Modelling of Forward Libor and Swap Rates

Transcription

1 Modelling of Forward Libor and Swap Raes Marek Rukowski Faculy of Mahemaics and Informaion Science Warsaw Universiy of Technology, -661 Warszawa, Poland Conens 1 Inroducion 2 2 Modelling of Forward Libor Raes Forward and Fuures Libor Raes Single-period Swaps Seled in Arrears Single-period Swaps Seled in Advance Eurodollar Fuures Conracs Lognormal Models of Forward Libor Raes Milersen-Sandmann-Sondermann Approach Brace-G aarek-musiela Approach Musiela-Rukowski Approach Jamshidian s Approach Dynamics of Libor Raes and Bond Prices Dynamics of L,T j under P Tj Dynamics of F B,T j+1,t j under P Tj Caps and Floors Marke Valuaion Formula for Caps and Floors Valuaion in he Lognormal Model of Forward Libor Raes Hedging of Caps and Floors Bond Opions Modelling of Forward Swap Raes Ineres Rae Swaps Lognormal Model of Forward Swap Raes Valuaion of Swapions Payer and Receiver Swapions Forward Swapions Valuaion in he Lognormal Model of Forward Libor Raes Marke Valuaion Formula for Swapions Valuaion in he Lognormal Model of Forward Swap Raes Hedging of Swapions Choice of Numeraire Porfolio Markov-Funcional Models Terminal Swap Rae Model Calibraion of Markov-Funcional Models

2 2 M.Rukowski 1 Inroducion The las decade was marked by a rapidly growing ineres in he arbirage-free modelling of bond marke. Undoubedly, one of he major achievemens in his area was a new approach o he erm srucure modelling proposed by Heah, Jarrow and Moron in heir work published in 1992, commonly known as he HJM mehodology. One of is main feaures is ha i covers a large variey of previously proposed models and provides a unified approach o he modelling of insananeous ineres raes and o he valuaion of ineres-rae sensiive derivaives. Le us give a very concise descripion of he HJM approach for a deailed accoun we refer, for insance, o Chaper 13 in Musiela and Rukowski 1997a. The HJM mehodology 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 df, T =α, T d + σ, T dw, 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 which plays he role of he real-world probabiliy. More formally, for every fixed T T, where T > is he horizon dae, we have f, T =f,t+ αu, T du + σu, T dw u for some Borel-measurable funcion f, :[,T ] R and sochasic processes applicaions α,t and σ,t. Le us noice ha, for any fixed mauriy dae T T, he iniial condiion f,tis deermined by he curren value of he coninuously compounded forward rae for he fuure dae T which prevails a ime. In pracical erms, he funcion f,t is deermined by he curren yield curve, which can be esimaed on he basis of observed marke prices of bonds and oher relevan insrumens. Le us denoe by B, T he price a ime T of a uni zero-coupon bond which maures a he dae T T. In he presen seup, he price B, T can be recovered from he formula T B, T =exp f, u du. The problem of he absence of arbirage opporuniies in he bond marke 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 measure he drif coefficien α in he dynamics of he insananeous forward rae is uniquely deermined by he volailiy coefficien σ, and a sochasic process which can be inerpreed as he marke price of he ineres-rae risk. If we denoe by P he maringale measure for he bond marke, and by W he associaed sandard Brownian moion, hen db, T =B, T r d + b, T dw, where r = f, is he shor-erm ineres rae, and he bond price volailiy b, T saisfies T b, T = σ, u du. 1

3 Modelling of Forward Libor and Swap Raes 3 Furhermore, i appears ha in he special case when he coefficien σ follows a deerminisic funcion, he valuaion formulae for ineres rae-sensiive derivaives are independen of he choice of he risk premium. In his sense, he choice of a paricular model from he broad class of HJM models hinges uniquely on he specificaion of he volailiy coefficien σ. The HJM mehodology appeared o be very successful boh from he heoreical and pracical viewpoins. Since he HJM approach o he erm srucure modelling is based on an arbirage-free dynamics of he insananeous coninuously compounded forward raes, i requires a cerain degree of smoohness wih respec o he enor of he bond prices and heir volailiies. For his reason, working wih such models is no always convenien. An alernaive consrucion of an arbirage-free family of bond prices, making no reference o he insananeous raes, is in some circumsances more suiable. The firs sep in his direcion was done by Sandmann and Sondermann 1993, who focused on he effecive annual ineres rae. This approach was furher developed in ground-breaking papers by Milersen e al and Brace e al. 1997, who proposed o model insead he family of forward Libor raes. The main goal was o produce an arbirage-free erm srucure model which would suppor he common pracice of pricing such ineres-rae derivaives as caps and swapions hrough a suiable version of Black s formula. This pracical requiremen enforces he lognormaliy of he forward Libor or swap rae under he corresponding forward maringale measure. Le us recall ha, by marke convenion, he forward Libor rae over he fuure accrual pariod [T,T + δ], as seen a ime, is se o saisfy 1+δL, T = B, T B, T + δ, or equivalenly, B, T B, T + δ L, T =. B, T + δ The las formula makes i obvious ha he volailiy of he forward Libor rae is no deerminisic if he bond price volailiy follows a deerminisic funcion. For his reason he Black s formula for caps is manifesly incompaible wih he Gaussian HJM model ha is, he HJM model in which he bond price volailiy b, T is deerminisic. Consequenly, he marke formula for caps canno be derived in his seup hough he value of a cap is given by a closed-form expression in he Gaussian HJM framework. On he oher hand, i is ineresing o noice ha Brace e al paramerize heir version of he lognormal forward Libor model inroduced by Milersen e al wih a piecewise consan volailiy funcion. They need o consider smooh volailiy funcions in order o analyse he model in he HJM framework, however. The backward inducion approach o he modelling of forward Libor and swap rae developed in Musiela and Rukowski 1997 and Jamshidian 1997 overcomes his echnical difficuly. In addiion, in conras o he previous papers, i allows also for he modelling of forward Libor and swap raes associaed wih accrual periods of differing lenghs. I should be sressed ha a similar bu no idenical approach o he modelling of marke rae was developed in a series of papers by Hun e al. 1996, 1997 and Hun and Kennedy 1997, Since special emphasis is pu here on he exisence of he underlying low-dimensional Markov process ha governs direcly he dynamics of ineres raes, his alernaive approach is ermed he Markov-funcional approach. This propery leads o a considerable simplificaion in numerical procedures associaed wih he model s implemenaion. Anoher imporan feaure of his approach is is abiliy of providing a perfec fi o marke prices of a given family of ineres-rae opions e.g., a family of caps wih a fixed mauriy and varying srike level. Anoher racable erm srucure model which is beyond he scope of he presen ex, however isheraional lognormal model proposed by Flesaker and Hughson 1996a, 1996b see also Rukowski 1997 and Jin and Glasserman 1997 in his regard. Le us finally menion ha we use hroughou he noaion adoped in Musiela and Rukowski 1997a. The ineresed reader is referred o his monograph for more deails on erm srucure modelling as well as for he general background.

4 4 M.Rukowski 2 Modelling of Forward Libor Raes In his secion, we presen various approaches o he modelling of forward Libor raes. Due o he limied space, we focus on model s consrucion and is basic properies and he valuaion of he mos ypical derivaives. For furher deails, he ineresed reader is referred o he original papers: Musiela and Sondermann 1993, Sandmann and Sondermann 1993, Goldys e al. 1994, Sandmann e al. 1995, Brace e al. 1997, Jamshidian 1997, 1999, Milersen e al. 1997, Musiela and Rukowski 1997b, Rady 1997, Sandmann and Sondermann 1997, Rukowski 1998, 1999, Yasuoka 1998, and Glasserman and Kou The issues relaed o he model s implemenaion 1 are exensively reaed in Brace 1996, Andersen and Andreasen 1997, Sidenius 1997, Brace e al. 1998, Musiela and Sawa 1998, Hull and Whie 1999, Schlögl 1999, Yasuoka 1999, Loz and Schlögl 1999, and Glasserman and Zhao 2, Brace and Womersley 2, and Dun e al Forward and Fuures Libor Raes Our firs ask is o examine hese properies of forward and fuures conracs relaed o he noion of he Libor rae which are universal; ha is, which do no rely on specific assumpions imposed on a paricular model of he erm srucure of ineres raes. To his end, we fix an index j, and we consider various ineres-rae sensiive derivaives relaed o he period [T j,t j+1 ]. To be more specific, we shall focus in his secion on single-period forward swaps ha is, forward rae agreemens. We need o inroduce some noaion. We assume ha we are given a prespecified collecion of rese/selemen daes <T <T 1 < <T n = T, referred o as he enor srucure. Also, we denoe δ j = T j T j 1 for j =1,...,n. We wrie B, T j o denoe he price a ime of a T j -mauriy zero-coupon bond. P is he spo maringale measure, while for any j =,...,n we wrie P Tj o denoe he forward maringale measure associaed wih he dae T j. The corresponding d-dimensional Brownian moions are denoed by W and W Tj, respecively. Also, we wrie F B, T, U =B, T /B, U soha F B, T j+1,t j = B, T j+1 B, T j, [,T j], is he forward price a ime of he T j+1 -mauriy zero-coupon bond for he selemen dae T j. We use he symbol π X o denoe he value i.e., he arbirage price a ime of a European coningen claim X. Finally, we shall use he leer E for he Doléans exponenial, for insance, E γ u dwu =exp γ u dwu 1 γ u 2 du, 2 where he do and sand for he inner produc and Euclidean norm in R d, respecively Single-period Swaps Seled in Arrears Le us firs consider a single-period swap agreemen seled in arrears; i.e., wih he rese dae T j and he selemen dae T j+1 muli-period ineres rae swaps are examined in Secion 3. By he conracual feaures, he long pary pays δ j+1 κ and receives B 1 T j,t j+1 1aimeT j+1. Equivalenly, he pays an amoun Y 1 =1+δ j+1 κ and receives Y 2 = B 1 T j,t j+1 a his dae. The values a ime T j of hese payoffs are π Y 1 =B, T j+1 1+δ j+1 κ, π Y 2 =B, T j. The second equaliy above is rivial, since he payoff Y 2 is equivalen o he uni payoff a ime T j. Consequenly, for any fixed T j, he value of he forward swap rae, which makes he conrac 1 In paricular, an arbirage-free discreizaion of he lognormal model of forward Libor raes.

5 Modelling of Forward Libor and Swap Raes 5 worhless a ime, can be found by solving for κ = κ, T j,t j+1 he following equaion I is hus apparen ha π Y 2 π Y 1 =B, T j B, T j+1 1+δ j+1 κ =. κ, T j,t j+1 = B, T j B, T j+1, [,T j ]. δ j+1 B, T j+1 Noe ha he forward swap rae κ, T j,t j+1 coincides wih he forward Libor rae L, T j which, by he marke convenion, is se o saisfy 1+δ j+1 L, T j = B, T j B, T j+1 = E P Tj+1 B 1 T j,t j+1 F 2 for every [,T j ]. Le us noice ha he las equaliy is a consequence of he definiion of he forward measure P Tj+1. We conclude ha in order o deermine he forward Libor rae L,T j, i is enough o find he forward price F X, T j+1 aime of he coningen claim X = B 1 T j,t j+1 in he forward conac ha seles a ime T j+1. Indeed, i is well known see, for insance, Musiela and Rukowski 1997a ha F X, T j+1 =B, T j+1 E PTj+1 B 1 T j,t j+1 F. Furhermore, i is eviden ha he process L,T j follows necessarily a maringale under he forward probabiliy measure P Tj+1. Recall ha in he Heah-Jarrow-Moron framework, we have, under P Tj+1, df B, T j,t j+1 =F B, T j,t j+1 b, T j b, T j+1 dw Tj+1, 3 where, for each mauriy dae T, he process b,t represens he price volailiy of he T -mauriy zero-coupon bond. On he oher hand, if he process L,T j is sricly posiive, i can be shown o admi he following represenaion 2 dl, T j =L, T j λ, T j dw Tj+1, where λ,t j is an adaped sochasic process which saifies mild inegrabiliy condiions. Combining he las wo formulae wih 2, we arrive a he following fundamenal relaionship, which plays an essenial role in he consrucion of he lognormal model of forward Libor raes, δ j+1 L, T j 1+δ j+1 L, T j λ, T j=b, T j b, T j+1, [,T j ]. 4 For insance, in he consrucion which is based on he backward inducion, relaionship 4 will allow us o deermine he forward measure for he dae T j, provided ha P Tj+1,W Tj+1 and he volailiy λ, T j of he forward Libor rae L,T j 1 are known. One may assume, for insance, ha λ,t j is a prespecified deerminisic funcion. Recall ha in he Heah-Jarrow-Moron framework 3 he Radon-Nikodým densiy of P Tj wih respec o P Tj+1 is known o saisfy dp Tj = E Tj dp Tj+1 b, Tj b, T j+1 dw Tj+1. 5 In view of 4, we hus have dp Tj δ j+1 L, T j = E Tj dp Tj+1 1+δ j+1 L, T j λ, T j dw Tj+1. 2 This represenaion is a consequence of he maringale represenaion propery of he sandard Brownian moion. 3 See Heah e al

6 6 M.Rukowski For our furher purposes, i is also useful o observe ha his densiy admis he following represenaion dp Tj = cf B T j,t j,t j+1 =c 1+δ j+1 LT j,t j, P Tj+1 -a.s., 6 dp Tj+1 where c> is he normalizing consan, and hus dp Tj dp Tj+1 F = cf B, T j,t j+1 =c 1+δ j+1 L, T j, P Tj+1 -a.s. Finally, he dynamics of he process L,T j under he probabiliy measure P Tj somewha involved sochasic differenial equaion δj+1 L, T j λ, T j 2 dl, T j =L, T j 1+δ j+1 L, T j d + λ, T j dw Tj. are given by a As we shall see in wha follows, i is neverheless no hard o deermine he probabiliy law of L,T j under he forward measure P Tj a leas in he case of he deerminisic volailiy λ,t j ofhe forward Libor rae Single-period Swaps Seled in Advance Consider now a similar swap which is, however, seled in advance ha is, a ime T j. Our firs goal is o deermine he forward swap rae implied by such a conrac. Noe ha under he presen assumpions, he long pary formally pays an amoun Y 1 =1+δ j+1 κ and receives Y 2 = B 1 T j,t j+1 aheselemendaet j which coincides here wih he rese dae. The values a ime T j of hese payoffs admi he following represenaions π Y 1 =B, T j 1+δ j+1 κ, π Y 2 =B, T j E PTj B 1 T j,t j+1 F. The value κ =ˆκ, T j,t j+1 of he modified forward swap rae, which makes he swap agreemen seled in advance worhless a ime, can be found from he equaliy π Y 2 π Y 1 =B, T j E PTj B 1 T j,t j+1 F 1 + δ j+1 κ =. I is clear ha ˆκ, T j,t j+1 =δ 1 j+1 E PTj B 1 T j,t j+1 F 1. We are in a posiion o inroduce he modified forward Libor rae L, T j by seing, for every [,T j ], L, T j :=δj+1 1 E PTj B 1 T j,t j+1 F 1. Le us make wo remarks. Firs, i is clear ha finding of he modified forward Libor rae L,T j is formally equivalen o finding he forward price of he claim B 1 T j,t j+1 for he selemen dae T j. 4 Second, i is useful o observe ha 1 BTj,T j+1 L, T j =E PTj F = E PTj LT j,t j F. 7 δ j+1 BT j,t j+1 In paricular, i is eviden ha a he rese dae T j he wo kinds of forward Libor raes inroduced above coincide, since manifesly LT j,t j = 1 BT j,t j+1 δ j+1 BT j,t j+1 = LT j,t j. 4 Recall ha in he case of a forward Libor rae, he selemen dae was T j+1.

7 Modelling of Forward Libor and Swap Raes 7 To summarize, he sandard forward Libor rae L,T j saisfies L, T j =E PTj+1 LT j,t j F, [,T j ], wih he iniial condiion L,T j = B,T j B,T j+1 δ j+1 B,T j+1. On he oher hand, for he modified Libor rae L,T j wehave L, T j =E PTj LT j,t j F, [,T j ], wih he iniial condiion L,T j =δ 1 j+1 E PTj B 1 T j,t j+1 1. The calculaion of he righ-hand side above involve no only on he iniial erm srucure, bu also he volailiies of bond prices for more deails, we refer o Rukowski Eurodollar Fuures Conracs The nex objec of our sudies is he fuures Libor rae. A Eurodollar fuures conrac is a fuures conrac in which he Libor rae plays he role of an underlying asse. By convenion, a he conrac s mauriy dae T j, he quoed Eurodollar fuures price, denoed by ET j,t j, is se o saisfy ET j,t j :=1 δ j+1 LT j,t j. Equivalenly, in erms of he zero-coupon bond price we have ET j,t j =2 B 1 T j,t j+1. From he general heory, i follows ha he Eurodollar fuures price a ime T j equals E, T j :=E P ET j,t j = 2 E P B 1 T j,t j+1 F 8 recall ha P represens he spo maringale measure in a given model of he erm srucure. I is hus naural o inroduce he concep of he fuures Libor rae, associaed wih he Eurodollar fuures conrac, hrough he following definiion. Definiion 2.1 Le E, T j be he Eurodollar fuures price a ime for he selemen dae T j. The implied fuures Libor rae L f, T j saisfies E, T j =1 δ j+1 L f, T j, [,T j ]. 9 I follows immediaely from 8 9 ha he following equaliy is valid Equivalenly, we have 1+δ j+1 L f, T j =E P B 1 T j,t j+1 F. 1 L f, T j =E P LT j,t j F =E P LT j,t j F. Noe ha in any erm srucure model, he fuures Libor rae necessarily follows a maringale under he spo maringale measure P provided, of course, ha P is well-defined in his model. 2.2 Lognormal Models of Forward Libor Raes We shall now describe alernaive approaches o he modelling of forward Libor raes in a coninuousand discree-enor seups.

8 8 M.Rukowski Milersen-Sandmann-Sondermann Approach The firs aemp o provide a rigorous consrucion a lognormal model of forward Libor raes was done by Milersen e al The ineresed reader is referred also o Musiela and Sondermann 1993, Goldys e al. 1994, and Sandmann e al for relaed previous sudies. As a saring poin in heir approach, Milersen e al posulae ha he forward Libor raes process L,T saisfies dl, T =µ, T d + L, T λ, T dw, wih a deerminisic volailiy funcion λ,t:[,t] R d. I is no difficul o deduce from he las formula ha he forward price of a zero-coupon bond saisfies df, T + δ, T = F, T + δ, T 1 F, T + δ, T λ, T dw T. Subsequenly, hey focus on he parial differenial equaion saisfied by he funcion v = v, x, which expresses he forward price of he bond opion in erms of he forward bond price. I is ineresing o noe ha he PDE 11 was previously solved by Rady and Sandmann 1994 who worked wihin a differen framework, however. 5 The PDE for he opion s price is v λ, T 2 x 2 1 x 2 2 v x 2 = 11 wih he erminal condiion vt,x=k x +. As a resul, Milersen e al obained no only he closed-form soluion for he price of a bond opion his was already achieved in Rady and Sandmann 1994, bu also he marke formula for he caple s price. The rigorous approach o he problem of exisence of such a model was presened by Brace e al. 1997, who also worked wihin he coninuous-ime Heah-Jarrow-Moron framework Brace-G aarek-musiela Approach To formally inroduce he noion of a forward Libor rae, we assume ha we are given a family B, T of bond prices, and hus also he collecion F B, T, U of forward processes. In conras o he previous secion, we shall now assume ha a sricly posiive real number δ<t, which represens he lengh of he accrual period, is fixed hroughou. By definiion, he forward δ-libor rae 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]. 12 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]. 13 B, T + δ In paricular, he iniial erm srucure of forward Libor raes saisfies L,T=δ 1 B,T B,T + δ Given a family F B, T, T of forward processes, i is no hard o derive he dynamics of he associaed family of forward Libor raes. For insance, one finds ha 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, 5 In fac, hey were concerned wih he valuaion of opions on zero-coupon bonds for he erm srucure model pu forward by Bühler and Käsler 1989.

9 Modelling of Forward Libor and Swap Raes 9 where P T +δ is he forward measure for he dae T + δ, and he asociaed Wiener process W T +δ equals = W bu, T + δ du, [,T + δ]. W T +δ Pu anoher way, he process L,T solves he equaion dl, T =δ δl, T γ, T, T + δ dw T +δ, 15 subjec o he iniial condiion 14. Suppose ha forward Libor raes L, T are sricly posiive. Then formula 15 can be rewrien as follows dl, T =L, T λ, T dw T +δ, 16 where for any [,T] λ, T = 1+δL, T γ, T, T + δ. 17 δl, T This shows ha he collecion of forward processes uniquely specifies he family of forward Libor raes. 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 6 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 B,T,T [,T ]. The associaed iniial erm srucure L,T of forward Libor raes saisfies, for every T [,T δ], B,T B,T + δ L,T=. 18 δb,t + δ To consruc a model saisfying LR.1 LR.2, Brace e al place hemselves in he Heah- Jarrow-Moron seup and hey assume ha for every T [,T ], he volailiy b, T vanishes for every [T δ,t]. In essence, he consrucion elaboraed in Brace e al is based on he forward inducion, as opposed o he backward inducion which we shall use in he nex secion. They sar by posulaing ha he dynamics of L, T under he spo maringale measure P are governed by he following SDE dl, T =µ, T d + L, T λ, T dw, where λ is a deerminisic funcion, and he drif coefficien µ is unspecified. arbirage-free dynamics of he insananeous forward rae f, T are Recall ha he df, T =σ, T σ, T d + σ, T dw, where σ, T = T σ, u du = b, T. On he oher hand, he relaionship cf. 13 T +δ 1+δL, T =exp f, u du T 19 is valid. Applying Iô s formula o boh sides of 19, and comparing he diffusion erms, we find ha T +δ σ, T + δ σ, T = σ, u du = δl, T λ, T. 1+δL, T T 6 Volailiy λ could well follow an adaped sochasic process; we deliberaely focus here on a lognormal model of forward Libor raes in which λ is deerminisic.

10 1 M.Rukowski To solve he las equaion for σ in erms of L, i is necessary o impose some sor of iniial condiion on σ. For insance, by seing σ, T =for T +δ, we obain he following relaionship [δ 1 T ] b, T = σ δl, T kδ, T = λ, T kδ. 2 1+δL, T kδ k=1 The exisence and uniqueness of soluions o SDEs which govern he insananeous forward rae f, T and he forward Libor rae L, T forσ given by 2 can be shown using forward inducion. Taking his resul for graned, we conclude ha L, T saisfies, under he spo maringale measure P dl, T =L, T σ, T + δ λ, T d + L, T λ, T dw. In his way, Brace e al are able o compleely specify heir model of forward Libor raes Musiela-Rukowski Approach In his secion, we describe an alernaive approach o he modelling of forward Libor raes; he consrucion presened below is a sligh modificaion of ha given by Musiela and Rukowski 1997b. Le us sar by inroducing some noaion. We assume ha we are given a prespecified collecion of rese/selemen daes <T <T 1 < <T n = T, referred o as he enor srucure by convenion, T 1 =. Le us denoe δ j = T j T j 1 for j =,...,n. Then obviously T j = j i= δ i for every j =,...,n. We find i convenien o denoe, for m =,...,n, T m = T j=n m+1 δ j = T n m. For any j =,...,n 1, we define he forward Libor rae L,T j by seing L, T j = B, T j B, T j+1, [,T j ]. δ j+1 B, T j+1 Definiion 2.2 For any j =,...,n, a probabiliy measure P Tj on Ω, F Tj, equivalen o P, is said o be he forward Libor measure for he dae T j if, for every k =,...,n he relaive bond price follows a local maringale under P Tj. U n j+1, T k := B, T k B, T j, [,T k T j ], I is clear ha he noion of forward Libor measure is in fac idenical wih ha of a forward probabiliy measure for a given dae. Also, i is rivial o observe ha he forward Libor rae L,T j necessarily follows a local maringale under he forward Libor measure for he dae T j+1. If, in addiion, i is a sricly posiive process, he exisence of he associaed volailiy process can be jusified by sandard argumens. In our furher developmen, we shall go he oher way around; ha is, we will assume ha for any dae T j, he volailiy λ,t j of he forward Libor rae L,T j is exogenously given. In principle, i can be a deerminisic R d -valued funcion of ime, an R d -valued funcion of he underlying forward Libor raes, or i can follow a d-dimensional adaped sochasic process. For simpliciy, we assume hroughou ha he volailiies of forward Libor raes are bounded processes or funcions. To be more specific, we make he following sanding assumpions. Assumpions LR. We are given a family of bounded adaped processes λ,t j,j =,...,n 1, which represen he volailiies of forward Libor raes L,T j. In addiion, we are given an iniial erm srucure of ineres raes, specified by a family B,T j,j =,...,n, of bond prices. We assume here ha B,T j >B,T j+1 forj =,...,n 1.

11 Modelling of Forward Libor and Swap Raes 11 Our aim is o consruc a family L,T j,j=,...,n 1 of forward Libor raes, a collecion of muually equivalen probabiliy measures P Tj,j =1,...,n, and a family W Tj,j =1,...,n of processes in such a way ha: i for any j =1,...,n he process W Tj follows a d-dimensional sandard Brownian moion under he probabiliy measure P Tj, ii for any j =,...,n 1, he forward Libor rae L,T j saisfies he SDE dl, T j =L, T j λ, T j dw Tj+1, [,T j ], 21 wih he iniial condiion L,T j = B,T j B,T j+1. δ j+1 B,T j+1 As already menioned, he consrucion of he model is based on backward inducion, herefore we sar by defining he forward Libor rae wih he longes mauriy, i.e., T n 1. We posulae ha L,T n 1 =L,T1 is governed under he underlying probabiliy measure P by he following SDE7 wih he iniial condiion Pu anoher way, we have dl, T 1 =L, T 1 λ, T 1 dw L,T 1 = B,T 1 B,T δ n B,T L, T 1 =B,T 1 B,T δ n B,T E λu, T1 dw u Since B,T1 >B,T, i is clear ha he L,T1 follows a sricly posiive maringale under P T = P. The nex sep is o define he forward Libor rae for he dae T2. For his purpose, we need o inroduce firs he forward probabiliy measure for he dae T1. By definiion, i is a probabiliy measure Q, which is equivalen o P, and such ha processes U 2, T k = B, T k B, T 1 are Q-local maringales. I is imporan o observe ha he process U 2,Tk admis he following represenaion U 2, Tk = U 1, Tk 1+δ n L, T1. Le us formulae an auxiliary resul, which is a sraighforward consequence of Iô s rule. Lemma 2.1 Le G and H be real-valued adaped processes, such ha dg = α dw, dh = β dw. Assume, in addiion, ha H > 1 for every and denoe Y =1+H 1. Then dy G =Y α Y G β dw Y β d. I follows immediaely from Lemma 2.1 ha du 2, Tk =η k dw δ nl, T1 1+δ n L, T1 λ, T 1 d 7 Noice ha, for simpliciy, we have chosen he underlying probabiliy measure P o play he role of he forward LibormeasureforhedaeT. This choice is no essenial, however...

12 12 M.Rukowski for a cerain process η k. Therefore i is enough o find a probabiliy measure under which he process W T 1 δ n Lu, T1 := W 1+δ n Lu, T1 λu, T 1 du = W γu, T 1 du, [,T 1 ], follows a sandard Brownian moion he definiion of γ,t 1 is clear from he conex. This can be easily achieved using Girsanov s heorem, as we may pu dp T 1 dp = E T 1 γu, T1 dw u, P-a.s. We are in a posiion o specify he dynamics of he forward Libor rae for he dae T2 under P T1, namely we posulae ha dl, T2 =L, T2 λ, T2 dw T 1 wih he iniial condiion L,T2 = B,T 2 B,T 1 δ n 1 B,T1. Le us now assume ha we have found processes L,T1,...,L,T m. This means, in paricular, ha he forward Libor measure P T m 1 and he associaed Brownian moion W T m 1 are already specified. Our aim is o deermine he forward Libor measure P T m. I is easy o check ha U m+1, T k := B, T k B, T m = U m, T k 1+δ n m L, T m. Using Lemma 2.1, we obain he following relaionship W T m = W T m 1 δ n m Lu, T m 1+δ n m Lu, Tm λu, T m du for [,Tm ]. The forward Libor measure P Tm can hus be easily found using Girsanov s heorem. Finally, we define he process L,Tm+1 as he soluion o he SDE dl, T m+1 =L, T m+1 λ, T m+1 dw T m wih he iniial condiion L,T m+1 = B,T m+1 B,T m δ n m B,T m. Remarks. i I is no difficul o check ha equaliy 6 is saisfied wihin he presen seup. ii If he volailiy coefficien λ,t m :[,T n ] R d is a deerminisic funcion, hen for each dae [,T m ] he random variable L, T m has a lognormal probabiliy law under he forward probabiliy measure P Tm+1. Le us now examine he exisence and uniqueness of he implied savings accoun, 8 in a discreeime seup. 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, i is clear ha 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. 8 The ineresed reader is referred o Musiela and Rukowski 1997b for he definiion of an implied savings accoun in a coninuous-ime seup. See also Döberlein and Schweizer 1998 and Döberlein e al for furher developmens and he general uniqueness resul.

13 Modelling of Forward Libor and Swap Raes 13 This in urn yields, upon seing = T j F B T j,t j,t j+1 =1/BT j,t j+1, 22 so ha he price BT j,t j+1 of a single-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, for k =,...,n recall ha T 1 =, B T k = k k F B Tj 1,T j 1,T j = B 1 T j 1,T j j= since, by convenion, we se B =1. Noe ha F B Tj 1,T j 1,T j =1+δLTj 1,T j > 1forj =,...,n, and since B T j = F B T j 1,T j 1,T j B T j 1, we find ha B T j >B T j 1 for every j =,...,n. We conclude ha he implied savings accoun B follows a sricly increasing discree-ime process. Le us define he probabiliy measure P, equivalen o P on Ω, F T, by he formula 9 dp dp = B T B,T, P-a.s. 23 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 24 for every l k n, hen in he case of l = k 1, equaliy 24 coincides wih 22. Le us observe ha i is no possible o uniquely deermine he coninuous-ime dynamics of a bond price B, T j wihin he framework of he discree-enor model of forward Libor raes he specificaion of forward Libor raes for all mauriies is necessary for his purpose Jamshidian s Approach The backward inducion approach o modelling of forward Libor raes presened in he preceding secion was re-examined and essenially generalized by Jamshidian In his secion, we presen briefly his approach o he modelling of forward Libor raes. As made apparen in he preceding secion, in he direc modelling of Libor raes, no explici reference is made o he bond price processes, which are used o formally define a forward Libor rae hrough equaliy 13. Neverheless, o explain he idea ha underpins Jamshidian s approach, we shall emporarily assume ha we are given a family of bond prices B, T j for he fuure daes T j,,...,n. By definiion, he spo Libor measure is ha probabiliy measure equivalen o P, under which all relaive bond prices are local maringales, when he price process obained by rolling over single-period bonds, is aken as a numeraire. The exisence of such a measure can be eiher posulaed, or derived from oher condiions. 1 Le us pu, for [,T ] as before T 1 = j= m G = B, T m B 1 T j 1,T j, 25 j= where m = inf {k =, 1,... k δ i } = inf {k =, 1,... T k }. i= 9 Recall ha P plays he role of he forward Libor measure for he dae T. Therefore, formula 23 is a consequence of sandard definiion of a forward measure. 1 One may assume, e.g., ha bond prices B, T j saisfyheweak no-arbirage condiion, meaning ha here exiss a probabiliy measure P, equivalen o P, and such ha all processes B, T k /B, T are P-local maringales.

14 14 M.Rukowski I is easily seen ha G represens he wealh a ime of a porfolio which sars a ime wih one uni of cash invesed in a zero-coupon bond of mauriy T, and whose wealh is hen reinvesed a each dae T j,j=,...,n 1, in zero-coupon bonds which maure a he nex dae; ha is, T j+1. Definiion 2.3 A spo Libor measure P L is a probabiliy measure on Ω, F T which is equivalen o P, and such ha for any j =,...,nhe relaive bond price B, T j /G follows a local maringale under P L. Noe ha B, T k+1 /G = m j= 1+δj LT j 1,T j 1 1 k j=m+1 1+δj L, T j 1 so ha all relaive bond prices B, T j /G,j=,...,n are uniquely deermined by a collecion of forward Libor raes. In his sense, G is he correc choice of he reference price process in he presen seing. We shall now concenrae on he derivaion of he dynamics under P L of forward Libor raes L,T j,j=,...,n 1. Our aim is o show ha hese dynamics involve only he volailiies of forward Libor raes as opposed o volailiies of bond prices or oher processes. Therefore, i is possible o define he whole family of forward Libor raes simulaneously under one probabiliy measure of course, his feaure can also be deduced from he preceding consrucion. To faciliae he derivaion of he dynamics of L,T j, we posulae emporarily ha bond prices B, T j follow Iô processes under he underlying probabiliy measure P, more explicily db, T j =B, T j a, T j d + b, T j dw 26 for every j =,...,n, where, as before, W is a d-dimensional sandard Brownian moion under an underlying probabiliy measure P i should be sressed, however, ha we do no assume here ha P is a forward or spo maringale measure. Combining 25 wih 26, we obain Furhermore, by applying Iô s rule o equaliy we find ha where µ, T j = dg = G a, Tm d + b, T m dw δ j+1 L, T j = B, T j B, T j+1, 28 dl, T j =µ, T j d + ζ, T j dw, B, T j a, Tj a, T j+1 ζ, T j b, T j+1 δ j+1 B, T j+1 and B, T j ζ, T j = b, Tj b, T j δ j+1 B, T j+1 Using 28 and he las formula, we arrive a he following relaionship b, T m b, T j+1 = j k=m δ k+1 ζ, T k 1+δ k+1 L, T k. 3 By definiion of a spo Libor measure P L, each relaive price B, T j /G follows a local maringale under P L. Since, in addiion, P L is assumed o be equivalen o P, i is clear ha i is given by he Doléans exponenial, ha is dp L dp = E T h u dw u, P-a.s.

15 Modelling of Forward Libor and Swap Raes 15 for some adaped process h. I i no hard o check, using Iô s rule, ha h necessarily saisfies, for [,T j ], a, T j a, T m = b, T m h b, Tj b, T m for every j =,...,n. Combining 29 wih he las formula, we obain and his in urn yields B, T j a, Tj a, T j+1 = ζ, T j b, T m h, δ j+1 B, T j+1 dl, T j =ζ, T j Using 3, we conclude ha process L,T j saisfies b, Tm b, T j+1 h d + dw. dl, T j = j k=m δ k+1 ζ, T k ζ, T j 1+δ k+1 L, T k d + ζ, T j dw L, where he process W L = W h u du follows a d-dimensional sandard Brownian moion under he spo Libor measure P L. To furher specify he model, we assume ha processes ζ, T j,j=,...,n 1, have he following form, for [,T j ], ζ, T j =λ j, L, Tj,L, T j+1,...,l, T n, where λ j :[,T j ] R n j+1 R d are given funcions. In his way, we obain a sysem of SDEs dl, T j = j k=m δ k+1 λ k, L k λ j, L j 1+δ k+1 L, T k d + λ j, L j dw L, wherewewriel j =L, T j,l, T j+1,...,l, T n. Under mild regulariy assumpions, his sysem can be solved recursively, saring from L,T n 1. The lognormal model of forward Libor raes corresponds o he choice of ζ, T j = λ, T j L, T j, where λ,t j : [,T j ] R d is a deerminisic funcion for every j. 2.3 Dynamics of Libor Raes and Bond Prices We assume ha he volailiies of processes L,T j follow deerminisic funcions. Pu anoher way, we place ourselves wihin he framework of he lognormal model of forward Libor raes. I is ineresing o noe ha in all approaches, here is a uniquely deermined correspondence beween forward measures and forward Brownian moions associaed wih differen daes T,...,T n. On he oher hand, however, here is a considerable degree of ambiguiy in he way in which he spo maringale measure is specified in some insances, i is no inroduced a all. Consequenly, he fuures Libor rae L f,t j, which equals cf. Secion L f, T j =E P LT j,t j F =E P LT j,t j F, 31 is no necessarily specified in he same way in various approaches o he lognormal model of forward Libor raes. For his reason, we sar by examining he disribuional properies of forward Libor raes, which are idenicall in all abovemenioned models. For a given funcion g : R R and a fixed dae u T j, we are ineresed in he following payoff of he form X = g Lu, T j which seles a ime T j. Paricular cases of such payoffs are X 1 = g B 1 T j,t j+1, X 2 = g BT j,t j+1,x 3 = g F B u, T j+1,t j.

16 16 M.Rukowski Recall ha B 1 T j,t j+1 =1+δ j+1 LT j,t j =1+δ j+1 LTj,T j =1+δ j+1 L f T j,t j. The choice of he pricing measure is hus largely he maer of convenience. Similarly, we have 1 BT j,t j+1 = 1+δ j+1 LT j,t j = F BT j,t j+1,t j. 32 More generally, he forward price of a T j+1 -mauriy bond for he selemen dae T j equals F B u, T j+1,t j = Bu, T j+1 Bu, T j = 1 1+δ j+1 Lu, T j. 33 Generally speaking, o value he claim X = glu, T j = gf B u, T j+1,t j which seles a ime T j we may use he formula π X =B, T j E PTj X F, [,T j ]. I is hus clear ha o value a claim in he case u T j, i is enough o know he dynamics of eiher L,T j orf B,T j+1,t j under he forward probabiliy measure P Tj. If u = T j, we may equally well use he he dynamics, under P Tj, of eiher L,T j orl f,t j. For insance, π X 1 = B, T j E PTj B 1 T j,t j+1 F = B, T j E PTj F 1 B T j,t j+1,t j F bu also π X 1 =B, T j 1+δ j+1 E PTj ZT j F, where ZT j =LT j,t j = LT j,t j =L f T j,t j Dynamics of L,T j under P Tj We shall now derive he ransiion probabiliy densiy funcion p.d.f. of he process L,T j under he forward probabiliy measure P Tj. Le us firs prove he following relaed resul, due o Jamshidian Proposiion 2.1 Le u T j. Then E PTj Lu, Tj F = L, Tj + δ j+1var PTj+1 Lu, Tj F δ j+1 L, T j In he case of he lognormal model of Libor raes, we have E PTj Lu, Tj F = L, Tj 1+ δ j+1l, T j e v 2 j,u 1, 35 1+δ j+1 L, T j where v 2 j, u =Var PTj+1 u λs, T j dws Tj+1 = u λs, T j 2 ds. 36 In paricular, he modified Libor rae L, T j saisfies 11 L, T j =E PTj LTj,T j F = L, Tj 1+ δ j+1l, T j e v 2 j,tj 1. 1+δ j+1 L, T j 11 This equaliy can be referred o as he convexiy correcion.

17 Modelling of Forward Libor and Swap Raes 17 Proof. Combining 6 wih he maringale propery of he process L,T j under P Tj+1, we obain E PTj Lu, Tj F = E PTj δj+1 Lu, T j Lu, T j F 1+δ j+1 L, T j so ha E PTj Lu, Tj F = L, Tj + δ j+1 E PTj+1 Lu, Tj L, T j 2 F. 1+δ j+1 L, T j In he case of he lognormal model, we have where Consequenly, Lu, T j =L, T j e ηj,u 1 2 v2 j,u, η j, u = u λs, T j dw Tj+1 s. 37 E PTj+1 Lu, Tj L, T j 2 F = L 2, T j e v2 j,u 1. This gives he desired equaliy 35. The las assered equaliy is a consequence of 7. To derive he ransiion probabiliy densiy funcion p.d.f. of he process L,T j, noice ha for any u T j, and any bounded Borel measurable funcion g : R R we have glu, T j 1+δ j+1 Lu, T j F E PTj glu, Tj F = E PTj+1 The following simple lemma appears o be useful. 1+δ j+1 L, T j Lemma 2.2 Le ζ be a nonnegaive random variable on a probabiliy space Ω, F, P wih he probabiliy densiy funcion f P. Le Q be a probabiliy measure equivalen o P. Suppose ha for any bounded Borel measurable funcion g : R R we have E P gζ = E Q 1 + ζgζ. Then he p.d.f. f Q of ζ under Q saisfies f P y =1+yf Q y.. ¾ Proof. The asserion is in fac rivial since, by assumpion, gyf P y dy = gy1 + yf Q y dy for any bounded Borel measurable funcion g : R R. ¾ Assume he lognormal model of Libor raes and fix x R. Recall ha for any u we have Lu, T j =L, T j e ηj,u 1 2 Var P Tj+1 η j,u, where η j, u is given by 37 so ha i is independen of he σ-field F. Markovian propery of L,T j under he forward measure P Tj+1 is hus apparen. Denoe by p L, x; u, y he ransiion p.d.f. under P Tj+1 of he process L,T j. Elemenary calculaions involving Gaussian densiies yield p L, x; u, y = P Tj+1 {Lu, T j =y L, T j =x} 1 lny/x+ 1 = { 2πvj, uy exp 2 v2 j, u 2 } 2vj 2, u

18 18 M.Rukowski for any x, y > and<u. Taking ino accoun Lemma 2.2, we conclude ha he ransiion p.d.f. of he process 12 L,T j, under he forward probabiliy measure P Tj, saisfies p L, x; u, y =P Tj {Lu, T j =y L, T j =x} = 1+δ j+1y 1+δ j+1 x p L, x; u, y. We are in a posiion o sae he following resul, which can be used, for insance, o value a coningen claim of he form X = hlt j which seles a ime T j cf. Schmid Corollary 2.1 The ransiion p.d.f. under P Tj of he forward Libor rae L,T j equals, for any <uand x, y >, 1+δ j+1 y lny/x+ 1 p L, x; u, y = { 2πvj, u y1 + δ j+1 x exp 2 v2 j, u 2 } 2vj 2., u Dynamics of F B,T j+1,t j under P Tj Observe ha he forward bond price F B,T j+1,t j saisfies F B, T j+1,t j = B, T j+1 1 = B, T j 1+δ j+1 L, T j. 38 Firs, his implies ha in he lognormal model of Libor raes, he dynamics of he forward bond price F B,T j+1,t j are governed by he following sochasic differenial equaion, under P Tj, df B = F B 1 F B λ, T j dw Tj, 39 wherewewrief B =F B, T j+1,t j. If he iniial condiion saisfies <F B < 1, his equaion can be shown o admi a unique srong soluion i saisfies <F B < 1 for every >. This makes clear ha he process F B,T j+1,t j and hus also he process L,T j are Markovian under P Tj. Using Corollary 2.1 and relaionship 38, one can find he ransiion p.d.f. of he Markov process F B,T j+1,t j under P Tj ;hais, p B, x; u, y =P Tj {F B u, T j+1,t j =y F B, T j+1,t j =x}. We have he following resul see Rady and Sandmann 1994, Milersen e al. 1997, and Jamshidian Corollary 2.2 The ransiion p.d.f. under P Tj of he forward bond price F B,T j+1,t j equals, for any <uand arbirary <x,y<1, 2 x ln x1 y p B, x; u, y = 2πvj, uy 2 1 y exp y1 x v2 j, u 2vj 2, u. Proof. Le us fix x, 1. Using 38, i is easy o show ha p B, x; u, y =δ 1 y 2 p L, 1 x δx ; u, 1 y, δy where δ = δ j+1. The formula now follows from Corollary 2.1. Le us observe ha he resuls of his secion can be applied o value he so-called irregular cash flows, such as caps or floors seled in advance for more deails on his issue we refer o Schmid The Markov propery of L,T j under P Tj can be easily deduced from he Markovian feaures of he forward price F B,T j,t j+1 under P Tj see formulae ¾

19 Modelling of Forward Libor and Swap Raes Caps and Floors An ineres rae cap known also as a ceiling rae agreemen 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 j he cap srike rae and he lengh of he accrual period, 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 j which maures a he caple end dae. Similarly o swap agreemens, ineres rae caps and floors may be seled eiher in arrears or in advance. In a forward cap or floor, which sars a ime T, and is seled in arrears a daes T j,j = 1,...,n, he cash flows a imes T j are N p LT j 1 κ + δ j and N p κ LT j 1 + δ j, respecively, where N p sands for he noional principal recall ha δ j = T j T j 1. As usual, he rae LT j 1 =LT j 1,T j 1 is deermined a he rese dae T j 1, and i saisfies BT j 1,T j 1 =1+δ j LT j 1. 4 The price a ime T of a forward cap, denoed by FC, is we se N p =1 FC = = E P B B Tj LT j 1 κ + δ j F B, T j E PTj LT j 1 κ + δ j F. 41 On he oher hand, since he cash flow of he j h caple a ime T j is manifesly a F Tj 1 -measurable random variable, we may direcly express he value of he cap in erms of expecaions under forward measures P Tj 1,,...,n. Indeed, we have FC = Consequenly, using 4 we ge equaliy FC = B, T j 1 E PTj 1 BT j 1,T j LT j 1 κ + δ j F B, T j 1 E PTj 1 δj BT j 1,T j + F, 43 which is valid for every [,T]. I is apparen ha a caple is essenially equivalen o a pu opion on a zero-coupon bond; i may also be seen as an opion on a single-period swap. 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 j = δ j.

20 2 M.Rukowski The las inequaliy holds whenever δ j BT j 1,T j < 1. This shows ha boh of he considered opions are exercised in he same circumsances. If exercised, he caple pays δ j LT j 1 κ a ime T j, or equivalenly δ j BT j 1,T j LT j 1 κ =1 δ j BT j 1,T j = δ δ 1 j j BT j 1,T j a ime T j 1. This shows once again ha he j h caple, wih srike level κ and nominal value 1, is essenially equivalen o a pu opion wih srike price 1 + κδ j 1 and nominal value δ j =1+κδ j wrien on he corresponding zero-coupon bond wih mauriy T j. The analysis of a floor conrac can be done long he simlar lines. By definiion, he j h floorle pays κ LT j 1 + a ime T j. Therefore, bu also FF = FF = E P B B Tj κ LT j 1 + δ j F, 44 B, T j 1 E PTj 1 δj BT j 1,T j 1 + F. 45 Combining 41 wih 44 or 43 wih 45, we obain he following cap-floor pariy relaionship FC FF = B, Tj 1 δ j B, T j 46 which is also an immediae consequence of he no-arbirage propery, so ha i does no depend on model s choice Marke Valuaion Formula for Caps and Floors 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 formulae. For his reason, we shall firs describe how marke praciioners value caps. 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. Le us consider an ineres rae cap wih expiry dae T andfixedsrikelevelκ. 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 accrual period of lengh δ follows a geomeric Brownian moion under he marke probabiliy, Q say. More specifically dl, T =L, T σdw, 47 where W follows a one-dimensional sandard Brownian moion under Q, and σ is a sricly posiive consan. The unique soluion of 47 is L, T =L,Texp σw 1 2 σ2 2, [,T], 48 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 FC = δb, T + δ E Q LT,T κ + F.

21 Modelling of Forward Libor and Swap Raes 21 More explicily, for any [,T]wehave FC = δb, T + δ L, T N ê 1, T κn ê 2, T, 49 where N is he sandard Gaussian cumulaive disribuion funcion Nx = 1 x e z2 /2 dz, x R, 2π and ê 1,2, T = lnl, T /κ ± 1 2 ˆv2, T ˆv, T wih 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,,...,n, where T j T j 1 = δ j, T = T, is priced by he formula FC = δ j B, T j L, T j 1 N ê j 1 κn ê j 2, 5 where for every j =,...,n 1 ê j 1,2 =lnl, T j 1/κ ± 1 2 ˆv2 j ˆv j 51 and ˆv 2 j =T j 1 σ 2 j for some consans σ j,,...,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. The value of a floor can be easily derived by combining 5 51 wih he cap-floor pariy relaionship 46. As we shall see in wha follows, he valuaion formulae obained for caps and floors in he lognormal model of forward Libor raes agree wih he marke pracice Valuaion in he Lognormal Model of Forward Libor Raes We shall now examine he valuaion of caps wihin he lognormal model of forward Libor raes of Secion 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, 52 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, for every [,T j 1 ] we have L, T j 1 =L,T j 1 E λu, T j 1 dwu Tj. In he presen seup, he cap valuaion formula 53 was firs esablished by Milersen e al. 1997, who focused on he dynamics of he forward Libor rae for a given dae. Equaliy 53 was subsequenly rederived hrough a probabilisic approach in Goldys 1997 and Rady Finally, he same resul was esablished by means of he forward measure approach in Brace e al The following proposiion is a consequence of formula 42, combined wih he dynamics 52. As before, N is he sandard Gaussian probabiliy disribuion funcion. Proposiion 2.2 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 FC = δ j B, T j L, T j 1 N ẽ j 1 κn ẽ j 2 = FC j, 53