaive non-cenral chi-square disribuion funcion, and depend on he deerminisic risk premium i is easily seen ha he bond price volailiy is now sochasic. Longsaff 199 has shown how o value European call and pu opions on yields in he CIR model. For a fixed ime o mauriy, he yield on a zero-coupon bond in he CIR framework is, of course, a linear funcion of he shor-erm rae, since Y, + τ =Ỹ r,τ= mτ+ñτr, where τ = T is fixed. The number Ỹ r, τ represens he yield a ime for zero-coupon bonds wih a consan mauriy τ, provided ha he curren level of he shor-erm rae is r = r. According o he conracual feaures, for a fixed τ, a European yield call opion eniles is owner o receive he payoff CT Y, which is expressed in moneary unis and equals CY T =Ỹ r T,τ K +, where K is he fixed level of he yield. Longsaff s model. Longsaff 1989 modified he CIR model by posulaing he following dynamics for he shor-erm rae dr = a b c r d + σ r dw, 8.26 referred o as he double square-roo DSR process. Longsaff derived a closed-form expression for he price of a zero-coupon bond B, T =vr,,t=e m,t n,t r p,t r for some explicily known funcions m, n and p, which are no reproduced here. The bond s yield is hus a non-linear funcion of he shor-erm rae. Also, he bond price is no a monoone decreasing funcion of he curren level of he shor-erm rae. This feaure makes he valuaion of a bond opion less sraighforward han usual. Indeed, ypically, i is possible o represen he exercise se of a bond opion in erms of r as he inerval [r, or,r ] for some consan r, depending on wheher an opion is a pu or a call see Sec Hull-Whie model. Noe ha boh Vasicek s and he CIR models are special cases of he following mean-revering diffusion process dr = a b cr d + σr β dw, where β 1 is a consan. These models of he shor-erm rae are hus buil upon a cerain diffusion process wih consan i.e., ime-independen coefficiens. In pracical applicaions, i is more reasonable o expec ha in some siuaions, he marke s expecaions abou fuure ineres raes involve ime-dependen coefficiens. Also, i would be a plausible feaure if a model fied no merely he iniial value of he shor-erm rae, bu raher he whole iniial yield curve. This desirable propery of a bond price model moivaed Hull and Whie 199a o propose an essenial modificaion of he models above. In is mos general form, he Hull-Whie mehodology assumes ha dr = a br d + σr β dw 8.27

135 134 CHAPTER 8. MODELS OF THE SHORT-TERM RATE for some consan β, where W is a one-dimensional Brownian moion, and a, b, σ : R + R are locally bounded funcions. By seing β = in 8.27, we obain he generalized Vasicek model, in which he dynamics of r are 4 dr = a br d + σ dw To explicily solve his equaion, le us denoe l = bu du. Then we have d e l r = e l a d + σ dw, so ha r = e l r + e lu au du + e lu σu dw u I is hus no surprising ha closed-form soluions for bond and bond opion prices are no hard o derive in his seing. On he oher hand, if we pu β =1/2, hen we obain he generalized CIR model dr = a br d + σ r dw. In his case, however, he closed-form expressions for he bond and opion prices are no easily available his would require solving wih ime-dependen coefficiens a,b and σ. The mos imporan feaure of he Hull-Whie approach is he possibiliy of he exac fi of he iniial erm srucure and, in some circumsances, also of he erm srucure of forward rae volailiies. This can be done, for insance, by means of a judicious choice of he funcions a, b and σ. Since he deails of he fiing procedure depend on he paricular model i.e., on he choice of β, le us illusrae his poin by resricing our aenion o he generalized Vasicek model. We sar by assuming ha he bond price B, T can be represened in he following way B, T =Br,,T=e m,t n,t r 8.29 for some funcions m and n, wih mt,t = and nt,t=. Plugging 8.29 ino he fundamenal PDE for he zero-coupon bond, which is. we obain v r, +1 2 σ2 2 v r 2 r, + a br v r, rvr, =, r m, T an, T σ2 n 2, T 1+n, T bn, T r =. Since he las equaion holds for every, T and r, we deduce ha m and n saisfy he following sysem of differenial equaions cf m, T =an, T 1 2 σ2 n 2, T, n, T =bn, T wih MT,T = nt,t =. Suppose ha an iniial erm srucure P,T is exogenously given. We adop he convenion o denoe by P,T he iniial erm srucure, which is given i can, for insance, be inferred from he marke daa, as opposed o he iniial erm srucure B, T, which is implied by a paricular sochasic model of he erm srucure. Assume also ha he forward rae volailiy is no prespecified. In his case, we may and do assume ha b =b and σ =σ are given consans. We shall hus search for he funcion a only. Since b and σ are consans, n is given by Furhermore, in view of 8.3, m equals m, T = 1 T T σ 2 n 2 u, T du aunu, T du A special case of such a model,wih b =, was considered in Meron 1973.

136 8.3. SINGLE-FACTOR MODELS 135 Since he forward raes implied by he model equal cf. 7.5 ln B,T f,t= = n T,Tr m T,T, 8.32 T easy calculaions involving 8.17 and 8.31 show ha ˆf,T def ln P,T T = = e bt r + e bt u au du σ2 1 e bt 2. T 2b 2 Pu anoher way, ˆf,T=gT ht, where g T = bgt +at, wih g = r, and ht =σ 2 1 e bt 2 /2b 2. Consequenly, we obain at =g T +bgt = ˆf T,T+h T +b ˆf,T+hT, and hus he funcion a is indeed uniquely deermined. This erminaes he fiing procedure. Though, a leas heoreically, his procedure can be exended o fi he volailiy srucure, is should be sressed ha he possibiliy of an exac mach wih he hisorical daa is only one of several desirable properies of a model of he erm srucure. Le us now summarize he mos imporan feaures of erm srucure models which assume he diffusion-ype dynamics of he shor-erm rae. Suppose ha he dynamics of r under he acual probabiliy P saisfy dr = µr, d + σr, dw 8.33 for some sufficienly regular funcions µ and σ. Assume, in addiion, ha he risk premium process equals λ = λr, for some funcion λ = λr,. In financial inerpreaion, he las condiion means ha he excess rae of reurn of a given zero-coupon bond depends only on he curren shorerm rae and he price volailiy of his bond. Using 8.33 and Girsanov s heorem, 5 we conclude ha under he maringale measure P = P λ, he process r saisfies dr = µ λ r,d + σr, dw, 8.34 where µ λ r, =µr, +λr, σr,. Le us sress once again ha i is essenial o assume ha he funcions µ, σ and λ are sufficienly regular for insance, locally Lipschiz wih respec o he firs variable, and saisfying he linear growh condiion, so ha he SDE 8.34, wih iniial condiion r >, admis a unique global srong soluion. Under such assumpions, he process r is known o follow, under he maringale measure P, a srong Markov process wih coninuous sample pahs. The arbirage price π X of any aainable coningen claim X, which is of he form X = gr T for some funcion g : R R, is given by he risk-neural valuaion formula π X =E P gr T e T r u du F = vr,, where v : R [,T ] R. I follows from he general heory of diffusion processes, more precisely from he resul known as he Feynman-Kac formula see Theorem in Karazas and Shreve 1988, ha under mild echnical assumpions, if a securiy pays coninuously a a rae hr, and yields a erminal payoff G T = gr T a ime T, hen he valuaion funcion v solves he following fundamenal PDE v r, +1 2 σ2 r, 2 v r 2 r, +µλ r, v r, rvr, +hr, =, r 5 We need,of course,o show ha an applicaion of Girsanov s heorem is jusified. Essenially,his means ha we need o impose cerain condiions which would guaranee ha P λ is indeed a probabiliy measure equivalen o P.

137 136 CHAPTER 8. MODELS OF THE SHORT-TERM RATE subjec o he erminal condiion vr, T = gr. Exisence of a closed-form soluion of his equaion for he mos ypical derivaive securiies in paricular, for a zero-coupon bond and an opion on such a bond is, of course, a desirable propery of a erm srucure model of diffusion ype. Oherwise, he efficiency of numerical procedures used o solve he fundamenal PDE becomes an imporan pracical issue American Bond Opions Le us denoe by P a r,,t he price a ime of an American pu opion wih srike price K and expiry dae T, wrien on a zero-coupon bond of mauriy U T. Arguing along similar lines as in Chap. 5, i is possible o show ha P a r,,t = ess sup τ T [,T ] E P e τ rv dv K Br τ,τ,u + r, where T [,T ] is he class of all sopping imes wih values in he inerval [, T ]. For a deailed jusificaion of he applicaion of sandard valuaion procedures o American coningen claims under uncerain ineres raes, we refer he reader o Amin and Jarrow For any [,T], he opimal exercise ime τ equals τ = inf {u [, T ] K Br u,u,u + = P a r u,u,t }. Assume ha he bond price is a decreasing funcion of he rae r his holds in mos, bu no all, single-facor models. Then τ = inf {u [, T ] r u ru } for a cerain process r, which represens he criical level of he shor-erm ineres rae. Using his, one can derive he following early exercise premium represenaion of he price of an American pu opion on a zero-coupon bond T P a r,,t=p r,,t+e P e u rv dv I {ru ru } r u Kdu F, where P r,,t sands for he price of he corresponding European-syle pu opion. The quasianalyical forms of his represenaion for Vasicek s model and he CIR model were found in Jamshidian More recenly, Chesney e al have sudied bond and yield opions of American syle for he CIR model, using he properies of he Bessel bridges Opions on Coupon-bearing Bonds A coupon-bearing bond is formally equivalen o a porfolio of discoun bonds wih differen mauriies. To value European opions on coupon-bearing bonds, we ake ino accoun he fac ha he zero-coupon bond price is ypically a decreasing funcion of he shor-erm rae r. This implies ha an opion on a porfolio of zero-coupon bonds is equivalen o a porfolio of opions on zero-coupon bonds wih appropriae srike prices. Le us consider, for insance, a European call opion wih exercise price K and expiry dae T on a coupon-bearing bond which pays coupons c 1,...,c m a daes T 1 <...<T m T. The payoff of he opion a expiry equals m +. C T = c j Br T,T,T j K j=1 Therefore he opion will be exercised if and only if r <r, where he criical ineres rae r solves he equaion m j=1 c j Br,T,T j =K. The opion s payoff can be represened in he following way m +, C T = c j BrT,T,T j K j j=1 where K j = Br,T,T j. The valuaion of a call opion on a coupon-bearing bond hus reduces o he pricing of opions on zero-coupon bonds.

138 Chaper 9 Models of Forward Raes The Heah, Jarrow and Moron approach o erm srucure modelling is based on an exogenous specificaion of he dynamics of insananeous, coninuously compounded forward raes f, T. For any fixed mauriy T T, he dynamics of he forward rae f, T are cf. Heah e al. 199, 1992a df, T =α, T d + σ, T dw, 9.1 where α and σ are adaped sochasic processes wih values in R and R d, respecively, and W is a d-dimensional sandard Brownian moion wih respec o he underlying probabiliy measure P o be inerpreed as he acual probabiliy. For any fixed mauriy dae T, he iniial condiion f,t is deermined by he curren value of he coninuously compounded forward rae for he fuure dae T which prevails a ime. The price B, T of a zero-coupon bond which maures a he dae T T can be recovered from he formula cf. 7.4 T B, T = exp f, u du, [,T], 9.2 provided ha he inegral on he righ-hand side of 9.2 exiss for almos all ω s. Leaving he echnical assumpions aside, he firs quesion ha should be addressed is he absence of arbirage in a financial marke model which involves all bonds, wih differing mauriies, as primary raded securiies. As expeced, he answer o his quesion can be formulaed in erms of he exisence of a suiably defined maringale measure. I appears ha in an arbirage-free seing ha is, under he maringale probabiliy he drif coefficien α in he dynamics 9.1 of he forward rae is uniquely deermined by he volailiy coefficien σ, and a sochasic process which can be inerpreed as he risk premium. More imporanly, if σ follows a deerminisic funcion, hen he valuaion resuls for ineres rae-sensiive derivaives appear o be independen of he choice of he risk premium. In his sense, he choice of a paricular model from he broad class of Heah-Jarrow-Moron HJM models hinges uniquely on he specificaion of he volailiy coefficien σ. I should be sressed ha for his specific feaure of coninuous-ime forward rae modelling o hold, we need o resric our aenion o he class of HJM models wih deerminisic coefficien σ; ha is, o Gaussian HJM models. 9.1 HJM Model Le W be a d-dimensional sandard Brownian moion given on a filered probabiliy space Ω, F, P. As usual, he filraion F = F W is assumed o be he righ-coninuous and P-compleed version of he naural filraion of W. We consider a coninuous-ime rading economy wih a rading inerval [,T ] for he fixed horizon dae T. We are in a posiion o formulae he basic posulaes of he HJM approach. 137

139 138 CHAPTER 9. MODELS OF FORWARD RATES HJM.1 For every fixed T T, he dynamics of he insananeous forward rae f, T are given by he inegraed version of 9.1 f, T =f,t+ αu, T du + σu, T dw u, [,T], 9.3 for a Borel-measurable funcion f, :[,T ] R, and some applicaions α : C Ω R, σ: C Ω R d, where C = { u, u T }. HJM.2 For any mauriy T, α,t and σ,t follow adaped processes, such ha T T αu, T du + σu, T 2 du <, P-a.s. Though unnecessary, one may find i useful o inroduce also a savings accoun as an addiional primary securiy. For his purpose, assume ha here exiss a measurable version of he process f,, [,T ]. I is hen naural o posulae ha he shor-erm ineres rae saisfies r = f, for every. Consequenly, he savings accoun equals cf. 7.8 B = exp fu, u du, [,T ]. 9.4 The following auxiliary lemma deals wih he dynamics of he bond price process B, T under he acual probabiliy P. I is ineresing o observe ha he drif and volailiy coefficiens in he dynamics of B, T can be expressed in erms of he coefficiens α and σ of forward rae dynamics, and he shor-erm ineres rae r = f,. Lemma The dynamics of he bond price B, T are deermined by he expression where a and b are given by he following formulae and for any [,T] we have db, T =B, T a, T d + b, T dw, 9.5 a, T =f, α, T σ, T 2, b, T = σ, T, 9.6 Proof. α, T = T α, u du, σ, T = T Le us denoe I =lnb, T. I follows direcly from ha σ, u du. 9.7 T T I = f,u du T αv, u dv du σv, u dw v du. Applying Fubini s sandard and sochasic heorems for he laer, see Theorem IV.45 in Proer 199, we find ha T T T I = f,u du αv, u du dv σv, u du dw v, or equivalenly T T T I = f,u du αv, u du dv σv, u du dw v + f,u du + v v αv, u du dv + v v σv, u du dw v.

140 9.1. HJM MODEL 139 Consequenly, I = I + T r u du where we have used he represenaion αu, v dv du T u u r u = fu, u =f,u+ Taking ino accoun equaions 9.7, we obain I = I + r u du u αv, u dv + u α u, T du σu, v dv dw u, σv, u dw v. 9.8 σ u, T dw u. To check ha 9.5 holds, i suffices o apply Iô s formula Absence of Arbirage In he presen seing, a coninuum of bonds wih differen mauriies is available for rade. We shall assume, however, ha any paricular porfolio involves invesmens in an arbirary, bu finie, number of bonds. An alernaive approach, in which infinie porfolios are also allowed, can be found in Björk e al. 1997a, 1997b. For any collecion of mauriies <T 1 <T 2 <...<T k = T, we wrie T o denoe he vecor T 1,...,T k ; similarly, B, T sands for he R k -valued process B, T 1,...,B, T k. We find i convenien o exend he R k -valued process B, T over he ime inerval [,T ] by seing B, T = for any T,T ] and any mauriy <T <T. By a bond rading sraegy we mean a pair φ, T, where φ is a predicable R k -valued sochasic process which saisfies φ i = for every T i,t ] and any i =1,...,k. A bond rading sraegy φ, T is said o be self-financing if he wealh process V φ, which equals V φ def = φ B, T = k φ i B, T i, saisfies k V φ =V φ+ φ u dbu, T =V φ+ φ i u dbu, T i i=1 for every [,T ]. To ensure he arbirage-free properies of he bond marke model, we need o examine he exisence of a maringale measure for a suiable choice of a numeraire; in he presen seup, we can ake eiher he bond price B, T or he savings accoun B. Assume, for simpliciy, ha he coefficien σ in 9.3 is bounded. We are looking for a condiion ensuring he absence of arbirage opporuniies across all bonds of differen mauriies. Le us inroduce an auxiliary process F B by seing i=1 F B, T, T def = B, T B, T, [,T]. In view of 9.5, he dynamics of he process F B,T,T are given by df B, T, T =F B, T, T ã, T d + b, T b, T dw, where for every [,T] ã, T =a, T a, T b, T b, T b, T.

141 14 CHAPTER 9. MODELS OF FORWARD RATES We know ha any probabiliy measure equivalen o P on Ω, F T saisfies d ˆP dp = E T h u dw u, P-a.s. 9.9 for some predicable R d -valued process h. Le us fix a mauriy dae T. I is easily seen from Girsanov s heorem and he dynamics of F B, T, T ha F B, T, T follows a maringale 1 under ˆP, provided ha for every [,T] a, T a, T = b, T h b, T b, T. 9.1 In order o exclude arbirage opporuniies beween all bonds wih differen mauriies, i suffices o assume ha a maringale measure ˆP can be chosen simulaneously for all mauriies. The following condiion is hus sufficien for he absence of arbirage beween all bonds. As usual, we resric our aenion o he class of admissible rading sraegies. Condiion M.1 There exiss an adaped R d -valued process h such ha E P {E } T h u dw u =1 and, for every T T, equaliy 9.1 is saisfied, or equivalenly T T α, u du T T σ, u du 2 + h T T σ, u du =. By aking he parial derivaive wih respec o T, we obain T α, T +σ, T h + σ, u du =, 9.11 T for every T T. For any process h of condiion M.1, he probabiliy measure ˆP given by 9.9 will laer be inerpreed as he forward maringale measure for he dae T see Sec Assume now, in addiion, ha one may inves also in he savings accoun given by 9.4. In view of 9.5, he relaive bond price Z, T =B, T /B saisfies under P dz, T = Z, T α, T 1 2 σ, T 2 d + σ, T dw. The following no-arbirage condiion excludes arbirage no only across all bonds, bu also beween bonds and he savings accoun. Condiion M.2 There exiss an adaped R d -valued process λ such ha E P {E } T λ u dw u =1 and, for any mauriy T T, we have α, T = 1 2 σ, T 2 σ, T λ. Differeniaion of he las equaliy wih respec o T gives α, T =σ, T σ, T λ, [,T], The maringale propery of F B, T, T, as opposed o he local maringale propery,follows from he assumed boundedness of σ.

142 9.1. HJM MODEL 141 which holds for any T T. A probabiliy measure P, which saisfies dp dp = E T λ u dw u, P-a.s. for some process λ saisfying M.2, can be seen as a spo maringale measure for he HJM model; in his conex, he process λ is associaed wih he risk premium. Define a P -Brownian moion W by seing W = W λ u du, [,T]. The nex resul, whose proof is sraighforward, deals wih he dynamics of bond prices and ineres raes under he spo maringale measure P. Corollary For any fixed mauriy T T, he dynamics of he bond price B, T under he spo maringale measure P are db, T =B, T r d σ, T dw, 9.13 and he forward rae f, T saisfies df, T =σ, T σ, T d + σ, T dw Finally, he shor-erm ineres rae r = f, is given by he expression r = f,+ σu, σ u, du + σu, dw u I follows from 9.15 ha he expecaion of he fuure shor-erm rae under he spo maringale measure P does no equal he curren value f,t of he insananeous forward rae; ha is, f,t E P r T, in general. We shall see soon ha f,t equals he expecaion of r T under he forward maringale measure for he dae T see Corollary In view of 9.13, he relaive bond price Z, T =B, T /B saisfies and hus or equivalenly Z, T =B,T exp ln B, T =lnb,t+ dz, T = Z, T σ, T dw, 9.16 σ u, T dw u 1 2 r u 1 2 σ u, T 2 du σ u, T 2 du, σ u, T dw u. I is no hard o check ha, under mild echnical assumpions, he no-arbirage condiions M.1 and M.2 are equivalen. We assume from now on ha he following assumpion is saisfied. HJM.3 No-arbirage condiion M.1 or equivalenly, M.2 is saisfied. I is no essenial o assume ha he maringale measure for he bond marke is unique, so long as we are no concerned wih he compleeness of he model. Recall ha if a marke model is arbiragefree, any aainable claim admis a unique arbirage price anyway i is uniquely deermined by he replicaing sraegy, wheher a marke model is complee or no.

143 142 CHAPTER 9. MODELS OF FORWARD RATES 9.2 Forward Measure Approach The aim of his secion is o describe he specific feaures ha disinguish he arbirage valuaion of coningen claims wihin he classic Black-Scholes framework from he pricing of opions on socks and bonds under sochasic ineres raes. We assume hroughou ha he price B, T of a zerocoupon bond of mauriy T T T > is a fixed horizon dae follows an Iô process under he maringale measure P 2 db, T =B, T r d + b, T dw, 9.17 wih BT,T = 1, where W denoes a d-dimensional sandard Brownian moion defined on a filered probabiliy space Ω, F, P, and r sands for he insananeous, coninuously compounded rae of ineres. In oher words, we ake for graned he exisence of an arbirage-free family B, T of bond prices associaed wih a cerain process r which models he shor-erm ineres rae. Moreover, i is implicily assumed ha we have already consruced an arbirage-free model of a marke in which all bonds of differen mauriies, as well as a cerain number of oher asses called socks in wha follows, are primary raded securiies. I should be sressed ha he way in which such a consrucion is achieved is no relevan for he resuls presened in wha follows. In paricular, he concep of he insananeous forward ineres rae, which is known o play an essenial role in he HJM mehodology, is no employed. As already menioned, in addiion o zero-coupon bonds, we shall also consider oher primary asses, referred o as socks in wha follows. The dynamics of a sock price S i,i=1,...,m, under he maringale measure P are given by he following expression ds i = S i r d + σ i dw, S i >, 9.18 where σ i represens he volailiy of he sock price S i. Unless explicily saed oherwise, for every T and i, he bond price volailiy b, T and he sock price volailiy σ i are assumed o be R d -valued, bounded, adaped processes. Generally speaking, we assume ha he prices of all primary securiies follow sricly posiive processes wih coninuous sample pahs. I should be observed, however, ha cerain resuls presened in his secion are independen of he paricular form of bond and sock prices inroduced above. We denoe by π X he arbirage price a ime of an aainable coningen claim X which seles a ime T. Therefore π X =B E P XB 1 T F, [,T], 9.19 by virue of he sandard risk-neural valuaion formula. In 9.19, B represens he savings accoun given by 7.8. Recall ha he price B, T of a zero-coupon bond which maures a ime T admis he following represenaion cf. 8.2 B, T =B E P B 1 T F, [,T], 9.2 for any mauriy T T. Suppose now ha we wish o price a European call opion, wih expiry dae T, which is wrien on a zero-coupon bond of mauriy U>T.The opion s payoff a expiry equals C T =BT,U K +, so ha he opion price C a any dae T is C = B E P B 1 T BT,U K+ F. To find he opion s price using he las equaliy, we need o know he join condiional probabiliy law of F T -measurable random variables B T and BT,U. The echnique which was developed o circumven his sep is based on an equivalen change of probabiliy measure. I appears ha i is possible o find a probabiliy measure P T such ha he following holds C = B, T E PT BT,U K + F. 2 The reader may find i convenien o assume ha he probabiliy measure P is he unique maringale measure for he family B, T, T T ; his is no essenial,however.

144 9.2. FORWARD MEASURE APPROACH 143 Consequenly, C = B, T E PT FB T,U,T K + F, where F B, U, T is he forward price a ime, for selemen a he dae T, of he U-mauriy zero-coupon bond see formula If b, U b, T is a deerminisic funcion, hen he forward price F B, U, T can be shown o follow a lognormal maringale under P T ; herefore, a Black-Scholes-like expression for he opion s price is available Forward Price Recall ha a forward conrac is an agreemen, esablished a he dae <T,o pay or receive on selemen dae T a preassigned payoff, say X, a an agreed forward price. I should be emphasized ha here is no cash flow a he conrac s iniiaion and he conrac is no marked o marke. We may and do assume, wihou loss of generaliy, ha a forward conrac is seled by cash on dae T. Therefore, a forward conrac wrien a ime wih he underlying coningen claim X and prescribed selemen dae T>may be summarized by he following wo basic rules: a a cash amoun X will be received a ime T, and a preassigned amoun F X, T of cash will be paid a ime T ; b he amoun F X, T should be predeermined a ime according o he informaion available a his ime in such a way ha he arbirage price of he forward conrac a ime is zero. In fac, since nohing is paid up fron, i is naural o admi ha a forward conrac is worhless a is iniiaion. We adop he following formal definiion of a forward conrac. Definiion Le us fix T T. A forward conrac wrien a ime on a ime T coningen claim X is represened by he ime T coningen claim G T = X F X, T ha saisfies he following condiions: a F X, T isaf -measurable random variable; b he arbirage price a ime of a coningen claim G T equals zero, i.e., π G T =. The random variable F X, T is referred o as he forward price of a coningen claim X a ime for he selemen dae T. The coningen claim X may be defined in paricular as a preassigned amoun of he underlying financial asse o be delivered a he selemen dae. For insance, if he underlying asse of a forward conrac is one share of a sock S, hen clearly X = S T. Similarly, if he asse o be delivered a ime T is a zero-coupon bond of mauriy U T, we have X = BT,U. Noe ha boh S T and BT,U are aainable coningen claims in our marke model. The following well-known resul expresses he forward price of a claim X in erms of is arbirage price π X and he price B, T of a zero-coupon bond which maures a ime T. Lemma The forward price F X, T a ime T, for he selemen dae T, of an aainable coningen claim X equals F X, T = E P XB 1 T F E P B 1 T F = π X B, T Proof. I is sufficien o observe ha π G T = B E P G T B 1 T F = B E P XB 1 T F F X, T E P B 1 T F where he las equaliy follows by condiion b of Definiion This proves he firs equaliy; he second follows immediaely from Le us examine he wo ypical cases of forward conracs menioned above. If he underlying asse for delivery a ime T is a zero-coupon bond of mauriy U T, hen 9.21 becomes F BT,U, T = =, B, U, [,T] B, T

145 144 CHAPTER 9. MODELS OF FORWARD RATES On he oher hand, he forward price of a sock S S sands hereafer for S i for some i equals F ST, T = S, [,T] B, T For he sake of breviy, we shall wrie F B, U, T and F S, T insead of F BT,U, T and F ST, T, respecively. More generally, for any radable asse Z, we wrie F Z, T o denoe he forward price of he asse ha is, F Z, T =Z /B, T for [,T] Forward Maringale Measure To he bes of our knowledge, wihin he framework of arbirage valuaion of ineres rae derivaives, he mehod of a forward risk adjusmen was pioneered under he name of a forward risk-adjused process in Jamshidian 1987 he corresponding equivalen change of probabiliy measure was hen used by Jamshidian 1989a in he Gaussian framework. The formal definiion of a forward probabiliy measure was explicily inroduced in Geman 1989 under he name of forward neural probabiliy. In paricular, Geman observed ha he forward price of any financial asse follows a local maringale under he forward neural probabiliy associaed wih he selemen dae of a forward conrac. Mos resuls in his secion do no rely on specific assumpions imposed on he dynamics of bond and sock prices. We assume ha we are given an arbirage-free family B, T of bond prices and he relaed savings accoun B. Noe ha by assumpion, <B,T=E P B 1 T <. Definiion A probabiliy measure P T on Ω, F T equivalen o P, wih he Radon-Nikodým derivaive given by he formula dp T dp = B 1 T E P B 1 T = 1 B T B,T, P -a.s., 9.24 is called he forward maringale measure or briefly, he forward measure for he selemen dae T. Noice ha he above Radon-Nikodým derivaive, when resriced o he σ-field F, saisfies for every [,T] def η = dp T dp 1 = E P F = B, T F B T B,T B B,T. When he bond price is governed by 9.17, an explici represenaion for he densiy process η is available. Namely, we have η = exp bu, T dwu 1 bu, T 2 du In oher words, η = E U T, where U T he formula W T = W = bu, T dw u. Furhermore, he process W T given by bu, T du, [,T], 9.26 follows a sandard Brownian moion under he forward measure P T. We shall someimes refer o W T as he forward Brownian moion for he dae T. The nex resul shows ha he forward price of a European coningen claim X which seles a ime T can be easily expressed in erms of he condiional expecaion under he forward measure P T. Lemma The forward price a for he dae T of an aainable coningen claim X which seles a ime T equals F X, T =E PT X F, [,T], 9.27 provided ha X is P T -inegrable. In paricular, he forward price process F X, T, [,T], follows a maringale under he forward measure P T.

146 9.2. FORWARD MEASURE APPROACH 145 Proof. The Bayes rule yields E PT X F = E P η T X F E P η T F = E P η T η 1 X F, 9.28 where η T = dp T dp = 1 B T B,T and η = E P η T F. Combining 9.21 wih 9.28, we obain he desired resul. Under 9.17, 9.28 can be given a more explici form, namely E PT X F =E P The following equaliies: { T X exp bu, T dwu 1 T } bu, T 2 du F. 2 F B, T, U =E PT BT,U F, < T U T, and F S, T =E PT S T F [,T], are immediae consequences of he las lemma. More generally, he relaive price of any raded securiy which pays no coupons or dividends follows a local maringale under he forward probabiliy measure P T, provided ha he price of a bond which maures a ime T is aken as a numeraire. The nex lemma esablishes a version of he risk-neural valuaion formula ha is ailored o he sochasic ineres rae framework. Lemma The arbirage price of an aainable coningen claim X which seles a ime T is given by he formula π X =B, T E PT X F, [,T] Proof. Equaliy 9.29 is an immediae consequence of 9.21 combined wih For a more direc proof, noe ha he price π X can be re-expressed as follows An applicaion of he Bayes rule yields π X =B E P XB 1 T F =B B,T E P η T X F. π X = B B,T E PT X F E P η T F = B B,T E PT X F E P = B, T E PT X F, 1 B T B,T F as expeced. The following corollary deals wih a coningen claim which seles a ime U T. Our aim is o express he value of his claim in erms of he forward measure for he dae T. Corollary Le X be an arbirary aainable coningen claim which seles a ime U. i If U T, hen he price of X a ime U equals π X =B, T E PT XB 1 U, T F. 9.3 ii If U T and X is F T -measurable, hen for any U we have π X =B, T E PT XBT,U F. 9.31

147 146 CHAPTER 9. MODELS OF FORWARD RATES Proof. Boh equaliies are inuiively clear. In case i, we inves a ime U a F U -measurable payoff X in zero-coupon bonds which maure a ime T. For he second case, observe ha in order o replicae a F T -measurable claim X a ime U, i is enough o purchase, a ime T, X unis of a zero-coupon bond mauring a ime U. Boh sraegies are manifesly self-financing, and hus he resul follows. An alernaive way of deriving 9.3 is o observe ha since X is F U -measurable, we have for every [,U] B E P X B T BU, T F { X = B E P B U BU, T E BU } P FU F B T X = B E P F. B U This means ha he claim X ha seles a ime U has, a any dae [,U], an idenical arbirage price o he claim Y = XB 1 U, T ha seles a ime T. Formula 9.3 now follows from relaion 9.29 applied o he claim Y. Similarly, o prove he second formula, we observe ha since X is F T -measurable, we have for [,T] X X BT } B E P F = B E P { E P FT F B U B T B U XBT,U = B E P F. B T We conclude once again ha a F T -measurable claim X which seles a ime U T is essenially equivalen o a claim Y = XBT,U which seles a ime T. 9.3 Gaussian HJM Model In his secion, we assume ha he volailiy σ of he forward rae is deerminisic; such a case will be referred o as he Gaussian HJM model. This erminology refers o he fac ha he forward rae f, T and he spo rae r have Gaussian probabiliy laws under he maringale measure P cf Our aim is o show ha he arbirage price of any aainable ineres rae-sensiive claim can be evaluaed by each of he following procedures. I We sar wih arbirary dynamics of he forward rae such ha condiion M.1 or M.2 is saisfied. We hen find a maringale measure P, and apply he risk-neural valuaion formula. II We assume insead ha he underlying probabiliy measure P is acually he spo forward, respecively maringale measure. In oher words, we assume ha condiion M.2 condiion M.1, respecively is saisfied, wih he process λ h, respecively equal o zero. Since boh procedures give he same valuaion resuls, we conclude ha he specificaion of he risk premium is no relevan in he conex of arbirage valuaion of ineres rae-sensiive derivaives in he Gaussian HJM framework. Pu anoher way, when he coefficien σ is deerminisic, we can assume, wihou loss of generaliy, ha α, T =σ, T σ, T. Observe ha by combining he las equaliy wih 9.6, we find immediaely ha a, T =f, =r. To formulae a resul which jusifies he consideraions above, we need o inroduce some addiional noaion. Le a funcion α be given by 9.12, wih λ =, i.e., α, T =σ, T σ, T, [,T], 9.32 so ha α, T = T α u, T du = 1 2 σ, T 2.

148 9.3. GAUSSIAN HJM MODEL 147 Finally, we denoe by B, T he bond price specified by he equaliy T B, T = exp f, u du, where he dynamics under P of he insananeous forward rae f, T are f, T =f,t+ α u, T du + σu, T dw u. Le us pu Z, T =B, T /B, Z, T =B, T /B, where T =T 1,...,T k is any finie collecion of mauriy daes. Proposiion Suppose ha he coefficien σ is deerminisic. Then for any choice T of mauriy daes and of a spo maringale measure P, he probabiliy law of he process Z, T, [,T ], under he maringale measure P coincides wih he probabiliy law of he process Z, T, [,T ], under P. Proof. The asserion easily follows by Girsanov s heorem. Indeed, for any fixed <T T, he dynamics of Z, T under a spo maringale measure P = P λ are dz, T = Z, T σ, T dw λ, 9.33 where W λ follows a sandard Brownian moion under P λ. On he oher hand, under P we have Moreover, for every <T T dz, T = Z, T σ, T dw Z,T=B,T=e T f,u du = B,T=Z,T. Since σ is deerminisic, he asserion follows easily from Example Le us assume ha he volailiy of each forward rae is consan, i.e., independen of he mauriy dae and he level of he forward ineres rae. Taking d =1, we hus have σ, T =σ for a sricly posiive consan σ>. By virue of 9.14, he dynamics of he forward rae process f, T under he maringale measure are given by he expression df, T =σ 2 T d + σdw, 9.35 so ha he dynamics of he bond price B, T are db, T =B, T r d σt dw, where he shor-erm rae of ineres r saisfies I follows from he las formula ha r = f,+ 1 2 σ2 2 + σw. dr = f T,+σ 2 d + σdw. Since his agrees wih he general form of he coninuous-ime Ho-Lee model, we conclude ha in he HJM framework, he Ho-Lee model corresponds o he consan volailiy of forward raes. Dynamics 9.35 make apparen ha he only possible movemens of he yield curve in he Ho-Lee model are parallel shifs; ha is, all raes along he yield curve flucuae in he same way. The price

149 148 CHAPTER 9. MODELS OF FORWARD RATES B, T of a bond mauring a T equals i follows from 9.36 ha bond prices of all mauriies are perfecly correlaed B, T = B,T B, exp 1 2 T Tσ2 T σw I can also be expressed in erms of r, namely B, T = B,T B, exp T f, 1 2 T 2 σ 2 T r. Example I is a convenional wisdom ha forward raes of longer mauriy flucuae less han raes of shorer mauriy. To accoun for his feaure in he HJM framework, we assume now ha he volailiy of a forward rae is a decreasing funcion of he ime o is effecive dae. For insance, we may assume ha he volailiy srucure is exponenially dampened: σ, T =σe γt, [,T], for sricly posiive real numbers σ, γ >. Then σ, T equals and consequenly σ, T = T σe γu du = σγ 1 e γt 1, 9.37 df, T = σ 2 γ 1 e γt e γt 1 d + σe γt dw I is hus clear ha for any mauriy T, he bond price B, T saisfies db, T =B, T r d + σγ 1 e γt 1 dw Subsiuing 9.37 ino 9.15, we obain r = f, σ 2 γ 1 e γ u e γ u 1 du + σe γ u dw u, so ha, as in he previous example, he negaive values of he shor-erm ineres rae are no excluded. Denoing m =f,+ σ2 1 e γ 2, 2γ 2 we arrive a he following formula r = m+ σe γ u dw u, so ha finally dr =a γr d + σdw, 9.4 where a =γm+m. This means ha 9.37 leads o a generalized version of Vasicek s model cf Noice ha in he presen framework, he perfec fi of he iniial erm srucure is rivially achieved.

150 9.4. MODEL OF LIBOR RATES Model of LIBOR Raes The Heah-Jarrow-Moron mehodology of erm srucure modelling presened in he previous secion is based on he arbirage-free dynamics of insananeous, coninuously compounded forward raes. The assumpion ha insananeous raes exis is no always convenien, since i requires a cerain degree of smoohness wih respec o he enor i.e., mauriy of bond prices and heir volailiies. An alernaive consrucion of an arbirage-free family of bond prices, making no reference o he insananeous, coninuously compounded raes, is in some circumsances more suiable. By definiion, he forward δ-libor rae 3 L, T for he fuure dae T T δ prevailing a ime is given by he convenional marke formula 1+δL, T =F B, T, T + δ, [,T] The forward LIBOR rae L, T represens he add-on rae prevailing a ime over he fuure ime inerval [T,T + δ]. We can also re-express L, T direcly in erms of bond prices, as for any T [,T δ], we have 1+δL, T = B, T, [,T] B, T + δ In paricular, he iniial erm srucure of forward LIBOR raes saisfies L,T=f s,t,t + δ =δ 1 B,T B,T + δ Under he forward measure P T +δ, we have dl, T =δ 1 F B, T, T + δ γ, T, T + δ dw T +δ, where W T +δ and P T +δ are ye unspecified. This means ha L,T solves he equaion dl, T =δ δl, T γ, T, T + δ dw T +δ, 9.44 subjec o he iniial condiion Suppose ha forward LIBOR raes L, T are sricly posiive. Then formula 9.44 can be rewrien as follows dl, T =L, T λ, T dw T +δ, 9.45 where for any [,T] λ, T = 1+δL, T δl, T γ, T, T + δ The consrucion of a model of forward LIBOR raes relies on he following assumpions. LR.1 For any mauriy T T δ, we are given a R d -valued, bounded, deerminisic funcion 4 λ,t, which represens he volailiy of he forward LIBOR rae process L,T. LR.2 We assume a sricly decreasing and sricly posiive iniial erm srucure P,T,T [,T ], and hus an iniial erm srucure L,T of forward LIBOR raes L,T=δ 1 P,T P,T + δ 1, T [,T δ] In pracice,several ypes of LIBOR raes occur,e.g.,3-monh LIBOR and 6-monh LIBOR. For ease of exposiion, we consider a fixed mauriy δ. 4 Volailiy λ could follow a sochasic process; we deliberaely focus here on a lognormal model of forward LIBOR raes in which λ is deerminisic.

151 15 CHAPTER 9. MODELS OF FORWARD RATES Discree-enor Case We sar by sudying a discree-enor version of a lognormal model of forward LIBOR raes. I should be sressed ha a so-called discree-enor model sill possesses cerain coninuous-ime feaures; for insance, forward LIBOR raes follow coninuous-ime processes. For ease of noaion, we shall assume ha he horizon dae T is a muliple of δ, say T = Mδ for a naural M. We shall focus on a finie number of daes, Tm = T mδ for m =1,...,M 1. The consrucion is based on backward inducion, herefore we sar by defining he forward LIBOR rae wih he longes mauriy, L, T1. We posulae ha he rae L, T1 is governed under he probabiliy measure P by he following SDE dl, T1 =L, T1 λ, T1 dw, 9.48 wih he iniial condiion L,T1 =δ 1 P,T1 P,T Pu anoher way, we posulae ha for every [,T1 ] L, T1 =δ 1 P,T1 P,T 1 E λu, T1 dw u. 9.5 Since P,T 1 >P,T, i is clear ha L, T 1 follows a sricly posiive coninuous maringale under P. Also, for any fixed T 1, he random variable L, T 1 has a lognormal probabiliy law under P. The nex sep is o define he forward LIBOR rae for he dae T 2, γ, T 1,T = δl, T 1 1+δL, T 1 λ, T 1, [,T 1 ] Given ha he volailiy γ, T 1,T is deermined by 9.51, he forward process F B, T 1,T is known o solve, under P df B, T 1,T =F B, T 1,T γ, T 1,T dw 9.52 and he iniial condiion is F B,T 1,T =P,T 1 /P,T. The forward process F B, T 1,T is a coninuous maringale under P, since he volailiy γ, T 1,T follows a bounded process. We inroduce a d-dimensional process W T 1, which corresponds o he dae T 1, by seing W T 1 = W γu, T 1,T du, [,T 1 ] Due o he boundedness of he process γ, T1,T, he exisence of he process W T 1 and of he associaed probabiliy measure P T 1, equivalen o P, under which he process W T 1 follows a Browinian moion, and which is given by he formula dp T 1 dp = E T 1 γu, T1,T dw u, P-a.s., 9.54 is rivial. The process W T 1 may be inerpreed as he forward Brownian moion for he dae T 1. We are in a posiion o specify he dynamics of he forward LIBOR rae for he dae T2 under he forward probabiliy measure P T 1. Analogously o 9.48, we se dl, T 2 =L, T 2 λ, T 2 dw T 1, 9.55 wih he iniial condiion L,T 2 =δ 1 P,T 2 P,T

152 9.4. MODEL OF LIBOR RATES 151 Solving equaion 9.55 and comparing wih 9.46 for T = T2, we obain γ, T2,T1 = δl, T 2 1+δL, T2 λ, T 2, [,T2 ] To find γ, T2,T, we make use of he relaionship γ, T2,T1 =γ, T2,T γ, T1,T, [,T2 ] Given he process γ, T2,T1, we can define he pair W T 2, PT 2 corresponding o he dae T2 and so forh. By working backwards o he firs relevan dae TM 1 = δ, we consruc a family of forward LIBOR raes L, Tm,m=1,...,M 1. Noice ha he lognormal probabiliy law of every process L, Tm under he corresponding forward probabiliy measure P T m 1 is ensured. Indeed, for any m =1,...,M 1, we have dl, Tm=L, Tm λ, Tm dw T m 1, 9.59 where W T m 1 is a sandard Brownian moion under PT m 1. This complees he derivaion of he lognormal model of forward LIBOR raes in a discree-enor framework. Noe ha in fac we have simulaneously consruced a family of forward LIBOR raes and a family of associaed forward processes. Le us now examine he exisence and uniqueness of he implied savings accoun, in a discree-ime seing. The implied savings accoun is hus seen as a discree-ime process, B,=,δ,...,T = Mδ. Inuiively, he value B of a savings accoun a ime can be inerpreed as he cash amoun accumulaed up o ime by rolling over a series of zero-coupon bonds wih he shores mauriies available. To find he process B in a discree-enor framework, we do no have o specify explicily all bond prices; he knowledge of forward bond prices is sufficien. Indeed, F B, T j,t j+1 = F B, T j,t F B, T j+1,t = B, T j B, T j+1, where we wrie T j = jδ. This in urn yields, upon seing = T j F B T j,t j,t j+1 =1/BT j,t j+1, 9.6 so ha he price BT j,t j+1 of a one-period bond is uniquely specified for every j. Though he bond ha maures a ime T j does no physically exis afer his dae, i seems jusifiable o consider F B T j,t j,t j+1 as is forward value a ime T j for he nex fuure dae T j+1. In oher words, he spo value a ime T j+1 of one cash uni received a ime T j equals B 1 T j,t j+1. The discree-ime savings accoun B hus equals B T k = k k F B Tj 1,T j 1,T j = B 1 T j 1,T j j=1 for k =,...,M 1, since by convenion B =1. Noe ha F B Tj,T j,t j+1 =1+δLTj,T j+1 > 1 for j =1,...,M 1, and since BT j+1 = F B T j,t j,t j+1 BT j, we find ha BT j+1 >BT j for every j =,...,M 1. We conclude ha he implied savings accoun B follows a sricly increasing discree-ime process. We define he probabiliy measure P P on Ω, F T by he formula dp dp = B T P,T, P-a.s The probabiliy measure P appears o be a plausible candidae for a spo maringale measure. Indeed, if we se BT l,t k =E P BT l /BT k F Tl 9.62 for every l k M, hen in he case of l = k 1, equaliy 9.62 coincides wih 9.6. j=1

153 152 CHAPTER 9. MODELS OF FORWARD RATES Coninuous-enor Case By a coninuous-enor model we mean a model in which all forward LIBOR raes L, T wih T [,T ] are specified. Given he discree-enor skeleon consruced in he previous secion, i is sufficien o fill he gaps beween he discree daes o produce a coninuous-enor model. To consruc a model in which each forward LIBOR rae L, T follows a lognormal process under he forward measure for he dae T + δ, we shall proceed by backward inducion. Firs sep. We consruc a discree-enor model using he previously described mehod. Second sep. We focus on he forward raes and forward measures for mauriies T T1,T. In his case we do no have o ake ino accoun he forward LIBOR raes L, T such raes do no exis in he presen model afer he dae T1. From he previous sep, we are given he values BT 1 and BT of a savings accoun. I is imporan o observe ha B T and B 1 T are F T1 -measurable random variables. We sar by defining a spo maringale measure P associaed wih he discreeenor model, using formula Since he model needs o mach a given iniial erm srucure, we search for an increasing funcion α :[T1,T ] [, 1] such ha αt1 =,αt =1, and he process ln B =1 α ln BT + 1 αlnb T, [T 1,T ], saisfies P,=E P 1/B for every [T1,T ]. Since <BT <B 1 T, and P,, [T 1,T ], is assumed o be a sricly decreasing funcion, a funcion α wih desired properies exiss and is unique. Third sep. In he previous sep, we have consruced he savings accoun BT for every T [T1,T ]. Hence he forward measure for any dae T T1,T can be defined by seing dp T dp = 1 BT P,T, P -a.s Combining 9.63 wih 9.61, we obain dp T dp = dp T dp dp dp = B T P,T BT P,T, P-a.s., for every T [T1,T ], so ha dp T B = E T P,T P dp F BT P,T F, [,T]. Exponenial represenaion of he above maringale ha is, he formula dp T dp F = P,T P,T E γu, T, T dwu, [,T], yields he forward volailiy γ, T, T for any mauriy T T1,T. This in urn allows us o define also he associaed P T -Brownian moion W T. Given he forward measure P T and he associaed Brownian moion W T, we define he forward LIBOR rae process L, T δ for arbirary T T1,T by seing cf where T δ = T δ, wih iniial condiion Finally, we se cf dl, T δ =L, T δ λ, T δ dw T, L,T δ =δ 1 P,T δ P,T 1. γ, T 1,T = δl, T 1 1+δL, T 1 λ, T 1, [,T 1 ],

154 9.5. MODEL OF FORWARD SWAP RATES 153 hence we are in a posiion o inroduce also he forward measure P T for he dae T = T1. To define he forward measure P U and he corresponding Brownian moion W U for any mauriy U T2,T1, we pu cf γ, U, T =γ, T δ,t= δl, T δ 1+δL, T δ λ, T δ, [,T δ ], where U = T δ so ha T = U + δ belongs o T1,T. The coefficien γ, U, T is found from he relaionship γ, U, T =γ, U, T γ, T, T, [,U]. Proceeding by backward inducion, we are able o specify a fully coninuous-ime family L, T of forward LIBOR raes wih desired properies. mild regulariy assumpions, his sysem can be solved recursively. 9.5 Model of Forward Swap Raes Le us now describe briefly he model of forward swap raes developed by Jamshidian Generally speaking, he goal is o consruc a lognormal model of he erm srucure of forward swap raes wih a fixed end dae. Le us observe ha he lognormal model of forward LIBOR raes of Secion 9.4 and he lognormal model of forward swap raes inroduced below are inconsisen wih each oher. Formally, for a given collecion of daes T j = jδ, j =1,...,M, we consider a forward sar fixed-for-floaing ineres rae swap see Sec which sars a ime T j and has M j accrual periods. The forward swap rae κ, T j,m j ha is, ha value of a fixed rae κ for which such a swap is worhless a ime is known o be given by he expression cf κ, T j,m j =B, T j B, T M δ M l=j+1 1 B, T l for every [,T j ] and every j =1,...,M 1. We consider a family of forward swap raes κ, T j =κ, T j,m j, [,T j ], for j =1,...,M 1; ha is, he underlying swaps differ in lengh, bu have a common expiry dae, T = T M. Le us denoe Tk = T kδ, in paricular T = T. The forward swap rae for he dae Tm equals κ, Tm= B, Tm B, T δ B, Tm B, T, [,T m] Suppose ha bond prices B, Tm, m=,...,m 1, are given on a probabiliy space Ω, F, P equipped wih a Brownian moion W. We find i convenien o assume ha P = P T is he forward measure for he dae T, and W = W T is he corresponding forward Brownian moion. For any m =1,...,M 1, we inroduce he coupon process Gm by seing G m = m 1 k= B, T k, [,T m 1] By definiion, he forward swap measure P T m 1 for he dae T m 1 is ha probabiliy measure equivalen o P, which corresponds o he choice of he process Gm as a numeraire. In oher words, for a fixed m and any k =,...,M 1, he relaive bond prices Z m, Tk def = B, T k B, Tk = G m B, Tm B, T

155 154 CHAPTER 9. MODELS OF FORWARD RATES for [,T k T m] are bound o follow a local maringale under he forward swap measure P T m 1. Since obviously G 1 = B, T, i is clear ha Z 1, T k =F B, T k,t, [,T k ], and hus he probabiliy measure P T can be chosen o coincide wih he forward maringale measure P T. More noiceably, i follows from ha he forward swap rae κ, T m is also a local maringale under P T m 1, since i equals κ, T m=δ 1 Z m, T m Z m, T, [,T m]. As already menioned, our aim is o direcly consruc a model of forward swap raes; he underlying bond price processes will no be explicily specified. For he sake of concreeness, we shall focus on he lognormal version of he model of forward swap raes; his resricion is no essenial, however. We assume ha we are given a family of bounded deerminisic funcions ν,tm:[,t m] R, m=1,...,m 1, which represen he volailiies of he forward swap raes. In addiion, an iniial erm srucure, represened by a family P,Tm, m=,...,m 1, of bond prices, is known. Our goal is o consruc a model of forward swap raes in such a way ha d κ, T m+1 = κ, T m+1ν, T m+1 d W T m 9.66 for every m =,...,M 2, where W T m is a Brownian moion under he corresponding forward swap measure P T m. The model should be consisen wih he iniial erm srucure, meaning ha κ,t m+1 = P,T m+1 P,T δ P,T m+...+ P,T We proceed by backward inducion. The firs sep is o inroduce he forward swap rae κ, T 1 by seing noe ha W T = W T = W d κ, T 1 = κ, T 1 ν, T 1 d W T 9.68 wih he iniial condiion κ,t1 = P,T 1 P,T δp,t. To specify he process κ,t2, we need firs o inroduce a forward swap measure P T 1 and an associaed Brownian moion W T 1. The following auxiliary lemma is a sraighforward consequence of Iô s formula. Lemma Le G and H be real-valued adaped processes, such ha dg = G g dw and dh = H h dw. Assume ha H> 1. Then he process Y = G /1 + H saisfies dy = Y g H h dw H h d. 1+H 1+H In he nex sep, our aim is o define he process κ,t2. Noice ha each process Z 1,Tk = F B,Tk,T follows a sricly posiive local maringale under P T = P T ; more precisely, we have dz 1, T k =Z 1, T k γ 1, T k dw T 9.69 for some process γ 1,Tk. According o he definiion of a forward swap measure, we posulae ha for every k, he process Z 2, T k = B, T k B, T 1 +B, T = Z 1, T k 1+Z 1, T 1

156 9.5. MODEL OF FORWARD SWAP RATES 155 follows a local maringale under P T 1. Applying Lemma o processes G = Z 1,Tk and H = Z 1,T1, we see ha for his propery o hold, i is enough o assume ha he process W T 1, which equals W T 1 = W T Z 1 u, T1 γ 1 u, T1 1+Z 1 u, T1 du, [,T1 ], follows a Brownian moion under P T 1 probabiliy measure P T 1 is ye unspecified, bu can be found easily using Girsanov s heorem. Noe ha Z 1, T 1 = B, T 1 B, T = δ κ, T 1 +Z 1, T =δ κ, T Differeniaing boh sides of he las equaliy, we ge cf Z 1, T 1 γ 1, T 1 =δ κ, T 1 ν, T 1. Consequenly, W T 1 is explicily given by he formula W T 1 = W T δ κu, T1 δ κu, T1 +2νu, T 1 du, [,T1 ]. We may now define, using Girsanov s heorem, he associaed forward swap measure P T 1. We are hus in a posiion o define he process κ,t 2, which solves he SDE d κ, T 2 = κ, T 2 ν, T 2 d W T wih he iniial condiion κ,t2 = P,T 2 P,T δ P,T1 +P,T. For he reader s convenience, le us consider one more inducive sep, in which we are looking for κ,t3. We now consider processes so ha Z 3, T k = B, Tk B, T2 +B, T 1 +B, T = Z 2, Tk 1+Z 2, T2, W T 2 = W T 1 for every [,T 2 ]. I is crucial o noe ha Z 2, T 2 = Z 2 u, T2 1+Z 2 u, T2 γ 2u, T2 du B, T 2 B, T 1 +B, T = δ κ, T 2 +Z 2, T, where in urn Z 2, T Z 1, T = δ κ, T1 +Z 1, T +1 and he process Z 1,T is already known from he previous sep.

157 156 CHAPTER 9. MODELS OF FORWARD RATES Le us now urn o he general case. We assume ha we have found forward swap raes κ,t1,..., κ,tm, he swap forward measure P T m 1, and he associaed Brownian moion W T m 1. Our aim is o deermine he forward swap measure P T m, he associaed Brownian moion W T m, and, of course, he forward swap rae κ,tm+1. We posulae ha processes Z m+1, T k = B, Tk B, Tm+...+ B, T = Z m, Tk 1+Z m, Tm follow local maringales under P T m. In view of Lemma 9.5.1, applied o processes G = Z m,t k and H = Z m,t m, i is clear ha we may se W T m = W T for [,T m]. Therefore, i is sufficien o analyze he process Z m, T m= Z m u, T m 1+Z m u, Tm γ mu, Tm du 9.71 B, T m B, T m B, T = δ κ, T m+z m, T. Observe ha Z m, T Z m 1, T = δ κ, Tm 1 +Z m 1, T +1 and, from he preceding sep, he process Z m 1,T is a raional funcion of forward swap raes κ,t1,..., κ,tm 1. Consequenly, he process under he inegral sign on he righ-hand side of 9.71 can be expressed using he erms κ,t1,..., κ,tm 1 and heir volailiies since he explici formula is raher involved, we do no repor i here. Having found he process W T m and probabiliy measure P T m, we inroduce he forward swap rae κ,tm+1 hrough , and so forh. Remarks. I is worhwhile o noice ha lognormal models of forward LIBOR and swap raes can be easily generalized o he case of accrual periods wih variable lengh. In marke pracice, he lengh of accrual periods of caps or swaps is known o vary slighly, even in he case when i is formally defined as a fixed ime inerval for insance, a quarer of he year may amoun equally well o 89, 9or 91 days.

158 Chaper 1 Opion Valuaion in Gaussian Models In his chaper, he forward measure mehodology is employed in arbirage pricing of ineres rae derivaive securiies in a Gaussian framework. By a Gaussian framework we mean any model of he erm srucure, eiher based on he shor-erm rae or on forward raes, in which all bond price volailiies as well as he volailiy of any oher underlying asse follow deerminisic funcions. This assumpion is made for exposiional simpliciy; i is no a necessary condiion in order o obain a closed-form soluion for he price of a paricular opion, however. For insance, when a European opion on a specific asse is examined in order o obain an explici expression for is arbirage price, i is in fac enough o assume ha he volailiy of he forward price of he underlying asse for he selemen dae coinciding wih he opion s mauriy dae is deerminisic. This chaper is organized as follows. In he firs secion, we examine ypical quesions relaed o he valuaion of European opions on socks, zero-coupon bonds and coupon-bearing bonds. As already indicaed, we posulae ha he bond price volailiies, as well as he volailiy of he opion s underlying asse, follow deerminisic funcions. The nex secion is devoed o he sudy of fuures prices and o arbirage valuaion of fuures opions. 1.1 European Spo Opions The firs sep owards explici valuaion of European opions is o observe ha Lemma provides a simple formula which expresses he price of a European call opion wrien on a radable asse, Z say, in erms of he forward price process F Z, T and he forward probabiliy measure P T. Indeed, we have for every [,T] π ZT K + = B, T E PT FZ T,T K + F, 1.1 since manifesly Z T = F Z T,T. To evaluae he condiional expecaion on he righ-hand side of 1.1, we need o find firs he dynamics, under he forward probabiliy measure P T, of he forward price F Z, T. The following auxiliary resul is an easy consequence of and Lemma For any fixed T>, he process W T given by he formula W T = W bu, T du, [,T], 1.2 follows a sandard d-dimensional Brownian moion under he forward measure P T. The forward price process for he selemen dae T of a zero-coupon bond which maures a ime U saisfies df B, U, T =F B, U, T b, U b, T dw T,

159 158 CHAPTER 1. OPTION VALUATION IN GAUSSIAN MODELS subjec o he erminal condiion F B T,U,T=BT,U. The forward price of a sock S saisfies F S T,T=S T, and df S, T =F S, T σ b, T dw T. 1.4 The nex resul, which uses he HJM framework, shows ha he yield-o-mauriy expecaions hypohesis cf. Sec. 8.1 is saisfied for any fixed mauriy T under he corresponding forward probabiliy measure P T. This feaure is merely a disan reminder of he classic hypohesis ha for every mauriy T, he insananeous forward rae f,t is an unbiased esimae, under he acual probabiliy P, of he fuure shor-erm rae r T. Corollary For any fixed T [,T ], he forward rae f,t is equal o he expeced value of he spo rae r T under he forward probabiliy measure P T. Proof. Observe ha in view of 9.15, we have r T = f,t+ T σ, T σ, T d + dw =f,t+ T since σ, T = b, T. Therefore, E PT r T =f,t, as expeced. σ, T dw T, Bond Opions For he reader s convenience, we shall examine separaely opions wrien on zero-coupon bonds and on socks a general valuaion resul is given in Proposiion A expiry dae T, he payoff of a European call opion wrien on a zero-coupon bond which maures a ime U T equals C T = BT,U K Since BT,U=F B T,U,T, he payoff C T can alernaively be re-expressed in he following way C T =F B T,U,T K + = F B T,U,TI D K I D, where D = {BT,U >K} = {F B T,U,T >K} is he exercise se. The nex proposiion provides a closed-form expression for he arbirage price of a European bond opion. Valuaion resuls of his form were derived previously by several auhors, including El Karoui and Roche 1989, Amin and Jarrow 1992, Brace and Musiela For he sake of exposiional simpliciy, we assume ha he volailiies are bounded; however, his assumpion can be weakened. Proposiion Assume ha he bond price volailiies b,t and b,u are bounded deerminisic funcions. The arbirage price a ime [,T] of a European call opion wih expiry dae T and srike price K, wrien on a zero-coupon bond which maures a ime U T, equals C = B, UN h 1 B, U,,T KB, T N h 2 B, U,,T, 1.6 where h 1,2 b,, T = lnb/k ln B, T ± 1 2 v2 U, T v U, T for b, R + [,T], and 1.7 v 2 U, T = T bu, U bu, T 2 du, [,T]. 1.8 The arbirage price of he corresponding European pu opion wrien on a zero-coupon bond equals P = KB, T N h 2 B, U,,T B, UN h 1 B, U,,T.

160 1.1. EUROPEAN SPOT OPTIONS 159 Proof. In view of he general valuaion formula 1.1, i is clear ha we have o evaluae he condiional expecaions C = B, T E PT FB T,U,TI D F KB, T PT {D F } = I 1 I 2. We known from Lemma ha he dynamics of F B, U, T under P T are given by formula 1.3, so ha T F B T,U,T=F B, U, T exp γu, U, T dwu T 1 2 where γu, U, T =bu, U bu, T. This can be rewrien as follows T F B T,U,T=F B, U, T exp ζ, T 1 2 v2 U, T, γu, U, T 2 du, where F B, U, T isf -measurable, and ζ, T = T γu, U, T dw T u is, under P T, a real-valued Gaussian random variable, independen of he σ-field F, wih zero expeced value and he variance Var PT ζ, T = vu 2, T. Using he properies of condiional expecaion, we obain P T {D F } = P T {ζ, T < ln F/K } 1 2 v2 U, T, where F = F B, U, T, so ha ln FB, U, T /K 1 2 I 2 = KB, T N v2 U, T. v U, T To evaluae I 1, we inroduce an auxiliary probabiliy measure P T P T on Ω, F T by seing d P T dp T T = exp γu, U, T dwu T 1 T γu, U, T 2 def du = η T. 2 By Girsanov s heorem, i is clear ha he process W T, which equals W T = W T γu, U, T du, [,T], follows a sandard Brownian moion under P T. Noe also ha he forward price F B T,U,T admis he following represenaion under P T T F B T,U,T=F B, U, T exp γu, U, T d W u T + 1 T γu, U, T 2 du, 2 so ha F B T,U,T=F B, U, T exp ζ, T v2 U, T, 1.9 where we denoe ζ, T = T γu, U, T d W T u. The random variable ζ, T has under P T a Gaussian law wih zero mean value and variance vu 2, T, and i is also independen of he σ-field F. Furhermore, once again using Lemma 1.1.1, we obain I 1 = B, UE PT {I T D exp γu, U, T dwu T 1 T } γu, U, T 2 du F, 2 ha is I 1 = B, U E PT ηt η 1 I D F.

161 16 CHAPTER 1. OPTION VALUATION IN GAUSSIAN MODELS By virue of he Bayes rule see 9.28, we find ha I 1 = B, U P T {D F }. Taking ino accoun 1.9, we conclude ha P T {D F } = P T { ζ, T ln FB, U, T /K } v2 U, T, and hus ln FB, U, T /K I 1 = B, U N v2 U, T. v U, T This complees he proof of he valuaion formula 1.6. The formula ha gives he price of he pu opion can be esablished along he same lines. Alernaively, o find he price of a European pu opion wrien on a zero-coupon bond, one may combine equaliy 1.6 wih he pu-call pariy relaionship Formula 1.6 can be re-expressed as follows C = B, T F N d1 F,,T KN d2 F,,T, 1.1 where we wrie briefly F o denoe he forward price F B, U, T, and d 1,2 F,, T = lnf/k ± 1 2 v2 U, T v U, T 1.11 for F, R + [,T], where v U, T is given by 1.8. Noe also ha we have d P T dp = d P T dp T T dp T dp = exp bu, U dw u 1 T bu, U 2 du. 2 I is hus apparen ha he auxiliary probabiliy measure P T is in fac he resricion of he forward measure P U o he σ-field F T. Since he exercise se D belongs o he σ-field F T, we have P T {D F } = P U {D F }. Therefore, formula 1.6 admis he following alernaive represenaion Sock Opions C = B, UP U {D F } KB, T P T {D F } The payoff a expiry of a European call opion wrien on a sock S equals C T = S T K +, where T is he expiry dae and K denoes he srike price. The nex resul, which is a sraighforward generalizaion of he Black-Scholes formula, provides an explici soluion for he arbirage price of a sock call opion. We assume ha he dynamics of S under he maringale measure P are ds = S r d + σ dw, where σ :[,T ] R is a deerminisic funcion. Proposiion Assume ha he bond price volailiy b,t and he sock price volailiy σ are bounded deerminisic funcions. Then he arbirage price of a European call opion wih expiry dae T and exercise price K, wrien on a sock S, equals C = S N h 1 S,,T KB, T N h 2 S,,T, 1.13 where h 1,2 s,, T = lns/k ln B, T ± 1 2 v2, T v, T for s, R + [,T], and 1.14 v 2, T = T σ u bu, T 2 du, [,T]. 1.15

162 1.1. EUROPEAN SPOT OPTIONS 161 Proof. The proof goes along he same lines as he proof of Proposiion Example Le us examine a very special case of he pricing formula esablished in Proposiion Le W =W 1,W 2 be a wo-dimensional sandard Brownian moion given on a probabiliy space Ω, F, P. We assume ha he bond price B, T saisfies, under P db, T =B, T r d + ˆb, T ρ, 1 ρ 2 dw, where ˆb,T:[,T] R is a real-valued, bounded deerminisic funcion, and he dynamics of he sock price S are ds = S r d +ˆσ, dw for some funcion ˆσ :[,T ] R. Le us inroduce he real-valued sochasic processes Ŵ 1 and Ŵ 2 by seing Ŵ 1 = W 1 and Ŵ 2 = ρw ρ 2 W 2. I is no hard o check ha Ŵ 1 and Ŵ 2 follow sandard one-dimensional Brownian moions under he maringale measure P, and heir cross-variaion equals Ŵ 1, Ŵ 2 = ρ for [,T ]. I is eviden ha db, T =B, T r d + ˆb, T dŵ and ds = S r d +ˆσ dŵ An applicaion of Proposiion yields he following resul, firs esablished in Meron Corollary Assume ha he dynamics of a bond and a sock price are given by 1.16 and 1.17, respecively. If he volailiy coefficiens ˆb and ˆσ are deerminisic funcions, hen he arbirage price of a European call opion wrien on a sock S is given by , wih v 2, T = T ˆσ 2 u 2ρˆσuˆbu, T +ˆb 2 u, T du We are in a posiion o formulae a resul which encompasses boh cases sudied above. The dynamics of he spo price Z of a radable asse are assumed o be given by he expression dz = Z r d + ξ dw I is essenial o assume ha he volailiy ξ b, T of he forward price of Z for he selemen dae T is deerminisic. Proposiion The arbirage price of a European call opion wih expiry dae T and exercise price K, wrien on an asse Z, is given by he expression C = B, T F Z, T N d1 F Z, T,,T KN d2 F Z, T,,T, where for F, R + [,T], and d 1,2 F,, T = lnf/k ± 1 2 v2, T v, T 1.2 v 2, T = T ξ u bu, T 2 du, [,T] Le P sand for he price a ime T of a European pu opion wrien on an asse Z, wih expiry dae T and srike price K. Then he following useful resul is valid. The reader may find i insrucive o derive 1.22 by consrucing paricular rading porfolios.

163 162 CHAPTER 1. OPTION VALUATION IN GAUSSIAN MODELS Corollary The following pu-call pariy relaionship is valid C P = Z B, T K, [,T] Proof. We make use of he forward measure mehod. We have and hus for every [,T]. C P = B, T E PT FZ T,U,T K F, C P = B, T F Z, U, T B, T K = Z B, T K Opion on a Coupon-bearing Bond Our aim is o value a European opion wrien on a coupon-bearing bond. For a given selecion of daes T 1 <...<T m T, we consider a coupon-bearing bond whose value Z a ime T 1 is m Z = c j B, T j, [,T 1 ], 1.23 j=1 where c j are real numbers. We shall sudy a European call opion wih expiry dae T T 1, whose payoff a expiry has he following form m +. C T =Z T K + = c j BT,T j K 1.24 j=1 Proposiion The arbirage price of a European call opion on a coupon-bearing bond is given by he formula m C = c j B, T j J j 1 KB, T J 2, 1.25 where for j =1,...,m, J j 1 = Q { m l=1 j=1 l=1 } c l B, T l e ζ l+v lj v ll /2 >KB, T 1.26 { m } J 2 = Q c l B, T l e ζ l v ll /2 >KB, T, 1.27 and ζ 1,...,ζ m is a random variable whose law under Q is Gaussian, wih zero expeced value, and which has he following variance-covariance marix Cov Q ζ k,ζ l =v kl = T for k, l =1,...,m, where γ, T k,t=b, T k b, T. γu, T k,t γu, T l,t du 1.28 Proof. We need o evaluae he condiional expecaion m C = B, T c j E PT F B T,T j,ti D F KB, T P T {D F } = I 1 I 2, j=1 where D sands for he exercise se m m D = { c j BT,T j >K} = { c j F B T,T j,t >K}. j=1 j=1

164 1.1. EUROPEAN SPOT OPTIONS 163 Le us firs examine he condiional probabiliy P T {D F }. By virue of Lemma 1.1.1, he process F B =F B, T l,t saisfies T F B T =F B exp γu, T l,t dwu T 1 T γu, T l,t 2 du, 2 where γu, T l,t=bu, T l bu, T. In oher words, F B T,T l,t=f B, T l,t e ξt l v ll/2, where ξl T is a random variable independen of he σ-field F, and such ha he probabiliy law of under P T is he Gaussian law N,v ll. Therefore ξ T l P T {D F } = P T { m l=1 } c l B, T l e ξt l v2 ll /2 >KB, T. This proves ha I 2 = KB, T J 2. Le us show ha I 1 = m j=1 c j B, T j J j 1. To his end, i is sufficien o check ha for any fixed j we have B, T E PT FB T,T j,ti D F = B, Tj J j This can be done by proceeding in much he same way as in he proof of Proposiion Le us fix j and inroduce an auxiliary probabiliy measure P Tj on Ω, F T by seing Then he process d P Tj dp T T = exp γu, T j,t dwu T 1 T γu, T j,t 2 du. 2 W j = W T γu, T j,t du follows a sandard Brownian moion under P Tj. Recall ha P Tj = P Tj on F T, hence we shall wrie simply P Tj in place of P Tj in wha follows. For any l, he forward price F B, T l,t has he following represenaion under P Tj df B, T l,t=f B, T l,tγ, T l,t W j + γ, T j,t d. 1.3 For a fixed j, we define he random variable ξ 1,...,ξ m by he formula ξ l = T γu, T l,t d W j u. I is clear ha he random variable ξ 1,...,ξ m is independen of F, wih Gaussian law under P Tj. More precisely, he expeced value of each random variable ξ i is zero, and for every k, l =1,...,m, we have T v kl =Cov PTj ξ k,ξ l = γu, T k,t γu, T l,t du. On he oher hand, using 1.3 we find ha F B T,T l,t=f B, T l,t exp ξ l 1 2 v ll + v lj for every l =1,...,m. The Bayes rule yields B, T E PT FB T,T j,ti D F = B, Tj P Tj {D F }.

165 164 CHAPTER 1. OPTION VALUATION IN GAUSSIAN MODELS Furhermore, { m P Tj {D F } = P Tj c l B, T l exp ξ l 1 2 v ll + v lj >KB, T }. l=1 By combining he las wo equaliies, we arrive a The nex resul suggess an alernaive way o prove Proposiion Lemma Le us denoe D = {Z T >K}. Then he arbirage price of a European call opion wrien on a coupon bond saisfies m C = c j B, T j P Tj {D F } KB, T P T {D F } j=1 Proof. We have Z T = m j=1 c jb T E P B 1 T j F T, and hus C = B E P = { I D B 1 T m c j B E P j=1 m j=1 } c j B T E P B 1 T j F T K F B 1 T j I D F KB E P B 1 T I D F, since D F T. Using 9.28, we ge for every j B E P B 1 T j I D F = B B,T j E P ηtj I D F = B E PTj I D F E P B 1 T j F = B, T j P Tj {D F }. Since a similar relaion holds for he las erm, his ends he proof Pricing of General Coningen Claims Le us consider a European coningen claim X, which seles a ime T, of he form X = gz 1 T,...,Zn T, where g : R n R is a bounded Borel-measurable funcion. Assume ha he price process Z i of he i h asse saisfies, under P dz i = Z i r d + ξ i dw Then T F Z it,t=f Z i, T exp γ i u, T dwu T 1 T γ i u, T 2 du, 2 where γ i u, T =ξu i bu, T, or in shor F Z it,t=f Z i, T exp ζ i, T 1 2 γ ii, where ζ i, T = T γ i u, T dwu T and γ ii = T γ i u, T 2 du. The forward price F Z i, T isa random variable measurable wih respec o he σ-field F, while he random variable ζ i, T is independen of his σ-field. Moreover, i is clear ha he probabiliy disribuion under he forward measure P T of he vecor-valued random variable T T ζ 1, T,...,ζ n, T = γ 1 u, T dwu T,..., γ n u, T dwu T

166 1.1. EUROPEAN SPOT OPTIONS 165 is Gaussian N, Γ, where he enries of he n n marix Γ are γ ij = T γ i u, T γ j u, T du. Le us inroduce a k n marix Θ = [θ 1,...,θ n ] such ha Γ = Θ Θ. Proposiion Assume ha γ i is a deerminisic funcion for i =1,...,n. Then he arbirage price a ime [,T] of a European coningen claim X = gzt 1,...,Zn T which seles a ime T equals Z 1 π X =B, T g n k x + θ 1 R B, T n k k x,...,zn n k x + θ n n k x dx, B, T n k x where n k is he sandard k-dimensional Gaussian densiy n k x =2π k/2 e x 2 /2, x R k, and he vecors θ 1,...,θ n R k are such ha for every i, j =1,...,n, we have Proof. θ i θ j = T γ i u, T γ j u, T du. We have π X = B E P B 1 T gz1 T,...,ZT n F = B, T E PT gf Z 1T,T,...,F Z nt,t F = B, T J. In view of he definiion of he marix Θ, i is clear ha J = g F Z 1, T e θ1 x θ1 2 /2,...,F Z n, T e θn x θn 2 /2 n k x dx R k Z 1 = g n k x + θ 1 R B, T n k k x,...,zn n k x + θ n n k xdx. B, T n k x This ends he proof of he proposiion Replicaion of Opions In preceding secions, we have valued opions using a risk-neural valuaion approach, assuming implicily ha opions correspond o aainable claims. In his secion, we focus on he consrucion of a replicaing porfolio. Consider a coningen claim X which seles a ime T, and is represened by a P T -inegrable, sricly posiive random variable X. The forward price of X for he selemen dae T saisfies F X, T =E PT X F =F X,T+ F X u, T γ u dw T u 1.33 for some predicable process γ. Assume, in addiion, ha γ is a deerminisic funcion. Le us denoe F = F X, T. Our aim is o show, by means of a replicaing sraegy, ha he arbirage price of a European call opion wrien on a claim X, wih expiry dae T and srike price K, equals C = B, T F N d1 F,,T KN d2 F,,T, 1.34 where d 1 and d 2 are given by 1.2 wih v 2, T =vx 2, T = T γ u 2 du. Equaliy 1.34 yields he following expression for he forward price of he opion F C, T =F N d1 F,,T KN d2 F,,T. 1.35

167 166 CHAPTER 1. OPTION VALUATION IN GAUSSIAN MODELS Noe ha by applying Iô s formula o 1.35, we obain df C, T =N d 1 F,,T df Forward asse/bond marke. Le us consider a T -forward marke, i.e., a financial marke in which he forward conracs for selemen a ime T play he role of primary securiies. Consider a forward sraegy ψ =ψ 1,ψ 2, where ψ 1 and ψ 2 sand for he number of forward conracs on he underlying claim X and on he zero-coupon bond wih mauriy T, respecively. Observe ha he T -forward marke differs essenially from a fuures marke. The forward wealh process Ṽ of a T -forward marke porfolio ψ equals Ṽ ψ =ψ 1 F X, T +ψ 2 F B, T, T Since clearly F B, T, T = 1 for any [,T], a porfolio ψ is self-financing in he T -forward marke if is forward wealh saisfies dṽψ =ψ 1 df X, T =ψ 1 F X, T γ dw T, where he las equaliy follows from Our aim is o find he forward porfolio ψ ha replicaes he forward conrac wrien on he opion, and o subsequenly rederive pricing formulae To replicae he forward conrac wrien on he opion, i is enough o ake posiions in forward conracs on a claim X and in forward conracs on T -mauriy bonds. Suppose ha he opion s forward price equals F C, T =uf X, T, for some funcion u. Arguing along similar lines as in he firs proof of Theorem 3.2.1, wih consan ineres rae r = and ime-variable deerminisic volailiy γ, one may derive he following PDE u x, γ x 2 u xx x, =, wih ux, T =x K + for x R +. The soluion u o his problem is given by he formula ux, =xn d1 x,, T KN d2 x,, T. The corresponding sraegy ψ =ψ 1,ψ 2 in he T -forward marke is ψ 1 = u x FX, T, = N d1 F X, T,,T 1.38 and ψ 2 = uf X, T, ψ 1 F X, T. I can be checked, using Iô s formula, ha he sraegy ψ is self-financing in he T -forward marke; moreover, ṼT ψ =V T ψ =X K +. The forward price of he opion is hus given by 1.35, and consequenly is spo price a ime equals C = B, T Ṽψ =B, T uf X, T, The las formula coincides wih Forward/spo asse/bond marke. I may be convenien o replicae he erminal payoff of an opion by means of a combined spo/forward rading sraegy. Le he dae be fixed, bu arbirary. Consider an invesor who purchases a ime he number F C, T oft -mauriy bonds and holds hem o mauriy. In addiion, a any dae s she akes ψs 1 posiions in T -mauriy forward conracs on he underlying claim, where ψs 1 is given by The erminal wealh of such a sraegy a he dae T equals T F C, T + ψs 1 df X s, T =F C, T +ṼT ψ Ṽψ =X K +, since Ṽψ =F C, T and ṼT ψ =X K +. Spo asse/bond marke. To replicae an opion in a spo marke, we need o assume ha i is wrien on an asse which is radable in he spo marke. As he second asse, we use a T -mauriy

168 1.1. EUROPEAN SPOT OPTIONS 167 bond, wih he spo price B, T. Assume ha a claim X corresponds o he value Z T of a radable asse, whose spo price a ime equals Z. To replicae an opion in he spo marke, we consider he spo rading sraegy φ = ψ, where Z and a T -mauriy bond are primary securiies. We deduce easily from 1.39 ha he wealh V φ equals V φ =φ 1 Z + φ 2 B, T =B, T V ψ =C, so ha he sraegy φ replicaes he opion value a any dae T. I remains o check ha φ is self-financing. The following propery is a general feaure of self-financing sraegies in he T -forward marke: a T -forward rading sraegy ψ is self-financing if and only if he spo marke sraegy φ = ψ is self-financing. Replicaion of a European call opion wih erminal payoff Z T K + can hus be done using he spo rading sraegy φ =φ 1,φ 2, where φ 1 = N d1 FZ, T,,T and φ 2 =C φ 1 Z /B, T = KN d2 F,,T. Here, φ 1 and φ 2 represen he number of unis of he underlying asse and of T -mauriy bonds held a ime, respecively. Spo asse/cash marke. Le us show ha since a savings accoun follows a process of finie variaion, replicaion of an opion wrien on Z in he spo asse/cash marke is no always possible. Suppose ha ˆφ =ˆφ 1, ˆφ 2 is an asse/cash self-financing rading sraegy which replicaes an opion. In paricular, we have ˆφ 1 dz + ˆφ 2 db = dc. 1.4 On he oher hand, from he preceding paragraph, we know ha φ 1 dz + φ 2 db, T =dc = C r d + ξ C dw, 1.41 where ξ C =φ 1 Z ξ + φ 2 Z b, T /C. A comparison of maringale pars in 1.4 and 1.41 yields φ 1 Z ξ dw + φ 2 B, T b, T dw = ˆφ 1 Z ξ dw. When he underlying Brownian moion is mulidimensional, we canno solve he las equaliy for ˆφ 1, in general. If, however, W is one-dimensional and processes Z and ξ are sricly posiive, hen we have ˆφ 1 = φ 1 + φ 2 b, T B, T /ξ Z. We pu, in addiion, ˆφ 2 = B 1 C ˆφ 1 Z. I is clear ha such a sraegy replicaes he opion. Moreover, i is self-financing, since simple calculaions show ha ˆφ 1 dz + ˆφ 2 db = r C d + ξ C C dw = dc = dv ˆφ. For insance, he sock/cash rading sraegy ha involves a ime ˆφ 1 = N d1 F,,T K b, T B, T ξ Z N d2 F,,T shares of sock, and he amoun C ˆφ 1 Z held in a savings accoun, is a self-financing sraegy replicaing a European call opion wrien on Z.

169 168 CHAPTER 1. OPTION VALUATION IN GAUSSIAN MODELS 1.2 Fuures Prices Our nex goal is o esablish he relaionship beween forward and fuures prices. We consider an arbirary radable asse, whose spo price Z has he dynamics given by he expression dz = Z r d + ξ dw. The forward price of Z for selemen a he dae T is already known o saisfy T F Z T,T=F Z, T exp γ Z u, T dwu T 1 2 T γ Z u, T 2 du, where γ Z u, T =ξ u bu, T, and W T = W bu, T du is a Brownian moion under he forward measure P T. Since he maringale measure P for he spo marke is assumed o be unique, i is naural o inroduce he fuures price by means of he following definiion. Definiion The fuures price f Z, T of an asse Z, in he fuures conrac ha expires a ime T, is given by he formula f Z, T =E P Z T F, [,T] Equaliy 1.42 defines he fuures price of a coningen claim Z T which seles a ime T ; hence i applies o any coningen claim which seles a ime T. We are in a posiion o esablish he relaionship beween he forward and fuures prices of an arbirary asse. Proposiion Assume ha he volailiy γ Z,T=ξ b,t of he forward price process F Z, T follows a deerminisic funcion. Then he fuures price f Z, T equals T f Z, T =F Z, T exp bu, T ξu bu, T du Proof. I is clear ha T F Z T,T=F Z, T ζ exp bu, T ξu bu, T du, where ζ sands for he following random variable T ζ = exp ξu bu, T dwu 1 2 T ξ u bu, T 2 du. The random variable ζ is independen of he σ-field F, and is expecaion under P is equal o 1 ha is, E P ζ =1. Since by definiion f Z, T =E P Z T F =E P F Z T,T F, using he well-known properies of condiional expecaion, we obain T f Z, T =F Z, T exp bu, T ξu bu, T du E P ζ, which is he desired resul. Observe ha he dynamics of he fuures price process f Z, T, [,T], under he maringale measure P are df Z, T =f Z, T ξ b, T dw I is ineresing o noe ha he dynamics of he forward price F Z, T under he forward measure P T are given by he analogous expression df Z, T =F Z, T ξ b, T dw T. 1.45

170 1.2. FUTURES PRICES Fuures Opions We shall now focus on an explici soluion for he arbirage price of a European call opion wrien on a fuures conrac on a zero-coupon bond. Le us denoe by f B, U, T he fuures price for selemen a he dae T of a U-mauriy zero-coupon bond. From 1.44, we have df B, U, T =f B, U, T b, U b, T dw, 1.46 subjec o he erminal condiion f B T,U,T=BT,U. The wealh process V f ψ of any fuures rading sraegy ψ =ψ 1,ψ 2 equals V f ψ =ψ 2 B, T, [,T] A fuures rading sraegy ψ =ψ 1,ψ 2 is said o be self-financing if is wealh process V f = V f ψ saisfies he sandard relaionship V f ψ =V f ψ+ ψ 1 u df u + ψ 2 u dbu, T We fix U and T, and we wrie briefly f insead of f B, U, T in wha follows. he relaive wealh process Ṽ f Le us consider = V f ψb 1, T. As one migh expec, he relaive wealh of a self-financing fuures rading sraegy follows a local maringale under he forward measure P T. Indeed, using Iô s formula we ge dṽ f = B 1, T dv f + V f db 1, T +d V f,b 1,T, so ha dṽ f = B 1, T ψ 1 df + B 1, T ψ 2 db, T +ψ 2 B, T db 1, T + ψ 1 d f,b 1,T + ψ 2 d B,T,B 1,T = B 1, T ψ 1 df + ψ 1 d f,b 1,T. On he oher hand, we have d f,b 1,T = f B 1, T b, U b, T b, T d. Combining hese formulae, we arrive a he expression dṽ f ψ =ψ 1 f B 1, T b, U b, T dw b, T d, which is valid under P, or equivalenly, a he formula dṽ f ψ =ψ 1 f B 1, T b, U b, T dw T, 1.49 which in urn is saisfied under he forward probabiliy measure P T. We conclude ha he relaive wealh of any self-financing fuures sraegy follows a local maringale under he forward measure for he dae T. Therefore, o find he arbirage price π f X a ime [,T]ofanyP T -inegrable coningen claim 1 X of he form X = gf T,T, we can make use of he equaliy π f X =B, T E PT X F, [,T]. 1.5 To check his, noe ha if ψ is a fuures rading sraegy replicaing X, hen he process Ṽ f ψ is a P T -maringale, and hus E PT X F = E PT Ṽ f T ψ F = B 1, T V f ψ =B 1, T π f X. 1 As usual,i is implicily assumed ha a claim X is also aainable.

171 17 CHAPTER 1. OPTION VALUATION IN GAUSSIAN MODELS Proposiion Assume ha U T. The arbirage price a ime [,T] of a European call opion wih expiry dae T and exercise price K, wrien on he fuures conrac for a U-mauriy zero-coupon bond wih delivery dae T, equals C f = B, T f h U, T N g 1 f,,t KN g 2 f,,t, 1.51 where g 1 f,, T = lnf/k+ 1 T 2 bu, U 2 bu, T 2 du, 1.52 v U, T g 2 f,, T =g 1 f,, T v U, T, he funcion v U, T is given by 1.8, and T h U, T = exp bu, U bu, T bu, T du Proof. We need o evaluae C f = B, T E PT fb T,U,T K + F = B, T E PT FB T,U,T K + F. Proceeding as in he proof of Proposiion 1.1.1, we find ha cf C f = B, T F N d1 F,,T KN d2 F,,T, 1.54 where F = F B, U, T, d 1 F,, T = lnf/k+ 1 2 v2 U, T, 1.55 v U, T d 2 F,, T = d 1 F,, T v U, T, and v U, T is given by 1.8. On he oher hand, 1.43 yields F B, U, T =f B, U, T exp T bu, T bu, U bu, T du + f B, U, T h U, T. Subsiuing he las formula ino 1.54, we find he desired formula. For ease of noaion, we wrie f o denoe he fuures price f Z, T of a radable securiy Z. We assume ha he volailiies b,t and γ Z,T are deerminisic funcions. The proof of he nex resul is similar o ha of Proposiion 1.2.2, and is hus omied. Proposiion The arbirage price of a European call opion wih expiry dae T and srike price K, wrien on a fuures conrac which seles a ime T for delivery of one uni of securiy Z, is given by he formula C f = B, T f h, T N g 1 f,,t KN g 2 f,,t, 1.56 where g 1,2 f,, T = lnf/k+lnh, T ± 1 2 v2, T 1.57 v, T for f, R + [,T], he funcion v, T is given by 1.21, and T h, T = exp γ Z u, T bu, T du Le us examine he pu-call pariy for fuures opions. We wrie P f o denoe he price of a European pu fuures opion wih expiry dae T and srike price K. Arguing as in he proof of Proposiion 1.1.3, one may esablish he following resul. Proposiion Under he assumpions of Proposiion 1.2.3, he following pu-call pariy relaionship is valid C f P f T = B, T f Z, T exp γ Z u, T bu, T du K.

172 1.3. PDE APPROACH TO INTEREST RATE DERIVATIVES PDE Approach o Ineres Rae Derivaives This secion presens he PDE approach o he hedging and valuaion of coningen claims in he Gaussian HJM seing. As discussed in Chap. 8, PDEs play an imporan role in pricing of ermsrucure derivaives in he framework of diffusion models of shor-erm ineres raes. In such a case, one works wih he PDE saisfied by he price process of an ineres rae-sensiive securiy, considered as a funcion of he ime parameer and he curren value of a shor-erm rae r. In he presen seing, however, i is no assumed ha he shor-erm rae follows a diffusion process. The PDEs examined in his secion are direcly relaed o he price dynamics of bonds and underlying asses. To be more specific, he arbirage price of a derivaive securiy is expressed in erms of he ime parameer, he curren price of an underlying asse and he price of a cerain zero-coupon bond. For ease of exposiion, we focus on he case of spo and fuures European call opions for he proofs, see Rukowski PDEs for Spo Derivaives We sar by examining he case of a European call opion wih expiry dae T wrien on a radable asse Z. We assume hroughou ha he dynamics of he spo price process of Z are governed under a probabiliy measure P by he expression 2 dz = Z µ d + ξ dw, 1.59 where µ is a sochasic process. For a fixed dae D T, he price of a bond which maures a ime D is assumed o follow, under P db, D =B, D κ d + b, D dw, 1.6 where κ is a sochasic process. Volailiies ξ and b,d can also follow sochasic processes; we shall assume, however, ha he volailiy ξ b, D of he forward price of Z is deerminisic. We consider a European opion, wih expiry dae T, wrien on he forward price of Z for he dae D, where D T. More precisely, by definiion he opion s payoff a expiry equals C T = BT,DF Z T,D K + =Z T KBT,D +. When D = T, we deal wih a sandard opion wrien on Z. For D > T, he opion can be inerpreed eiher as an opion wrien on he forward price of Z, wih deferred payoff a ime D, or simply as an opion o exchange one uni of an asse Z for K unis of D-mauriy bonds. Proposiion Assume ha he price processes Z and B, D follow 1.59 and 1.6, respecively, and he volailiy ξ b, D of he forward price is deerminisic. Consider a European coningen claim X, of he form X = BT,DgZ T /BT,D, which seles a ime T. The arbirage price of X equals π X =v Z,B, D, = B, DH Z B 1, D, for every [,T], where he funcion H : R + [,T] R solves he following PDE H z,+1 2 ξ b, D 2 z 2 2 H z, =, z2 wih he erminal condiion Hz,T =gz for every z R +. Remarks. A savings accoun can be used in he replicaion of European claims which sele a ime T and have he form X = B T gz T /B T for some funcion g. Le us consider, for insance, 2 We assume implicily ha Z follows a sricly posiive process. I should be sressed ha P is no necessarily a maringale measure.

173 172 CHAPTER 1. OPTION VALUATION IN GAUSSIAN MODELS a European opion wih expiry dae T and erminal payoff Z T KB T +. If he volailiy of he underlying asse is deerminisic, Proposiion is in force, and hus replicaion of such an opion involves N k 1 Z,B,,T unis of he underlying asse, combined wih he amoun KB N k 2 Z,B,,T held in a savings accoun. In general, a sandard European opion canno be replicaed using a savings accoun PDEs for Fuures Derivaives Le us fix hree daes T,D and R, such ha T min {D, R}. The fuures price of an asse Z in a conrac which seles a ime R saisfies df Z, R =f Z, R ξ b, R dw We have assumed ha he drif coefficien in he dynamics of f Z, R vanishes. This is no essenial, however. Indeed, suppose ha a non-zero drif in he dynamics of he fuures price is presen. Then we may eiher modify all foregoing consideraions in a suiable way, or, more convenienly, we may firs make, using Girsanov s heorem, an equivalen change of an underlying probabiliy measure in such a way ha he drif of he fuures price will disappear. The drif of bond price will hus change however, i is arbirary, and i does no ener he final resul anyway. I is convenien o assume ha he volailiy ζ of he fuures price and he bond price volailiy b,r are deerminisic funcions. For convenience, we shall wrie f insead of f Z, R. Consider a European opion wih he erminal payoff C f T = BT,Df T K + a ime T, or equivalenly f T K + a ime D hence i is a sandard fuures opion wih deferred payoff. We are in a posiion o formulae a counerpar of Proposiion Proposiion Suppose ha he fuures price f = F Z, R of an asse Z saisfies 1.61, where he volailiy ζ = ξ b, R is such ha ζ b, D is a deerminisic funcion. Le X be a European coningen claim which seles a ime T and has he form X = BT,Dgf T for some funcion g : R + R. The arbirage price of X equals π f X =v f,b, D, = B, DL f e η,t, for every [,T], where η, T = T and he funcion L = Lz, solves he PDE ξ u bu, R bu, D du, [,T], L z,+1 2 ζ 2 z 2 2 L z, =, z2 wih he erminal condiion Lz,T =gz for every z R +. Example Assume ha a fuures conrac has a zero-coupon bond which maures a ime U R as he underlying asse. Then ξ u = bu, U and hus v 2 f, T = T Moreover, in his case we have wriing f in place of x bu, U bu, R 2 du. l 1 f,, T = lnf/k+ T γu, U, R bu, D du v, T T γu, U, R 2 du, where γu, U, R =bu, U bu, R. In paricular, if D = R = T we obain l 1 f,, T =g 1 f,, T and η, T =h, T, where g 1 and h are given by 1.52 and 1.53 respecively.

174 Chaper 11 Valuaion of Swap Derivaives The aim of his chaper is wofold. Firs, we give a general descripion of basic swap derivaives. We shall examine he mos ypical examples of ineres rae derivaives such as ineres rae swaps, caps, floors and swapions. Second, we provide he explici valuaion soluions for some of hese insrumens wihin he Gaussian HJM framework and in he case of lognormal models of LIBOR. In he las secion, an alernaive model of he erm srucure, pu forward by Jamshidian 1997, is presened Ineres Rae Swaps Le us consider a forward sar payer swap seled in arrears, wih noional principal N. We consider a finie collecion of daes T j,j=,...,n, where, for simpliciy, T j T j 1 = δ for every j =1,...,n. The floaing rae LT j received a ime T j+1 is se a ime T j by reference o he price of a zerocoupon bond over ha period namely, LT j saisfies BT j,t j+1 1 =1+T j+1 T j LT j =1+δLT j Formula 11.1 agrees wih marke quoaions of LIBOR; indeed, LT j is he spo LIBOR rae prevailing a ime T j for he period of lengh δ = T j+1 T j. More generally, he forward LIBOR rae L, T j for he fuure ime period [T j,t j+1 ] of lengh δ saisfies 1+δL, T j = B, T j B, T j+1 = F B, T j,t j+1, 11.2 so ha LT j coincides wih LT j,t j. A any dae T j,j=1,...n, he cash flows of a payer swap are LT j 1 δn and κδn, where κ is a preassigned fixed rae of ineres he cash flows of a receiver swap have he same size, bu opposie signs. The number n, which coincides wih he number of paymens, is referred o as he lengh of a swap, he daes T,...,T n 1 are known as rese daes, and he daes T 1,...,T n as selemen daes. We shall refer o he firs rese dae T as he sar dae of a swap. Finally, he ime inerval [T j 1,T j ] is referred o as he j h accrual period. We may and do assume, wihou loss of generaliy, ha he noional principal N = 1. The value a ime of a forward sar payer swap, which is denoed by FS or FS κ, equals { n B } FS κ = E P LT j 1 κδ F B Tj = n j=1 j=1 E P { B BT j 1,T j 1 B δ } F, Tj 173

175 174 CHAPTER 11. VALUATION OF SWAP DERIVATIVES where we wrie δ =1+κδ. Consequenly, FS κ = n j=1 E P n j=1 { BT j 1,T j 1 δ E P B B Tj F = B E P B Tj 1 BTj 1 B Tj FTj 1 F } n B, T j 1 δb, T j, j=1 which, afer rearranging, yields FS κ =B, T n c j B, T j 11.3 for every [,T], where c j = κδ for j =1,...,n 1, and c n = δ =1+κδ. The las equaliy makes clear ha a forward swap seled in arrears is, essenially, a conrac o deliver a specific coupon-bearing bond and o receive in he same ime a zero-coupon bond. This relaionship, which in fac may be inferred by a sraighforward comparison of he fuure cash flows from hese bonds, provides a simple mehod for he replicaion of a swap conrac. As menioned, a swap agreemen is worhless a iniiaion. This imporan feaure of a swap leads o he following definiion, which refers in fac o he more general concep of a forward swap. Basically, a forward swap rae is ha fixed rae of ineres which makes a forward swap worhless. Definiion The forward swap rae κ, T, n a ime for he dae T is ha value of he fixed rae κ which makes he value of he forward swap zero, i.e., ha value of κ for which FS κ =. Using 11.3, we obain j=1 κ, T, n =B, T B, T n δ n 1. B, T j 11.4 A swap swap rae, respecively is he forward swap forward swap rae, respecively wih = T. The swap rae, κt, T, n, equals κt,t,n=1 BT,T n δ j=1 n 1. BT,T j 11.5 Noe ha he definiion of a forward swap rae implicily refers o a swap conrac of lengh n which sars a ime T. I would hus be more correc o refer o κ, T, n as he n-period forward swap rae prevailing a ime, for he fuure dae T. A forward swap rae is a raher heoreical concep, as opposed o swap raes, which are quoed daily subjec o an appropriae bid-ask spread by financial insiuions who offer ineres rae swap conracs o heir insiuional cliens. In pracice, swap agreemens of various lenghs are offered. Also, ypically, he lengh of he reference period varies over ime; for insance, a 5-year swap may be seled quarerly during he firs hree years, and semi-annually during he las wo. For he sake of simpliciy, we assume hroughou ha all reference periods T j T j 1 of a swap are of he same lengh, which is denoed by δ. Swap raes also play an imporan role as a basis for several derivaive insrumens. For insance, an appropriae swap rae is commonly used as a srike level for an opion wrien on he value of a swap; ha is, a swapion. Remarks. Le us examine one leg of a swap ha is, an ineres rae swap agreemen wih only one paymen dae. For n =1, 11.4 gives κ, T, U = j=1 B, T B, U B, UU T,

176 11.2. GAUSSIAN MODEL 175 where we wrie U and κ, T, U insead of T 1 and κ, T, 1, respecively. Using 7.3, we find ha UY,U TY,T κ,t,u = f,t,u, U T where denoes approximae equaliy. This shows ha he swap rae does no coincide wih he forward ineres rae f, T, U deermined by a forward rae agreemen Gaussian Model The basic swap derivaives we shall now examine are: caps, floors, capions, swapions, opions on a swap rae spread, and yield curve swaps. Furher examples of swap derivaives will be discussed in he nex chaper Forward Caps and Floors An ineres rae cap known also as a ceiling rae agreemen, or briefly CRA is a conracual arrangemen where he granor seller has an obligaion o pay cash o he holder buyer if a paricular ineres rae exceeds a muually agreed level a some fuure dae or daes. Similarly, in an ineres rae floor, he granor has an obligaion o pay cash o he holder if he ineres rae is below a preassigned level. When cash is paid o he holder, he holder s ne posiion is equivalen o borrowing or deposiing a a rae fixed a ha agreed level. This assumes ha he holder of a cap or floor agreemen also holds an underlying asse such as a deposi or an underlying liabiliy such as a loan. Finally, he holder is no affeced by he agreemen if he ineres rae is ulimaely more favorable o him han he agreed level. This feaure of a cap or floor agreemen makes i similar o an opion. Specifically, a forward sar cap or a forward sar floor is a srip of caples floorles, each of which is a call pu opion on a forward rae, respecively. Le us denoe by κ and by δ he cap srike rae and he lengh of a caple, respecively. We shall check ha an ineres rae caple i.e., one leg of a cap may also be seen as a pu opion wih srike price 1 per dollar of noional principal which expires a he caple sar day on a discoun bond wih face value 1 + κδ which maures a he caple end dae. This propery makes he valuaion of a cap relaively simple; essenially, i can be reduced o he problem of opion pricing on zero-coupon bonds. Similarly o swap agreemens, ineres rae caps and floors may be seled eiher in arrears or in advance. In a forward cap or floor wih he noional principal N seled in arrears a daes T j,j=1,...,n, where T j T j 1 = δ and T = T, he cash flows a imes T j are NLT j 1 κ + δ and Nκ LT j 1 + δ, respecively. As usual, he rae LT j 1 is deermined a he rese dae T j 1, and i saisfies BT j 1,T j 1 =1+LT j 1 T j T j The arbirage price a ime T of a forward cap, denoed by FC, is we assume ha N =1 FC = n j=1 E P LT j 1 κ + δ F B Tj B Le us consider a caple i.e., one leg of a cap wih rese dae T and selemen dae T + δ. The value a ime of a caple equals { B +δ } Cpl = E P BT,T + δ 1 1δ 1 κ F B T +δ { B 1 = E P B T +δ BT,T + δ δ + } F { B 1 = E P B T BT,T + δ δ + B } E P FT F B T +δ

177 176 CHAPTER 11. VALUATION OF SWAP DERIVATIVES = E P { B 1 B δbt,t + } + δ F T = B, T E PT { 1 δbt,t + δ + F }, where he las equaliy was deduced from Lemma I is apparen ha a caple is a pu opion on a zero-coupon bond; i is also an opion on a one-period swap. Remarks. Since he cash flow of he j h caple a ime T j is manifesly a F Tj 1 -measurable random variable, we may use Corollary o express he value of he cap in erms of expecaions under forward measures. Indeed, from 9.31 we have FC = n B, T j 1 E PTj 1 BT j 1,T j LT j 1 κ + δ F j=1 Consequenly, using 11.6 we ge equaliy FC = n 1 B, T j 1 E PTj 1 δbtj 1,T j + F, 11.9 j=1 which is valid for every [,T]. The equivalence of a cap and a pu opion on a zero-coupon bond can be explained in an inuiive way. For his purpose, i is enough o examine wo basic feaures of boh conracs: he exercise se and he payoff value. Le us consider he j h caple. A caple is exercised a ime T j 1 if and only if LT j 1 κ>, or equivalenly, if BT j 1,T j 1 =1+LT j 1 T j T j 1 > 1+κδ = δ. The las inequaliy holds whenever δbt j 1,T j < 1. This shows ha boh of he considered opions are exercised in he same circumsances. If exercised, he caple pays δlt j 1 κ a ime T j, or equivalenly δbt j 1,T j LT j 1 κ =1 δbt j 1,T j = δ δ 1 BT j 1,T j a ime T j 1. This shows once again ha he caple wih srike level κ and nominal value 1 is essenially equivalen o a pu opion wih srike price 1 + κδ 1 and nominal value 1 + κδ wrien on he corresponding zero-coupon bond wih mauriy T j. We assume ha he bond price volailiy is a deerminisic funcion as before, such an assumpion is referred o as he Gaussian model. The following lemma is an immediae consequence of Proposiion Recall ha for any daes U, T [,T ], we denoe F B, U, T =B, U/B, T and γ, U, T =b, U b, T. Lemma For any T T δ, he arbirage price a ime [,U] of a caple wih expiry dae T, selemen dae T + δ, and srike level κ equals Cpl = B, T N e 1, T δf B, T + δ, T N e 2, T, 11.1 where and e 1,2, T = ln F B, T, T + δ ln δ ± 1 2 v2, T v, T v 2, T = T γu, T, T + δ 2 du. The nex resul provides a general pricing formula for a forward cap in he Gaussian case.

178 11.2. GAUSSIAN MODEL 177 Proposiion The price a ime T of an ineres rae cap wih srike level κ, seled in arrears a imes T j,j=1,...,n, equals FC = n j=1 B, T j 1 N e j 1 δf B, T j,t j 1 N e j 2, where and Proof. e j 1,2 =ln F B, T j 1,T j ln δ ± 1 2 v2 j, v j v 2 j = Tj 1 We represen he price of a forward cap in he following way n { B } FC = E P LT j 1 κ + δ F = = j=1 n j=1 n j=1 E P E P B Tj { B B Tj γu, T j 1,T j 2 du BTj 1,T j 1 1 +δ } δ 1 κ F { B 1 B δbt + } j 1,T j F = Tj 1 n Cpl j, j=1 where Cpl j sands for he price a ime of he j h caple. The asserion now follows from Lemma The price of a forward floor a ime [,T] equals n B FF = E P κ LT j 1 + δ F B Tj Using a rivial equaliy j=1 κ LT j 1 + δ =LT j 1 κ + δ LT j 1 κδ, we find ha he following cap-floor pariy relaionship is saisfied a any ime [,T] he hree conracs are assumed o have he same paymen daes Forward Cap Forward Floor = Forward Swap. This relaionship can also be verified by a sraighforward comparison of he corresponding cash flows of boh porfolios. By combining he valuaion formulae for caps and swaps, we find easily ha n FF = FC FS = δb, Tj N e j 2 B, T j 1 N e j 1. j=1 Le us menion ha by a cap floor, respecively, we mean a forward cap forward floor, respecively wih = T Capions Since a caple is essenially a pu opion on a zero-coupon bond, a European call opion on a caple is an example of a compound opion. More exacly, i is a call opion on a pu opion wih a zerocoupon bond as he underlying asse of he underlying pu opion. Therefore, he valuaion of a call

179 178 CHAPTER 11. VALUATION OF SWAP DERIVATIVES opion on a caple can be done along he same lines as in Chap. 6 provided, of course, ha he model of a zero-coupon bond price has sufficienly good properies. A call opion on a cap, or a capion, is hus a call on a porfolio of pu opions. To price a capion observe ha is payoff a expiry is n +, CC T = Cpl j T K j=1 where as usual Cpl j T sands for he price a ime T of he jh caple of he cap, T is he call opion s expiry dae and K is is srike price. Suppose ha we place ourselves wihin he framework of he spo rae models of Chap. 8 for insance, he Hull-Whie model. Typically, he caple price is an increasing funcion of he curren value of he spo rae r. Le r be he criical level of ineres rae, which is implicily deermined by he equaliy n j=1 Cpl j T r =K. I is clear ha he opion is exercised when he rae r T is greaer han r. Le us inroduce numbers K j by seing K j = Cpl j T r for j =1,...,n. I is easily seen ha he capion s payoff is equal o he sum of he payoffs of n call opions on paricular caples, wih K j being he corresponding srike prices. Consequenly, he capion s price CC a ime T is given by he formula n CC = C Cpl j,t,k j, j=1 where C Cpl j,t,k j is he price a ime of a call opion wih expiry dae T and srike level K j wrien on he j h caple see Hull and Whie An opion on a cap or floor can also be sudied wihin he Gaussian HJM framework see Brace and Musiela However, resuls concerning capion valuaion wihin his framework are less explici han in he case of he Hull-Whie model Swapions The owner of a payer receiver, respecively swapion wih srike rae κ, mauring a ime T = T, has he righ o ener a ime T he underlying forward payer receiver, respecively swap seled in arrears. Because FS T κ is he value a ime T of he payer swap wih he fixed ineres rae κ, i is clear ha he price of he payer swapion a ime equals More explicily, we have PS = E P For he receiver swapion, we have ha is RS = E P PS = E P { B n E P B T RS = E P { B + } FS T κ F. B T j=1 { B n E P B T B T + LT j 1 κδ F T F } B Tj { B + } FS T κ F, B T j=1 B T + κ LT j 1 δ F T F }, B Tj where we wrie RS o denoe he price a ime of a receiver swapion. We will now focus on a payer swapion. In view of 11.15, i is apparen ha a payer swapion is exercised a ime T if and only if he value of he underlying swap is posiive a his dae. I should be made clear ha a swapion may be exercised by is owner only a is mauriy dae T. If exercised, a swapion

180 11.2. GAUSSIAN MODEL 179 gives rise o a sequence of cash flows a prescribed fuure daes. By considering he fuure cash flows from a swapion and from he corresponding marke swap 1 available a ime T, i is easily seen ha he owner of a swapion is proeced agains he adverse movemens of he swap rae ha may occur before ime T. Suppose, for insance, ha he swap rae a ime T is greaer han κ. Then by combining he swapion wih a marke swap, he owner of a swapion wih srike rae κ is eniled o ener a ime T, a no addiional cos, a swap conrac in which he fixed rae is κ. If, on he conrary, he swap rae a ime T is less han κ, he swapion is worhless, bu is owner is, of course, able o ener a marke swap conrac based on he curren swap rae κt,t,n κ. Concluding, he fixed rae paid by he owner of a swapion who inends o iniiae a swap conrac a ime T will never be above he preassigned level κ. Since we may rewrie as follows PS = E P { B B T 1 n j=1 + } c j BT,T j F, he payer swapion may also be seen as a pu opion on a coupon-bearing bond wih he coupon rae κ. Similar remarks are valid for he receiver swapion. I follows easily from ha PS RS = FS, i.e., Payer Swapion Receiver Swapion = Forward Swap provided ha boh swapions expire a he same dae T and have he same conracual feaures. We shall now show ha a payer receiver, respecively swapion can also be viewed as a sequence of call pu, respecively opions on a swap rae which are no allowed o be exercised separaely. A ime T he long pary receives he value of a sequence of cash flows, discouned from ime T j,j=1,...,n, o he dae T, defined by d p j = δ κt,t,n κ+, d r j = δ κ κt,t,n +, for he payer opion and he receiver opion, respecively, where κt,t,n=1 BT,T n δ n j=1 1 BT,T j is he corresponding swap rae a he opion s expiry. Indeed, he price a ime of he call payer opion on a swap rae is C = E P = E P = E P { B n E P B T { B B T 1 j=1 B } T κt,t,n κ + δ F T F B Tj n + } c j BT,T j F j=1 { B n E P B T j=1 B T + LT j 1 κδ F T F }, B Tj which is he payer swapion price PS. Equaliy C = PS may also be derived by direcly verifying ha he fuure cash flows from he following porfolios esablished a ime T are idenical: porfolio A aswapion and a marke swap; and porfolio B anopion on a swap rae and a marke swap. Indeed, boh porfolios correspond o a payer swap wih he fixed rae equal o κ. Similarly, for every T, he price of he pu receiver opion on a swap rae is as before, c j = κδ, j =1,...,n 1, 1 A any ime, a marke swap is ha swap whose curren value equals zero. Pu more explicily,i is he swap in which he fixed rae κ equals he curren swap rae.

181 18 CHAPTER 11. VALUATION OF SWAP DERIVATIVES and c n =1+κδ P = E P = E P = E P { B n E P B T { B j=1 B } T κ κt,t,n + δ F T F B Tj n + } c j BT,T j 1 F B T j=1 { B n E P B T j=1 B T + κ LT j 1 δ F T F }, B Tj which equals he price RS of he receiver swapion. As menioned earlier, a payer receiver, respecively swapion may be seen as a pu call, respecively opion on a coupon bond wih srike price 1 and coupon rae equal o he srike rae κ of he underlying forward swap. Therefore, he arbirage price of payer and receiver swapions can be evaluaed by applying he general valuaion formula o he funcions g p x 1,...,x n = 1 n +, n + c j x j g r x 1,...,x n = c j x j 1 j=1 for a payer and a receiver swapion, respecively. Le us rederive he valuaion formula for he payer swapion in a more inuiive way. Recall ha a payer swapion is essenially a sequence of fixed paymens d p j = δκt,t,n κ+ which are received a selemen daes T 1,...,T n, bu whose value is known already a he expiry dae T. Therefore, he random variable d p j is F T -measurable, and hus we may direcly apply Corollary 9.2.1, which gives PS = B, T n j=1 j=1 E PT δbt,t j κt,t,n κ + F for every [,T]. Afer simple manipulaions, his yields, as expeced PS = B, T E PT {1 n j=1 + } c j BT,T j F Le us now consider a forward swapion. In his case, we assume ha he expiry dae ˆT of he swapion precedes he iniiaion dae T of he underlying payer swap ha is, ˆT T. Noice ha if κ is a fixed srike level, hen we have always FS κ =FS κ FS κ, T, n, as by he definiion of he forward swap rae we have FS κ, T, n=. A direc applicaion of valuaion resul 11.3 o boh members on he righ-hand side of he las equaliy yields FS κ = n κ, T, n κ B, Tj j=1 for [,T]. I is hus clear ha he payoff PS ˆT a expiry ˆT of he forward swapion wih srike is eiher, if κ κ ˆT,T,n, or PS ˆT = n κ ˆT,T,n κ B ˆT,Tj j=1

182 11.3. MODEL OF FORWARD LIBOR RATES 181 if, on he conrary, inequaliy κ ˆT,T,n >κholds. We conclude ha he payoff PS ˆT of he forward swapion can be represened in he following way n +B PS ˆT = κ ˆT,T,n κ ˆT,Tj j=1 This means ha, if exercised, he forward swapion gives rise o a sequence of equal paymens κ ˆT,T,n κ a each selemen dae T 1,...,T n. By subsiuing ˆT = T we recover, in a more inuiive way and in a more general seing, he previously observed dual naure of he swapion: i may be seen eiher as an opion on he value of a paricular forward swap or, equivalenly, as an opion on he corresponding forward swap rae. I is also clear ha he owner of a forward swapion is able o ener a ime ˆT a no addiional cos ino a forward payer swap wih preassigned fixed ineres rae κ. The following resul provides a quasi-explici formula for he arbirage price of a payer swapion in he Gaussian framework he price of a receiver swapion is given by an analogous expression. Formula 11.2 can be easily generalized o he case of a forward swapion. To his end, i is enough o consider he following claim which seles a ime ˆT cf n PS ˆT = B ˆT,T B j ˆT,T B ˆT,T + n j=1 δ n i=1 B ˆT,T κ. i To value such a claim wihin he Gaussian framework, i is enough o apply Proposiion As usual, we wrie n k o denoe he sandard k-dimensional Gaussian densiy funcion. Proposiion Assume he Gaussian model of he erm srucure of ineres raes. For [,T], he arbirage price of a payer swapion equals n +dx, PS = B, T n k x c i B, T i n k x + θ i 11.2 R k i=1 where n k is he sandard k-dimensional Gaussian probabiliy densiy funcion, and vecors θ 1,...,θ n R k saisfy for every i, j =1,...,n θ i θ j = T γu, T i,t γu, T j,t du Remarks. Traded caps and swapions are of American raher han European syle. More exacly, hey ypically have semi-american feaures, since exercising is allowed on a finie number of daes for insance, on rese daes. As a simple example of such a conrac, le us consider a Bermudan swapion. Consider a fixed collecion of rese daes T,...,T n 1 and an associaed family of exercise daes T 1,..., T k wih T i [T ji,t ji+1. I should be sressed ha he exercise daes are known in advance; ha is, hey canno be chosen freely by he long pary. A Bermudan swapion gives is holder he righ o ener a ime T m a forward swap which sars a T ji+1 and ends a ime T n, provided ha his righ has no already been exercised a a previous ime T p for some p<m. Le us observe ha Bermudan swapions arise as embedded opions in cancellable swaps Model of Forward LIBOR Raes The main moivaion for he inroducion of a lognormal model of LIBOR raes was he marke pracice of pricing caps and swapions by means of Black-Scholes-like expressions. For his reason, we shall firs describe how marke praciioners value caps and swapions. The formulae commonly used by praciioners assume ha he underlying insrumen follows a geomeric Brownian moion under some probabiliy measure, Q say. Since he formal definiion of his probabiliy measure is no available, we shall informally refer o Q as he marke probabiliy.

183 182 CHAPTER 11. VALUATION OF SWAP DERIVATIVES Caps Le us consider an ineres rae cap wih expiry dae T and fixed srike level κ. Marke pracice is o price he opion assuming ha he underlying forward ineres rae process is lognormally disribued wih zero drif. Le us firs consider a caple ha is, one leg of a cap. Assume ha he forward LIBOR rae L, T, [,T], for he period of lengh δ follows a geomeric Brownian moion under he marke probabiliy, Q say. More specifically dl, T =L, T σdw, where W follows a one-dimensional sandard Brownian moion under Q, and σ is a sricly posiive consan. The unique soluion of is L, T =L,T exp σw 1 2 σ2 2, [,T], where he iniial condiion is derived from he yield curve Y, T, namely 1+δL,T= B,T B,T + δ = exp T + δy,t + δ TY,T. The marke price a ime of a caple wih expiry dae T and srike level κ is calculaed by means of he formula Cpl = δb, T + δ E Q LT,T κ + F. More explicily, for any [,T]wehave Cpl = δb, T + δ L, T N ê 1, T κn ê 1, T, where ê 1,2, T = lnl, T /κ ± 1 2 ˆv2, T ˆv, T and ˆv, 2 T =σ 2 T. This means ha marke praciioners price caples using Black s formula, wih discoun from he selemen dae T + δ. A cap seled in arrears a imes T j,j=1,...,n, where T j T j 1 = δ, T = T, is priced by he formula FC = δ where for every j =,...,n 1 n j=1 B, T j L, T j 1 N ê j 1 κn ê j 2, ê j 1,2 =lnl, T j 1/κ ± 1 2 ˆv2 j ˆv j and ˆv 2 j =σ2 j T j 1 for some consans σ j,j=1,...,n. Apparenly, he marke assumes ha for any mauriy T j, he corresponding forward LIBOR rae has a lognormal probabiliy law under he marke probabiliy. As we shall see in wha follows, he valuaion resul obained for caps in he lognormal case agrees wih marke pracice. Recall ha in a general framework of sochasic ineres raes, he price of a forward cap equals see formula 11.7 FC = n E P j=1 B B Tj LTj 1 κ + δ F = n Cpl j, j=1 where Cpl j = B, T j E PTj LTj 1,T j 1 κ + δ F

184 11.3. MODEL OF FORWARD LIBOR RATES 183 for every j =1,...,n. We shall now examine he valuaion of caps wihin he lognormal model of forward LIBOR raes. The dynamics of he forward LIBOR rae L, T j 1 under he forward probabiliy measure P Tj are dl, T j 1 =L, T j 1 λ, T j 1 dw Tj, where W Tj follows a d-dimensional Brownian moion under he forward measure P Tj, and λ,t j 1 : [,T j 1 ] R d is a deerminisic funcion. Consequenly, under P Tj we have L, T j 1 =L,T j 1 E λu, T j 1 dwu Tj for [,T j 1 ]. Le us firs consider a paricular caple, wih expiry dae T and srike rae κ. Since he proof of he nex resul is sandard, i is lef o he reader. Lemma The price a ime [,T] of a caple wih srike rae κ, mauring a T, equals Cpl = δb, T + δ L, T N ẽ 1, T κn ẽ 2, T, where ẽ 1,2, T = lnl, T /κ ± 1 2 ṽ2, T ṽ, T and ṽ, 2 T = T λu, T 2 du. To he bes of our knowledge, he cap pricing formula was firs esablished in a formal way in Sandmann e al. 1995, who focused on he dynamics of he forward LIBOR rae for a given dae. Equaliy was subsequenly rederived in Brace e al. 1997, where a mehod of coninuous-ime modelling of all forward LIBOR raes was presened. The reader may find i insrucive o compare valuaion formula wih formula 11.1, which holds for a Gaussian case. The following proposiion is an immediae consequence of Lemma and formula Proposiion Consider an ineres rae cap wih srike level κ, seled in arrears a imes T j,j=1,...,n. Assuming he lognormal model of LIBOR raes, he price of a cap a ime [,T] equals n FC = δ B, T j L, T j 1 N ẽ j 1 κn ẽ j 2, where and ṽ 2 j = T j Swapions j=1 λu, T j 1 2 du. ẽ j 1,2 =lnl, T j 1/κ ± 1 2 ṽ2 j ṽ j The commonly used formula for pricing swapions, based on he assumpion ha he underlying swap rae follows a geomeric Brownian moion under he marke probabiliy Q, is given by he Black swapion formula PS = δ n j=1 B, T j κ, T, nn h 1, T κn h 2, T, 11.3 where h 1,2, T = lnκ, T, n/κ ± 1 2 σ2 T σ T

185 184 CHAPTER 11. VALUATION OF SWAP DERIVATIVES for some consan σ>. To examine equaliy 11.3 in an inuiive way, le us assume, for simpliciy, ha =. In his case, using general valuaion resuls, we obain he following equaliy PS = δ n B,T j E PTj κt,t,n κ +. j=1 Apparenly, marke praciioners assume lognormal probabiliy law for he forward swap rae κt,t,n under P Tj. The swapion valuaion formula obained in he framework of he lognormal model of LIBOR raes appears o be more involved. I reduces o he marke formula 11.3 only in very special circumsances. On he oher hand, he swapion price derived wihin he lognormal model of forward swap raes agrees wih he marke formula. To be more precise, his holds for a specific family of swapions. This is by no means surprising, as he model was exacly ailored o handle a paricular family of swapions, or raher, o analyze cerain pah-dependen swapions such as Bermudan swapions. The price of a cap in he lognormal model of swap raes is no given by a closed-form expression, however. Recall ha wihin he general framework, he price a ime [,T] of a payer swapion wih expiry dae T = T and srike level κ equals PS = E P { B n E P B T j=1 B T + LT j 1 κδ F T F }, B Tj or equivalenly PS = E P { B n E P B T j=1 B T κt,t,n κ + δ F T F }. B Tj Le D F T be he exercise se of a swapion; ha is n D = {ω Ω κt,t,n κ + > } = {ω Ω c j BT,T j < 1}. Lemma The following equaliy holds for every [,T] j=1 PS = δ n j=1 B, T j E PTj LT,T j 1 κi D F Proof. we have Since PS = E P PS = δ E P = δ { B n I D E P B T n j=1 j=1 B T LT j 1 κδ F T F }, B Tj { n B } E P LT j 1 κi D FT F B Tj j=1 B, T j E PTj LT j 1 κi D F, where LT j 1 =LT j 1,T j 1. For any j =1,...,n, we have E PTj LT j 1 κi D F = E PTj E PTj LTj 1 κ F T ID F = E PTj LT,T j 1 κi D F, since F F T and he process L, T j 1 isap Tj -maringale.

186 11.3. MODEL OF FORWARD LIBOR RATES 185 For any k =1,...,n, we define he random variable ζ k by seing We wrie also ζ k = λ 2 k = T T Noe ha for every k =1,...,n and [,T], we have λu, T k 1 dw T k u, [,T] λu, T k 1 2 du, [,T] LT,T k 1 =L, T k 1 e ζ k λ 2 k /2. Recall also ha he processes W T k saisfy he following relaionship W T k+1 = W T k + δlu, T k 1+δLu, T k λu, T k du, [,T k ] For ease of noaion, we formulae he nex resul for = only; a general case can be reaed along he same lines. For any fixed j, we denoe by G j he join probabiliy disribuion funcion of he n-dimensional random variable ζ 1,...,ζ n under he forward measure P Tj. Proposiion Assume he lognormal model of LIBOR raes. The price a ime of a payer swapion wih expiry dae T = T and srike level κ equals PS = δ n B,T j j=1 where I D =I Dy 1,...,y n, and D sands for he se R n L,T j 1 e yj λ2 j /2 κ I D dg j y 1,...,y n, D = {y 1,...,y n R n n j=1 c j j k=1 1+δL,T k 1 e y k λ 2 k /2 1 < 1 }. Proof. Le us sar by considering arbirary [,T]. Noice ha B, T j B, T = and hus, in view of 9.42, we have j k=1 BT,T j = B, T k B, T k 1 = j k=1 j F B, T k 1,T k 1 k=1 1+δLT,T k 1 1. Consequenly, he exercise se D can be re-expressed in erms of forward LIBOR raes. Indeed, we have { n j 1 } D = ω Ω 1+δLT,T k 1 < 1, or more explicily j=1 c j k=1 { n D = ω Ω j=1 c j j k=1 1+δL, T k 1 e ζ k λ 2 k /2 1 < 1 }.

187 186 CHAPTER 11. VALUATION OF SWAP DERIVATIVES Le us pu =. In view of Lemma , o find he arbirage price of a swapion a ime, i is sufficien o deermine he join law under he forward measure P Tj of he random variable ζ 1,...,ζ n, where ζ 1,...,ζ n are given by Noe also ha { n j D = ω Ω 1+δL,T k 1 e ζ k λ 2 /2 1 } k < 1. j=1 c j k=1 This shows he validiy of he swapion valuaion formula for =. I is clear ha his resul admis a raher sraighforward generalizaion o arbirary < T. When >, one needs o examine he condiional probabiliy law of ζ 1,...,ζ n wih respec o he σ-field F Model of Forward Swap Raes For any fixed, bu oherwise arbirary, dae T j,j=1,...,m 1, we consider a swapion wih expiry dae T j, wrien on a payer swap seled in arrears, wih fixed rae κ, which sars a dae T j and has M j accrual periods. The j h swapion may be seen as a conrac which pays o is owner he amoun δκt j,t j,m j κ + a each selemen dae T j+1,...,t M. Equivalenly, we may assume ha i pays an amoun Y = M k=j+1 BT j,t k δ κt j,t j,m j κ + a ime T j. Noe ha Y admis he following represenaion cf Y = δg Tj M j κt j,t j,m j κ +. Recall ha he lognormal model of forward swap raes specifies he dynamics of κ, T j,m j by means of he following SDE see Sec. 9.5 Tj+1 dκ, T j,m j =κ, T j,m jν, T j d W, where ν,t j :[,T j ] R d is a bounded deerminisic funcion, and W Tj+1 follows a sandard d-dimensional Brownian moion under P Tj+1. Furhermore, by he definiion of he forward swap measure P Tj+1, any process of he form B, T k /G M j is a local maringale under P Tj+1. From he general consideraions concerning he choice of a numeraire, i follows ha for any aainable claim X = gbt j,t j+1,...,bt j,t M, which seles a ime T j, he arbirage price π X saisfies π X =G M j E PTj+1 XG 1 T j M j F, [,Tj ]. Applying his equaliy o he swapion s payoff Y, we obain PS j = π Y =G M j E PTj+1 κtj,t j,m j κ + F, where we wrie PS j o denoe he price a ime of he j h swapion. Proposiion Assume he lognormal model of forward swap raes. For any j =1,...,M 1, he arbirage price PS j of he j h swapion a ime T j equals where PS j = δ and v 2 S, T j= T j M k=j+1 B, T k κ, T j,m jn h 1, T j κn h 2, T j, h 1,2, T j = lnκ, T j,m j/κ ± 1 2 v2 S, T j v S, T j νu, T j 2 du.

188 Bibliography [1] Adams, P.D., Wya, S.B Biases in opion prices: evidence from he foreign currency opion marke. J. Banking Finance 11, [2] Amin, K On he compuaion of coninuous ime opion prices using discree approximaions. J. Finan. Quan. Anal. 26, [3] Amin, K., Jarrow, R Pricing opions on risky asses in a sochasic ineres rae economy. Mah. Finance 2, [4] Arzner, P., Delbaen, F Term srucure of ineres raes: he maringale approach. Adv. in Appl. Mah. 1, [5] Baxer, M.W., Rennie A Financial Calculus. An Inroducion o Derivaive Pricing. Cambridge Universiy Press, Cambridge. [6] Bensoussan, A On he heory of opion pricing. Aca Appl. Mah. 2, [7] Bensoussan, A., Lions, J.-L Applicaions des inéquaions variaionelles en conrôle sochasique. Dunod, Paris. [8] Biger, N., Hull, J The valuaion of currency opions. Finan. Manag. 12, [9] Bergman, Y.Z., Grundy, D.B., Wiener, Z General properies of opion prices. J. Finance 51, [1] Björk, T., Di Masi, G., Kabanov, Y., Runggaldier, W. 1997a Towards a general heory of bond marke. Forhcoming in Finance Sochas. [11] Björk, T., Kabanov, Y., Runggaldier, W. 1997b Bond marke srucure in he presence of marked poin processes. Forhcoming in Mah. Finance. [12] Black, F Fac and fanasy in he use of opions. Finan. Analyss J. 314, 36 41, [13] Black, F The pricing of commodiy conracs. J. Finan. Econom. 3, [14] Black, F., Karasinski, P Bond and opion pricing when shor raes are lognormal. Finan. Analyss J. 474, [15] Black, F., Scholes, M The valuaion of opion conracs and a es of marke efficiency. J. Finance 27, [16] Black, F., Scholes, M The pricing of opions and corporae liabiliies. J. Poliical Econom. 81, ibiembd Black, F., Derman, E., Toy, W. 199 A one-facor model of ineres raes and is applicaion o Treasury bond opions. Finan. Analyss J. 461, [17] Booksaber, R.M Opion Pricing and Sraegies in Invesing. Addison-Wesley, Reading Mass. [18] Booksaber, R., Clarke, R Opion porfolio sraegies: measuremen and evaluaion. J. Business 57,

189 188 BIBLIOGRAPHY [19] Booksaber, R., Clarke, R Problems in evaluaing he performance of porfolios wih opions. Finan. Analyss J. 411, [2] Bouaziz, L., Briys, E., Crouhy, M The pricing of forward-saring asian opions. J. Banking Finance 18, [21] Boyle, P.P Opions: a Mone-Carlo approach. J. Finan. Econom. 4, [22] Boyle, P.P Opion valuaion using a hree jump process. Inerna. Opions J. 3, [23] Boyle, P.P A laice framework for opion pricing wih wo sae variables. J. Finan. Quan. Anal. 23, [24] Boyle, P.P., Evnine, J., Gibbs, S Numerical evaluaion of mulivariae coningen claims. Rev. Finan. Sud. 2, [25] Brace, A., Musiela, M A mulifacor Gauss Markov implemenaion of Heah, Jarrow, and Moron. Mah. Finance 4, [26] Brace, A., Musiela, M Swap derivaives in a Gaussian HJM framework. Forhcoming in Mahemaics of Derivaive Securiies, M.A.H.Dempser, S.R.Pliska, eds. Cambridge Universiy Press, Cambridge. [27] Brace, A., G aarek, D., Musiela, M The marke model of ineres rae dynamics. Mah. Finance 7, [28] Brennan, M.J The pricing of coningen claims in discree ime models. J. Finance 34, [29] Brennan, M.J., Schwarz, E.S The valuaion of American pu opions. J. Finance 32, [3] Brennan, M.J., Schwarz, E.S Finie-difference mehods and jump processes arising in he pricing of coningen claims: a synhesis. J. Finan. Quan. Anal. 13, [31] Brenner, M., Subrahmanyam, M.G A simple formula o compue he implied sandard deviaion. Finan. Analyss J. 44, [32] Broadie, M., Deemple, J American opion valuaion: new bounds, approximaions, and a comparison of exising mehods. Rev. Finan. Sudies 9, [33] Bunch, D., Johnson, H.E A simple and numerically efficien valuaion mehod for American pus, using a modified Geske-Johnson approach. J. Finance 47, [34] Carr, P Two exensions o barrier opion valuaion. Appl. Mah. Finance 2, [35] Carr, P., Jarrow, R., Myneni, R Alernaive characerizaions of American pu opions. Mah. Finance 2, [36] Carverhill, A.P., Clewlow, L.J. 199 Flexible convoluion. Risk 34, [37] Chance, D An Inroducion o Opions and Fuures. Dryden Press, Orlando. [38] Chen, R., Sco, L Pricing ineres rae opions in a wo-facor Cox-Ingersoll-Ross model of he erm srucure. Rev. Finan. Sud. 5, [39] Chesney, M., Ellio, R., Gibson, R Analyical soluions for he pricing of American bond and yield opions. Mah. Finance 3, [4] Cheuk, T.H.F., Vors, T.C.F Complex barrier opions. J. Porfolio. Manag. 4, [41] Cheyee, O. 199 Pricing opions on muliple asses. Adv. in Fuures Opions Res. 4, [42] Conze, A., Viswanahan 1991 Pah dependen opions: he case of lookback opions. J. Finance 46,

190 BIBLIOGRAPHY 189 [43] Cornell, B., Reinganum, M.R Forward and fuures prices: evidence from he foreign exchange markes. J. Finance 36, [44] Corrado, C.J., Miller, T.W A noe on a simple, accurae formula o compue implied sandard deviaions. J. Banking Finance 2, [45] Couradon, G A more accurae finie-difference approximaion for he valuaion of opions. J. Finan. Quan. Anal. 18, [46] Cox, J.C., Ross, S.A The pricing of opions for jump processes. Working Paper, Universiy of Pennsylvania. [47] Cox, J.C., Ross, S.A The valuaion of opions for alernaive sochasic processes. J. Finan. Econom. 3, [48] Cox, J.C., Rubinsein, M Opions Markes. Prenice-Hall, Englewood Cliffs New Jersey. [49] Cox, J.C., Ross, S.A., Rubinsein, M Opion pricing: a simplified approach. J. Finan. Econom. 7, [5] Cox, J.C., Ingersoll, J.E., Ross, S.A. 1985a An ineremporal general equilibrium model of asse prices. Economerica 53, [51] Cox, J.C., Ingersoll, J.E., Ross, S.A. 1985b A heory of he erm srucure of ineres raes. Economerica 53, [52] Cvianić, J Nonlinear financial markes: hedging and porfolio opimizaion. Forhcoming in Mahemaics of Derivaive Securiies, M.A.H.Dempser, S.R.Pliska, eds. Cambridge Universiy Press, Cambridge. [53] Culand, N.J., Kopp, P.E., Willinger, W From discree o coninuous financial models: new convergence resuls for opion pricing. Mah. Finance 3, [54] Dalang, R.C., Moron, A., Willinger, W. 199 Equivalen maringale measures and noarbirage in sochasic securiies marke model. Sochasics Sochasics Rep. 29, [55] Das, S Swaps and Financial Derivaives: The Global Reference o Producs, Pricing, Applicaions and Markes, 2nd ed. Law Book Co., Sydney. [56] Derman, E., Kani, I., Chriss, N The volailiy smile and is implied ree. J. Derivaives, [57] Derman, E., Karasinski, P., Wecker, J. 199 Undersanding guaraneed exchange-rae conracs in foreign sock invesmens. Working paper, Goldman Sachs. [58] Dohan, M.U. 199 Prices in Financial Markes. Oxford Universiy Press, New York. [59] Dohan, U On he erm srucure of ineres raes. J. Finan. Econom. 6, [6] Dubofsky, D.A Opions and Financial Fuures: Valuaion and Uses. McGraw-Hill, New York. [61] Duffie, D Securiy Markes: Sochasic Models. Academic Press, Boson. [62] Duffie, D Fuures Markes. Prenice-Hall, Englewood Cliffs New Jersey. [63] Duffie, D Sae-space models of he erm srucure of ineres raes. In: Sochasic Analysis and Relaed Topics V, H.Körezlioglu, B.Øksendal, A. Üsünel, eds. Birkhäuser, Boson Basel Berlin, pp [64] Duffie, D., Glynn, P Efficien Mone Carlo esimaion of securiy prices. Ann. Appl. Probab. 5,

191 19 BIBLIOGRAPHY [65] Duffie, D., Proer, P From discree o coninuous ime finance: weak convergence of he financial gain process. Mah. Finance 2, [66] Durre, R Sochasic Calculus: A Pracical Inroducion. CRC Press. [67] Dybvig, P.H., Ingersoll, J.E., Ross, S.A Long forward and zero-coupon raes can never fall. J. Business 69, [68] Eberlein, E On modeling quesions in securiy valuaion. Mah. Finance 2, [69] Edwards, F.R., Ma, C.W Fuures and Opions. McGraw-Hill, New York. [7] Ellio, R.J Sochasic Calculus and Applicaions. Springer, Berlin Heidelberg New York. [71] El Karoui, N., Roche, J.C A pricing formula for opions on coupon bonds. Working paper, SDEES. [72] El Karoui, N., Karazas, I A new approach o he Skorohod problem and is applicaions. Sochasics Sochasics Rep. 34, [73] Elon, E.J., Gruber, M.J Modern Porfolio Theory and Invesmen Analysis, 5h ed. J.Wiley, New York. [74] Flesaker, B., Hughson, L. 1996a Posiive ineres. Risk 91, [75] Flesaker, B., Hughson, L. 1996b Posiive ineres: foreign exchange. In: Vasicek and Beyond, L.Hughson, ed. Risk Publicaions, London, pp [76] French, K.R A comparison of fuures and forward prices. J. Finan. Econom. 12, [77] French, K.R., Schwer, W.G., and Sambaugh, R.F Expeced sock reurns and volailiy. J. Finan. Econom. 19, [78] Galai, D Tess of marke efficiency of he Chicago Board Opions Exchange. J. Business 5, [79] Garman, M., Kohlhagen, S Foreign currency opion values. J. Inerna. Money Finance 2, [8] Geman, H The imporance of he forward neural probabiliy in a sochasic approach of ineres raes. Working paper, ESSEC. [81] Geman, H., Yor, M Bessel processes, Asian opions and perpeuiies. Mah. Finance 3, [82] Geman, H., Yor, M Pricing and hedging double-barrier opions: a probabilisic approach. Mah. Finance 6, [83] Geman, H., El Karoui, Roche, J.C Changes of numeraire, changes of probabiliy measures and pricing of opions. J. Appl. Probab. 32, [84] Genle, D Baske weaving. Risk 66, [85] Geske, R The valuaion of corporae liabiliies as compound opions. J. Finan. Quan. Anal. 12, [86] Geske, R. 1979a The valuaion of compound opions. J. Finan. Econom. 7, [87] Geske, R. 1979b A noe on an analyical valuaion formula for an unproeced American call opions wih known dividends. J. Finan. Econom. 7, [88] Geske, R., Johnson, H.E The American pu opion valued analyically. J. Finance 39,

192 BIBLIOGRAPHY 191 [89] Goldman, B., Sosin, H., Gao, M Pah dependen opions: buy a he low, sell a he high. J. Finance 34, [9] Grabbe, J. Orlin 1983 The pricing of call and pu opions on foreign exchange. J. Inerna. Money Finance 2, [91] Grabbe, J. Orlin 1995 Inernaional Financial Markes. 3rd ed. Prenice-Hall, Englewood Cliffs New Jersey. [92] Harrison, J.M Brownian Moion and Sochasic Flow Sysems. Wiley, New York. [93] Harrison, J.M., Kreps, D.M Maringales and arbirage in muliperiod securiies markes. J. Econom. Theory 2, [94] Harrison, J.M., Pliska, S.R Maringales and sochasic inegrals in he heory of coninuous rading. Sochasic Process. Appl. 11, [95] Harrison, J.M., Pliska, S.R A sochasic calculus model of coninuous rading: complee markes. Sochasic Process. Appl. 15, [96] Heah, D., Jarrow, R., Moron, A. 199 Bond pricing and he erm srucure of ineres raes: a discree ime approximaion. J. Finan. Quan. Anal. 25, [97] Heah, D., Jarrow, R., Moron, A. 1992a Bond pricing and he erm srucure of ineres raes: a new mehodology for coningen claim valuaion. Economerica 6, [98] Heah, D., Jarrow, R., Moron, A., Spindel, M. 1992b Easier done han said. Risk 59, [99] Heynen, R.C., Ka, H.M Parial barrier opions. J. Finan. Engrg 2, [1] Ho, T.S.Y., Lee, S.-B Term srucure movemens and pricing ineres rae coningen claims. J. Finance 41, [11] Huang, C.-F., Lizenberger, R.H Foundaions for Financial Economics. Norh-Holland, New York. [12] Hull, J Inroducion o Fuures and Opions Markes. 2nd ed. Prenice-Hall, Englewood Cliffs New Jersey. [13] Hull, J Opions, Fuures, and Oher Derivaives. 3rd ed. Prenice-Hall, Englewood Cliffs New Jersey. [14] Hull, J., Whie, A The use of he conrol variae echnique in opion pricing. J. Finan. Quan. Anal. 23, [15] Hull, J., Whie, A. 199a Pricing ineres-rae derivaive securiies. Rev. Finan. Sud. 3, [16] Hull, J., Whie, A. 199b Valuing derivaive securiies using he explici finie difference mehod. J. Finan. Quan. Anal. 25, [17] Hull, J., Whie, A The pricing of opions on ineres-rae caps and floors using he Hull-Whie model. J. Financial Engrg 2, [18] Ingersoll, J.E., Jr Theory of Financial Decision Making. Rowman and Lilefield, Toowa New Jersey. [19] Jacka, S.D Opimal sopping and he American pu. Mah. Finance 1, [11] Jacka, S.D A maringale represenaion resul and an applicaion o incomplee financial markes. Mah. Finance 2, [111] Jacka, S.D Local imes, opimal sopping and semimaringales. Ann. Probab. 21,

193 192 BIBLIOGRAPHY [112] Jaille, P., Lamberon, D., Lapeyre, B. 199 Variaional inequaliies and he pricing of American opions. Aca Appl. Mah. 21, [113] Jamshidian, F Pricing of coningen claims in he one facor erm srucure model. Working paper, Merrill Lynch Capial Markes. [114] Jamshidian, F. 1989a An exac bond opion pricing formula. J. Finance 44, [115] Jamshidian, F. 1989b The mulifacor Gaussian ineres rae model and implemenaion. Working paper, Merrill Lynch Capial Markes. [116] Jamshidian, F An analysis of American opions. Rev. Fuures Markes 11, [117] Jamshidian, F Hedging quanos, differenial swaps and raios. Appl. Mah. Finance 1, 1 2. [118] Jamshidian, F LIBOR and swap marke models and measures. Finance Sochas. 1, [119] Jarrow, R Modelling Fixed Income Securiies and Ineres Rae Opions. McGraw-Hill, New York. [12] Johnson, H An analyic approximaion o he American pu price. J. Finan. Quan. Anal. 18, [121] Johnson, H Opions on he maximum or minimum of several asses. J. Finan. Quan. Anal. 22, [122] Johnson, H., Shanno, D Opion pricing when he variance is changing. J. Finan. Quan. Anal. 22, [123] Karazas, I On he pricing of American opions. Appl. Mah. Opim. 17, [124] Karazas, I Opimizaion problems in he heory of coninuous rading. SIAM J. Conrol Opim. 27, [125] Karazas, I Lecures on he Mahemaics of Finance. CRM Monograph Series, Vol. 8, American Mahemaical Sociey, Providence Rhode Island. [126] Karazas, I., Shreve, S Brownian Moion and Sochasic Calculus. Springer, Berlin Heidelberg New York. [127] Kemna, A.G.Z., Vors, T.C.F. 199 A pricing mehod for opions based on average asse values. J. Bank. Finance 14, [128] Kim, I.J. 199 The analyic valuaion of American opions. Rev. Finan. Sud. 3, [129] Kuniomo, N., Ikeda, M Pricing opions wih curved boundaries. Mah. Finance 2, [13] Lamberon, D Convergence of he criical price in he approximaion of American opions. Mah. Finance 3, [131] Lamberon, D., Lapeyre, B Inroducion o Sochasic Calculus Applied o Finance. Chapman and Hall, London. [132] Leland, H.E. 198 Who should buy porfolio insurance? J. Finance 35, [133] Levy, E The valuaion of average rae currency opions. J. Inerna. Money Finance 11, [134] Longsaff, F.A A nonlinear general equilibrium model of he erm srucure of ineres raes. J. Finan. Econom. 23, [135] Longsaff, F.A. 199 The valuaion of opions on yields. J. Finan. Econom. 26,

194 BIBLIOGRAPHY 193 [136] Longsaff, F.A The valuaion of opions on coupon bonds. J. Bank. Finance 17, [137] Longsaff, F.A., Schwarz, E.S. 1992a Ineres rae volailiy and he erm srucure: a wofacor general equilibrium model. J. Finance 47, [138] Manaser, S., Koehler, G The calculaion of implied variances from he Black-Scholes model: a noe. J. Finance 37, [139] Markowiz, H.M Porfolio selecion. J. Finance 7, [14] Markowiz, H.M Mean-Variance Analysis in Porfolio Choice and Capial Markes. Basil Blackwell, Cambridge, Mass. [141] McKean, H.P., Jr Appendix: A free boundary problem for he hea equaion arising from a problem in mahemaical economics. Indus. Manag. Rev. 6, [142] MacMillan, L.W Analyic approximaion for he American pu opion. Adv. in Fuures Opions Res. 1, [143] Meron, R.C Theory of raional opion pricing. Bell J. Econom. Manag. Sci. 4, [144] Milersen, K., Sandmann, K., Sondermann, D Closed form soluions for erm srucure derivaives wih log-normal ineres raes. J. Finance 52, [145] Musiela, M., Rukowski, M Coninuous-ime erm srucure models: forward measure approach. Finance Sochas. 1, [146] Musiela, M., Rukowski, M Maringale Mehods in Financial Modelling. Springer, Berlin Heidelberg New York. [147] Myneni, R The pricing of he American opion. Ann. Appl. Probab. 2, [148] Park, H.Y., Chen, A.H Differences beween fuures and forward prices: a furher invesigaion of he marking-o-marke effecs. J. Fuures Markes 5, [149] Pliska, S.R Inroducion o Mahemaical Finance: Discree Time Models. Blackwell Publishers, Oxford. [15] Proer, P. 199 Sochasic Inegraion and Differenial Equaions. Springer, Berlin Heidelberg New York. [151] Ray, C.I The Bond Marke. Trading and Risk Managemen. Irwin, Homewood Illinois. [152] Redhead, K Financial Derivaives: An Inroducion o Fuures, Forwards, Opions and Swaps. Prenice-Hall, Englewood Cliffs New Jersey. [153] Rendleman, R., Barer, B Two-sae opion pricing. J. Finance 34, [154] Revuz, D., Yor, M Coninuous Maringales and Brownian Moion. Springer, Berlin Heidelberg New York. [155] Richard, S.F An arbirage model of he erm srucure of ineres raes. J. Finan. Econom. 6, [156] Rogers, L.C.G Which model for erm-srucure of ineres raes should one use? In: IMA Vol.65: Mahemaical Finance, M.H.A.Davis e al., eds. Springer, Berlin Heidelberg New York, pp [157] Roll, R An analyic valuaion formula for unproeced American call opions on socks wih known dividends. J. Finan. Econom. 5, [158] Rubinsein, M The valuaion of uncerain income sreams and he pricing of opions. Bell J. Econom. 7,

195 194 BIBLIOGRAPHY [159] Rubinsein, M Somewhere over he rainbow. Risk 411, [16] Rubinsein, M Implied binomial rees. J. Finance 49, [161] Rubinsein, M., Reiner, E Breaking down he barriers. Risk 48, [162] Rukowski, M Valuaion and hedging of coningen claims in he HJM model wih deerminisic volailiies. Appl. Mah. Finance 3, [163] Rukowski, M A noe on he Flesaker-Hughson model of erm srucure of ineres raes. Appl. Mah. Finance 4, [164] Rukowski, M Models of forward Libor and swap raes. Appl. Mah. Finance 6, [165] Ruiens, A. 199 Currency opions on average exchange raes pricing and exposure managemen. 2h Annual Meeing of he Decision Science Insiue, New Orleans 199. [166] Samuelson, P.A Raional heory of warran prices. Indus. Manag. Rev. 6, [167] Sandmann, K., Sondermann, D., Milersen, K.R Closed form erm srucure derivaives in a Heah-Jarrow-Moron model wih log-normal annually compounded ineres raes. In: Proceedings of he Sevenh Annual European Fuures Research Symposium Bonn, Chicago Board of Trade, pp [168] Schwarz, E.S The valuaion of warrans: implemening a new approach. J. Finan. Econom. 4, [169] Sharpe, W Invesmens. Prenice-Hall, Englewood Cliffs New Jersey. [17] Shiryayev, A.N Probabiliy. Springer, Berlin Heidelberg New York. [171] Shiryayev, A.N., Kabanov, Y.M., Kramkov, D.O., Melnikov, A.V. 1994a Toward he heory of pricing of opions of boh European and American ypes. I. Discree ime. Theory Probab. Appl. 39, [172] Shiryayev, A.N., Kabanov, Y.M., Kramkov, D.O., Melnikov, A.V. 1994b Toward he heory of pricing of opions of boh European and American ypes. II. Coninuous ime. Theory Probab. Appl. 39, [173] Sulz, R.M Opions on he minimum or maximum of wo risky asses. J. Finan. Econom. 1, [174] Sucliffe, C.M.S Sock Index Fuures. Chapman & Hall, London. [175] Taqqu, M.S., Willinger, W The analysis of finie securiy markes using maringales. Adv. in Appl. Probab. 19, [176] Turnbull, S.M., Wakeman, L.M A quick algorihm for pricing European average opions. J. Finan. Quan. Anal. 26, [177] van Moerbeke, P On opimal sopping and free boundary problem. Arch. Raional Mech. Anal. 6, [178] Vasicek, O An equilibrium characerisaion of he erm srucure. J. Finan. Econom. 5, [179] Vors, T Prices and hedge raios of average exchange rae opions. Inerna. Rev. Finan. Anal. 1, [18] Whaley, R.E Valuaion of American call opions on dividend-paying socks: empirical ess. J. Finan. Econom. 1, [181] Willinger, W., Taqqu, M.S Toward a convergence heory for coninuous sochasic securiies marke models. Mah. Finance 1,