Spot, Forward, and Futures Libor Rates

Transcription

1 Spo, Forward, Fuures Libor Raes MAREK RUTKOWSKI Insiue of Mahemaics, Poliechnika Warszawska, -66 Warszawa, Pol Absrac The properies of forward fuures ineres-rae conracs associaed wih a given collecion of rese daes are sudied wihin he frameworks of he Gaussian HJM model he lognormal model of Libor raes. We focus on he dynamics disribuional properies of spo, forward, fuures Libor raes under spo forward maringale measures. Keywords: zero-coupon bond, Libor rae, forward conrac, fuures conrac, Eurodollar fuures Inroducion In a series of recen papers o menion a few: Brace e al. [2], Goldys [5], Jamshidian [7], Milersen e al. [8], Musiela Rukowski [9], Rady [], Rukowski [3] various approaches o he problem of modelling of Libor raes were presened. In paricular, quesions relaed o he valuaion hedging of ineres raes derivaives, such as caps swapions, were examined in some deail. In he presen noe, a slighly differen perspecive is adoped, namely, he emphasis is on he dynamics disribuional properies of various spo, forward fuures raes under spo forward maringale measures. Though he so-called lognormal model of Libor raes is aken as a benchmark, o faciliae he comparison, he mos basic resuls relaed o behaviour of Libor raes wihin he Gaussian HJM framework are also given. A large par of he presen work is devoed he sudy of he dynamics he ransiion probabiliy densiy funcion of he forward Libor rae under spo forward maringale measures noe ha hese resuls are also closely relaed o he behavior of he forward bond price. Le us sress ha he disribuional properies of forward Libor raes are relaively easy o esablish in he presen framework. On he conrary, he sudy of he behavior of spo fuures Libor raes poses some nonrivial difficulies. This specific feaure is mainly due o he fac ha he before-menioned mehods of modelling focus on he deerminaion of he dynamics of forward Libor raes, hus ypically leaving some degree of freedom in he specificaion of oher Libor raes. We hope ha his research will prove useful as a saring poin for furher sudies of sill unsolved problem of exogenous modelling of eiher spo or fuures Libor raes. 2 Forward Fuures Libor Raes We assume ha we are given a prespecified collecion of rese/selemen daes T =<T < < T M = T, referred o as he enor srucure. Le us denoe δ j = T j T j for j =,...,M. We wrie B, T o denoe he price a ime of a T -mauriy zero-coupon bond. P is he spo probabiliy measure, while P Tj P Tj+, respecively is he forward probabiliy measure associaed wih he dae T j T j+, respecively. The corresponding d-dimensional Brownian moions are denoed by W W Tj W Tj+, respecively. Also, we wrie F B, T, U =B, T /B, U odenoehe forward price of a bond his inerpreaion is valid provided ha U T. Finally, π X denoes he value ha is, he arbirage price a ime of European coningen claim X see Musiela markru@alpha.im.pw.edu.pl.

2 2 M.Rukowski: Spo, Forward, Fuures Libor Raes Rukowski [] for deails. Our firs aim is o examine hese properies of forward fuures conracs or raes which are universal; ha is, which do no rely on specific assumpions imposed on a paricular model of he erm srucure of ineres raes. We fix an index j, we inroduce various kinds of ineres raes relaed o he period [T j,t j+ ]. 2. Forward Libor Rae Le us firs consider a one-period swap agreemen seled in arrears; i.e., wih he rese dae T j he selemen dae T j+. By he conracual feaures, he long pary pays δ j+ κ receives B T j,t j+ aimet j+. Equivalenly, he pays an amoun Y =+δ j+ κ receives Y 2 = B T j,t j+ a his dae. The values of hese payoffs a ime T j are π Y =B, T j+ +δ j+ 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 worhless a ime, can be found by solving for κ he following equaion π Y =B, T j+ +δ j+ κ = B, Tj =π Y 2. I is apparen ha κ = B, T j B, T j+, [,T j ]. δ j+ B, T j+ Noe ha κ coincides wih he forward Libor rae L, T j overheperiod [T j,t j+ ], which is given by he convenional formula +δ j+ L, T j def = B, T j B, T j+ = E P Tj+ B T j,t j+ F, Noe ha he second equaliy in is a sraighforward consequence of he definiion of he forward measure P Tj+. I is hus clear ha in order o deermine he forward Libor rae L,T j, i is enough o find he forward price of he claim B T j,t j+ for he selemen dae T j+. Furhermore, he process L,T j necessarily follows a maringale under he forward probabiliy measure P Tj+. Recall also ha wihin he HJM framework we have df B, T j,t j+ =F B, T j,t j+ b, T j b, T j+ dw Tj+ 2 under P Tj+, where b,t ss for he price volailiy of he T -mauriy zero-coupon bond. On he oher h, L,T j can be shown o admi he following represenaion dl, T j =L, T j λ, T j dw Tj+, where λ,t j is a d-dimensional adaped process. Combining he las wo formulae wih, we arrive a he following fundamenal relaionship δ j+ L, T j +δ j+ L, T j λ, T j=b, T j b, T j+, [,T j ]. 3 Le us sress ha equaliy 3 is an essenial ingredien in various consrucions of he lognormal model of forward Libor raes. For insance, if we work by he backward inducion, relaionship 3 allows us o deermine he forward measure for he dae T j, provided ha he forward measure P Tj+, he associaed Brownian moion W Tj+, he volailiy λ, T j are known i is cusomary o assume ha λ,t j is an exogenously given deerminisic funcion. Indeed, in he HJM framework, he Radon-Nikodým densiy of P Tj wih respec o P Tj+ is known o saisfy EZ ss for he Doléans exponenial of he process Z dp Tj = E Tj dp Tj+ b, Tj b, T j+ dw Tj+.

3 UNSW, Repor No.S97-, Sepember 997 his version: June In view of 3, we also have dp Tj δ j+ L, T j = E Tj dp Tj+ +δ j+ L, T j λ, T j dw Tj+. 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+ =c +δ j+ LT j, P Tj+ -a.s., 4 dp Tj+ where LT j =LT j,t j c> is he normalizing consan. Therefore dp Tj dp Tj+ F = cf B, T j,t j+ =c +δ j+ L, T j, P Tj+ -a.s. Finally, he process L,T j is known o saisfy he following sochasic differenial equaion, under he probabiliy measure P Tj, δj+ L, T j λ, T j 2 dl, T j =L, T j +δ j+ L, T j 2.2 Adjused Forward Libor Rae d + λ, T j dw Tj. 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 =+δ j+ κ receives Y 2 = B T j,t j+ aheselemendaet j which coincides here wih he rese dae. The values of hese payoffs a ime T j admi he following represenaions π Y =B, T j +δ j+ κ, π Y 2 =B, T j E PTj B T j,t j+ F. The value κ of he adjused forward swap rae, which makes he swap agreemen seled in advance worhless a ime, can be found from he equaliy π Y =B, T j +δ j+ κ = B, T j E PTj B T j,t j+ F =π Y 2. I is clear ha κ = δ j+ E PTj B T j,t j+ F. We are in a posiion o inroduce he adjused forward Libor rae L, T j by seing L, T j def = δ j+ E PTj B T j,t j+ F, [,T j ]. Le us make wo remarks. Firs, i is clear ha finding of he adjused Libor rae L,T j is essenially equivalen o pricing of he claim B T j,t j+ which seles a T j more precisely, we need o know he forward price of his claim for he dae T j. Second, i is useful o observe ha L, T j =E PTj BTj,T j BT j,t j+ δ j+ BT j,t j+ F = E PTj LT j,t j F. In paricular, i is eviden ha a he rese dae T j he wo forward Libor raes inroduced above coincide, since manifesly LT j,t j = BT j,t j+ δ j+ BT j,t j+ = LT j,t j.

4 4 M.Rukowski: Spo, Forward, Fuures Libor Raes To summarize, he sard forward Libor rae L,T j saisfies L, T j =E PTj+ LT j,t j F, [,T j ], wih he iniial condiion L,T j = B,T j B,T j+ δ j+ B,T j+, for he adjused 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 j+ E PTj B T j,t j+. Noe ha he las condiion depends no only on he iniial erm srucure, bu also on he volailiies of bond prices see, e.g., formula 9 below. 2.3 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 ET j,t j is se o saisfy cf. Amin Ng [] ET j,t j def = δ j+ LT j,t j. Equivalenly, in erms of he zero-coupon bond price we have ET j,t j =2 B T j,t j+. From he general heory, i follows ha he Eurodollar fuures price a ime T j equals E, T j def = E P ET j,t j F = δ j+ E P LTj,T j F =2 E P B T j,t j+ F 5 recall ha P represens he spo maringale measure in a given model of he erm srucure. I seems naural o inroduce he concep of he fuures Libor rae, associaed wih he Eurodollar fuures conrac, hrough he following definiion. Definiion 2. 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 = δ j+ L f, T j, [,T j ]. 6 I follows immediaely from 5 6 ha he following equaliy is valid +δ j+ L f, T j =E P B T j,t j+ F. Equivalenly, we have L f, T j =δj+ E P B T j,t j+ F = E P LT j,t j F =E P LT j,t j F. Noe ha he fuures Libor rae follows a maringale under he spo maringale measure P.

5 UNSW, Repor No.S97-, Sepember 997 his version: June Gaussian HJM Model In his secion, we focus on he Heah e al. [6] approach o he erm srucure modelling. We denoe by f, T he insananeous forward rae prevailing a ime for he fuure infiniesimal ime period [T,T + dt ], we wrie r, T =f, +T. Noe ha r = f, =r, represens he shor-erm rae. Consequenly, he savings accoun B saisfies B =exp r u du =exp fu, u du =exp ru, du. We assume ha for any mauriy dae T he insananeous forward rae saisfies df, T =α, T d + σ, T dw where σ,t is an adaped sochasic process. The process W is a d-dimensional sard Brownian moion defined on a filered probabiliy space Ω, F, P, where P is inerpreed as he spo maringale measure. We find i convenien o assume ha he underlying filraion is generaed by W. I is well known ha in he so-called risk-neural world, ha is, under P, we have see [6] T α, T =σ, T σ, u du = b, T b, T, T where b, T ss for he bond price volailiy, ha is, T b, T = σ, u du. In he HJM framework, he price process B,TofaT -mauriy discoun bond is inroduced hrough he formula T T B, T =exp f, u du =exp r, u du for every T. In paricular, he price process D,T of a sliding bond equals +T T D, T =exp f, u du =exp r, u du for any, T. Assume, in addiion, ha he bond price volailiies b,t j follow deerminisic funcions. In his case, i is no hard o express forward fuures Libor raes in erms of bond prices bond price volailiies. Furhermore, as soon as he dynamics of various raes under forward probabiliy measures are explicily known, i is sraighforward o value ineres-rae sensiive derivaives. As already menioned, in he HJM framework we have df B, T j,t j+ =F B, T j,t j+ b, T j b, T j+ dw Tj+ wih he erminal condiion F B T j,t j,t j+ =B T j,t j+. Also, he spo forward Brownian moions are known o saisfy dw Tj = dw Tj+ b, T j b, T j+ d, 7 dw = dw Tj + b, T j d. for every j =,...,M. Le us emphasize ha in he sard HJM framework here is no ambivalence in he definiion of he spo probabiliy measure. This should be conrased wih he case of he discree-enor lognormal model of forward Libor raes, in which he spo maringale measure is

6 6 M.Rukowski: Spo, Forward, Fuures Libor Raes no uniquely defined, hus here is some degree of freedom in he choice of he spo maringale measure, when a family of forward measures is previously specified. For sake of simpliciy, we denoe γ, T j,t j+ =b, T j b, T j+. We are in a posiion o esablish he following proposiion see Flesaker [4] for relaed resuls. Proposiion 3. Assume he Gaussian HJM model of he erm srucure of ineres raes. Then he following relaionships are valid +δ j+ L, T j =E PTj+ B T j,t j+ F = FB, 8 +δ j+ L, Tj =E PTj B Tj γu,t T j,t j+ F = FB e j,t j+ 2 du, 9 +δ j+ L f, T j =E P B T j,t j+ F = FB e Tj bu,t j+ γu,t j,t j+ du. Proof. The firs assered formula is in fac universal see. For he second, noe ha cf. 2 dw Tj df B =F B γ, T j,t j+ + γ, T j,t j+ d. Consequenly, T j dw Tj F B T j =F B exp γu, T j,t j+ u + γu, T j,t j+ du 2 Tj γu, T j,t j+ 2 du. Since B T j,t j+ =F B T j, upon aking condiional expecaion wih respec o he σ-field F, we obain 9. Finally, we have df B =F B γ, T j,t j+ dw b, T j+ d hus T j F B T j =F B exp γu, T j,t j+ dw u bu, T j+ du Tj γu, T j,t j+ 2 du. 2 This leads o equaliy. ¾ The dynamics of various forward Libor raes are also easy o find, as he following corollary shows. Corollary 3. We have dl, T j =δj+ +δj+ L, T j γ, T j,t j+ dw Tj+, d L, T j =δ j+ +δj+ L, T j γ, T j,t j+ dw Tj, dl f, T j =δj+ +δj+ L f, T j γ, T j,t j+ dw. Proof. The firs formula is an immediae consequence of combined wih 2. The nex wo expressions can be derived by applying Iô s rule o equaliies 9 respecively. Remark. Noice ha +δ j+ L f, T j = +δ j+ L, Tj e Tj bu,t j γu,t j,t j+ du. Therefore, L f,t j = L,T j if he bond price volailiy b,t j vanishes idenically. On he oher h, equaliy L f,t j =L,T j holds provided ha he volailiy b,t j+ vanishes idenically. For obvious reasons hese wo cases are of minor ineres. On he oher h, i is clear from Corollary 3. ha closed-form soluions for opions wrien on he forward or fuures LIBOR raes are no available in he presen seup. ¾

7 UNSW, Repor No.S97-, Sepember 997 his version: June Dynamics of Libor Raes Recall ha for any daes <T <U he forward Libor rae L, T, U is given by he formula cf. L, T, U = B, T B, U. U T B, U Forafixedδ>, we shall focus on he case when U = T + δ i.e., we examine he case of fixedlengh accrual period. As before, we prefer o wrie L, T raher han L, T, T + δ. Also, we shall denoe K, T =L, + T for any T. Finally, he spo Libor rae is defined by seing ˆL = L, =K,. Le us denoe by D, δ =B, + δ he price a ime of a + δ-mauriy zero-coupon bond. The process D,T will be referred o as he price of a sliding bond, he yield Z,T of a sliding bond as a sliding yield. Noe ha K, T = B, + T B, + T + δ δb, + T + δ = D, T D, T + δ δd, T + δ D, D, δ ˆL = = δ D, δ. 2 δd, δ For sake of convenience, we shall someimes wrie c, δ =b, + δ in wha follows. The following resul which holds wihin he Heah, Jarrow, Moron [6] seup will prove useful see, e.g., Brace e al. [2] or Rukowski [4] for he proof. Proposiion 4. Assume he HJM arbirage-free framework. The dynamics of he price process D,δ of a sliding bond, under he spo maringale measure P, are r dd, δ =D, δ r, δ d + c, δ dw, 3 where r = f, r, δ =f, + δ. The nex proposiion deals wih he dynamics of spo forward Libor raes in a risk-neural world. Proposiion 4.2 Assume ha he lengh δ> of he accrual period is fixed. The dynamics of he spo Libor rae ˆL are governed by he expression, under he spo maringale measure P, r, dˆl = δ + δ ˆL δ r + c, δ 2 d c, δ dw. 4 Furhermore, for any fixed mauriy dae T, he forward Libor raes K,T L,T saisfy dk, T = K, T δ + δk, T c, T c, T + δ c, T + δ d T + δ + δk, T c, T c, T + δ dw dl, T =δ + δl, T b, T b, T + δ dw b, T + δ d, 5 respecively. Proof. Applying Iô s formula o 3, we obain dd, δ =D, δ r, δ r + c, δ 2 d c, δ dw.

8 8 M.Rukowski: Spo, Forward, Fuures Libor Raes Since + δ ˆL = D, δ sohadˆl = δ dd, δ, he las formula immediaely yields 4. For he forward Libor rae K,T we have cf. dk, T = D, T δ d. D, T + δ An applicaion of Iô s formula yields dk, T = D, T r, T + δ r, T c, T c, T + δ c, T + δ d δd, T + δ + c, T c, T + δ dw. From he definiion of he process D,T, i is easily seen ha D, T + δ T = D, T + δr, T + δ D, T = D, T r, T. T Therefore K, T is necessarily differeniable wih respec o he second argumen, K, T = D, T = r, T + δ r, T D, T T δ T D, T + δ δd, T + δ. The formula for he dynamics of K,T now follows by elemenary algebra. Similar, bu simpler, calculaions show ha L,T saisfy 5. This complees he proof of he proposiion. ¾ Example 4. We shall examine he behavior of he spo Libor rae process wihin he framework of he Vasicek model. We assume ha he shor-erm rae r solves he SDE dr = a br d σdw, where a, b σ are sricly posiive consans. Le us denoe g = e b. The dynamics of D,T under he maringale measure P are r dd, δ =gδd, δ ab + 2 σ2 b 2 gδ d + σb dw. Le us firs focus on he yield process Z, δ =Y, + δ of he sliding bond. Since Z, δ = δ ln D, δ, one can show ha see Rukowski [4] dz, δ = hδ bz, δ d σb δ gδ dw, where hδ =b 2 b δ gδ ab 2 σ2 + 4 b 2 σ 2 δ g 2 δ+ab δ. Pu anoher way, he dynamics of he sliding yield Z,δ are dz, δ = â ˆbZ, δ d +ˆσdW, where â, ˆb ˆσ are consans depending on T. On he oher h, i is clear ha dd, δ =gδd, δ 2 σ2 b 2 gδ+ab r d + σb dw. Furhermore, we have +δ ˆL = B, + δ =exp m, + δ+n, + δr,

9 UNSW, Repor No.S97-, Sepember 997 his version: June hus where n, + δ =b gδ r = m, + δ + ln + δ ˆL n, + δ, I is no hard o check ha m, + δ =b 3 gδ bδ ab 2 σ2 4 b 3 σ 2 g 2 δ. gδr = b 2 gδ bδ ab 2 σ2 4 b 2 σ 2 g 2 δ+b ln + δ ˆL. Consequenly, for any fixed δ we have dˆl = δ dd, δ =δ + δ ˆL hδ b ln + δ ˆL d σb gδ dw where hδ is a consan, namely, hδ =ab gδ+b 2 bδ gδ ab 2 σ b 2 σ 2 g 2 δ. We conclude ha wihin Vasicek s framework, he spo Libor rae ˆL follows a diffusion process under he spo maringale measure P. Oher classic examples of shor-erm rae models can be sudied along he same lines. An ineresing problem which arises in his conex is hus he characerizaion of hose diffusion processes which may play he role of he spo Libor rae. 4. Lognormal Model of Forward Libor Raes From now on, he bond price volailiies b,t j are no longer assumed o be deerminisic. On he oher h, we shall frequenly assume ha he volailiies of processes L,T j follow deerminisic funcions. For various approaches o such a model, he reader is referred o Milersen e al. [8], Brace e al. [2], Musiela Rukowski [9], Jamshidian [7], Rukowski [3]. In his secion, we focus on he original mehod, due o Brace e al. [2]. I appears, ha formula 5 is a convenien saring poin in a consrucion of he lognormal model of forward Libor raes. Assume, for insance, ha for any mauriy dae T we are given a funcion λ,t:[,t] R d. We posulae ha δ + δl, T b, T b, T + δ dw = λ, T L, T dw, or equivalenly, b, T b, T + δ = δl, T λ, T. 6 +δl, T Equaliy 5 hen becomes dl, T =L, T λ, T dw b, T + δ d. To solve explicily he las equaion, we need o specify b, T + δ. To his end, i is enough o assume, for insance, ha he process b, T = b, T isknownforanyt every [T δ, T ]. Such an approach o he modelling of forward Libor raes was adoped by Brace e al. [2]. Discree-enor case. Le us firs focus on a discree se of rese daes T j = jδ, j =, 2,... his is commonly referred o as he discree-enor case. By assumpion, he funcion b,t =ˆb,T is exogenously specified. Equaliy 6 implies ha b, T 2 =ˆb, T δl, T +δl, T λ, T, [,T ].,

10 M.Rukowski: Spo, Forward, Fuures Libor Raes moreover b, T 2 =ˆb, T 2 for T,T 2 ]. Therefore, L,T saisfies dl, T =L, T λ, T dw ˆb, T d + δl, T +δl, T λ, T d. We shall now deermine he dynamics of L,T 2. For his purpose, noe ha b, T 3 =ˆb, T 2 i= δl, T i +δl, T i λ, T i for [,T ], b, T 3 =ˆb, T 2 δl, T 2 +δl, T 2 λ, T 2 for T,T 2 ], finally b, T 3 =ˆb, T 3 for T 2,T 3 ]. Consequenly, L,T 2 saisfies dl, T 2 =L, T 2 λ, T 2 dw ˆb, T d + 2 i= δl, T i +δl, T i λ, T i d for [,T ], dl, T 2 =L, T 2 λ, T 2 dw ˆb, T 2 d + δl, T 2 +δl, T 2 λ, T 2 d for [T,T 2 ]. In general, for any j =, 2,... he process L,T j solves on each inerval [T k,t k ],k=,...,j, he following SDE dl, T j =L, T j λ, T j dw ˆb, T k d + j i=k δl, T i +δl, T i λ, T i d. We have hus found a sysem of SDEs, which can be solved recursively. We firs solve his sysem in fac, a paricular SDE for L,T on [,T ] ; subsequenly, we solve i for L,T 2 ; firs on he inerval [,T ], hen [T,T 2 ], so forh. Remarks. The procedure above can be easily exended o he case of variable accrual periods. To be more specific, we may focus on an arbirary discree collecion of rese daes <T <T 2 <... For any j we are now preoccupied wih he forward Libor rae L,T j,t j+ which corresponds o he rese dae T j he accrual period [T j,t j+ ]. Noe ha in conras o he previously examined case, he lengh δ j+ = T j+ T j is no longer assumed o be consan, i.e., independen of j. If we wish o exend he consrucion discussed above o he presen case, i is sufficien o assume ha for any j he volailiy funcion b,t j is exogenously given on he inerval [T j,t j ]. I is worhwhile o menion ha an alernaive approach o he discree-enor case based on he backward inducion was developed in [7], [9], [3]. An advanage of his alernaive approach is ha one deals wih he dynamics of forward Libor raes under forward maringale measures; he need o specify he bond price volailiy ˆb,T j is hus avoided. Once he consrucion of a discree-enor model is achieved, he spo maringale measure can also be inroduced see [7] or [3]. Coninuous-enor case. Le us now consider he case of an arbirary rese dae T>. Suppose firs ha T δ. Then clearly dl, T =L, T λ, T dw ˆb, T d + δl, T +δl, T λ, T d. Suppose now ha T m <T <T m+ for some m. We denoe a = T T m, we se T =, T j =j δ + a

11 UNSW, Repor No.S97-, Sepember 997 his version: June 998 for j =, 2,... To find L,T=L, T m i is enough o make use of he following sysem of SDEs dl, T j =L, T j λ, T j dw ˆb, T k d + j i=k δl, T i +δl, T i λ, T i d on [ T k, T k ], also o be solved recursively we firs solve he sysem above for L, T on [, T ], hen for for L, T 2 on [, T ] [ T, T 2 ], ec. In his way we are able o specify a leas in principle a forward Libor rae L,T for any dae T >. Since K, T =L, + T for any, T >, we have defined also he family K,T of processes. I is ineresing o noe, however, ha i is unclear in general wheher his family of processes solves he equaion of Proposiion 4.2. The posiive answer o his quesion relies on he smoohness of K, T wih respec o he second argumen. Equivalenly, i hinges on he differeniabiliy of he bond price volailiy b, T wih respec T. The las propery is in urn direcly relaed o he exisence of he coefficien σ, T in he dynamics of he underlying forward rae f, T ; ha is, o he possibiliy of sudying he lognormal model of forward Libor raes wihin he general HJM framework. I is ineresing o noe ha in all mehods menioned above, here is a uniquely deermined correspondence beween forward measures forward Brownian moions associaed wih differen daes i is based on relaionships 3 7. On he oher h, 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 2.3 L f, T j =E P LT j,t j F =E P LT j,t j F, 7 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, of course, hold in all hese models. The properies of fuures Libor raes are examined in Secion Valuaion of European Coningen Claims For a given funcion g : R R, a fixed dae u T j, we are ineresed in European payoffs of he form X = g Lu, T j 8 which sele a ime T j. Paricular cases of such payoffs are recall ha all hese payoffs are seled a ime T j Recall ha X = g B T j,t j+, X 2 = g BT j,t j+, X 3 = g F B u, T j+,t j. B T j,t j+ =+δ j+ LT j,t j =+δ j+ LTj,T j =+δ j+ L f T j,t j. The choice of he pricing measure is hus largely he maer of convenience. Noe ha BT j,t j+ = +δ j+ LT j,t j = F BT j,t j+,t j,, more generally, he forward price of a T j+ -mauriy bond for he selemen dae T j equals F B u, T j+,t j = Bu, T j+ Bu, T j = +δ j+ Lu, T j. To value a European coningen claim X = glu, T j = gf B u, T j+,t j, which seles a ime T j, we may use he forward-risk adjused formula π X =B, T j E PTj X F, [,T j ].

12 2 M.Rukowski: Spo, Forward, Fuures Libor Raes To price X when u T j, i suffices o deermine he dynamics of eiher L,T j orf B,T j+,t j under he forward measure P Tj. When u = T j, we may equally well refer o he dynamics, under P Tj, of he adjused Libor rae L,T j or he fuures Libor rae L f,t j. Indeed, we have π X =B, T j E PTj B T j,t j+ F =B, T j E PTj F B T j,t j+,t j F, bu also π X =B, T j +δ j+ E PTj ZT j F, where ZT j =LT j,t j = LT j,t j =L f T j,t j. 4.3 Dynamics of L,T j under P Tj In his secion, we shall 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 of independen ineres i is due o Jamshidian [7]. Proposiion 4.3 Le u T j. Then E PTj Lu, Tj F = L, Tj + δ j+var PTj+ Lu, Tj F. +δ j+ L, T j In he case of he lognormal model of Libor raes, we have E PTj Lu, Tj F = L, Tj + δ j+l, T j e v 2 j,u, 9 +δ j+ L, T j where In paricular, v 2 j, u =Var PTj+ u λs, T j dws Tj+ = u λs, T j 2 ds. L, T j =L, T j + δ j+l, T j e v 2 j,tj. +δ j+ L, T j Proof. Combining 4 wih he maringale propery of L,T j under P Tj+, we obain E PTj Lu, Tj F = E PTj+ + δj+ Lu, T j Lu, T j F +δ j+ L, T j so ha E PTj Lu, Tj F = L, Tj + δ j+ E PTj+ Lu, Tj L, T j 2 F. +δ j+ L, T j In he case of he lognormal model, we have Lu, T j =L, T j e η,u 2 v2 j,u, where Consequenly, we have η, u = u λs, T j dw Tj+ s. 2 E PTj+ Lu, Tj L, T j 2 F = L 2, T j e v2 j,u, his proves equaliy 9. ¾

13 UNSW, Repor No.S97-, Sepember 997 his version: June To derive he ransiion probabiliy densiy funcion p.d.f. of he process L,T j, noice ha for any u T j, any bounded Borel measurable funcion g : R R we have E PTj glu, Tj F = E PTj+ The following simple lemma appears o be useful. glu, T j +δ j+ Lu, T j F +δ j+ L, T j. Lemma 4. Le ζ be a nonnegaive rom 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 + ζgζ. Then he p.d.f. f Q of ζ under Q saisfies f P y =+yf Q y. Proof. The asserion is in fac rivial since, by assumpion, gyf P y dy = gy + yf Q y dy for any bounded Borel measurable funcion g : R R. ¾ Assume he lognormal model of Libor raes fix x R. Recall ha for any u we have Lu, T j =L, T j e η,u 2 Var P T j+ η,u, where η, u is given by 2 so ha i is independen of he σ-field F. Markovian propery of L,T j under he forward measure P Tj+ is hus apparen. Denoe by p L, x, u, y he ransiion p.d.f. under P Tj+ of he process L,T j. Elemenary calculaions involving Gaussian densiies yield p L, x, u, y =P Tj+ {Lu, T j =y L, T j =x} = lny/x+ { 2πvj, uy exp 2 v2 j, u 2 } 2vj 2, u for any x, y > arbirary <u. Taking ino accoun Lemma 4., we conclude ha he ransiion p.d.f. of he process 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} = +δ j+y +δ j+ x p L, x, u, y. We are in a posiion o sae he following resul. Corollary 4. The ransiion p.d.f. under P Tj p L, x, u, y = of he forward Libor rae L,T j equals +δ j+ y lny/x+ { 2πvj, u y + δ j+ x exp 2 v2 j, u 2 } 2vj 2, u 2 for any <u arbirary x, y >. The Markovian propery of L,T j under P Tj follows from he properies of he forward price see he nex secion.

14 4 M.Rukowski: Spo, Forward, Fuures Libor Raes 4.4 Dynamics of F B,T j+,t j under P Tj Observe ha he forward bond price F B,T j+,t j saisfies F B, T j+,t j = B, T j+ = B, T j +δ j+ L, T j. 22 Firs, his implies ha in he lognormal model of Libor raes, he dynamics of he forward bond price F B,T j+,t j are governed by he following sochasic differenial equaion, under P Tj, df B = F B F B λ, T j dw Tj, 23 wherewewrief B =F B, T j+,t j. If he iniial condiion saisfies <F B <, his equaion can be shown o admi a unique srong soluion i saisfies <F B < for every >. This makes clear ha he process F B,T j+,t j hus also he process L,T j are Markovian under P Tj. Using formula 2 relaionship 22, one can find he ransiion p.d.f. of he Markov process F B,T j+,t j under P Tj ;hais, p B, x, u, y =P Tj {F B u, T j+,t j =y F B, T j+,t j =x}. We have he following resul see Rady Smann [2], Milersen e al. [8], Jamshidian [7]. Corollary 4.2 The ransiion p.d.f. under P Tj of he forward bond price F B,T j+,t j equals 2 x ln x y p B, x, u, y = 2πvj, uy 2 y exp y x + 2 v2 j, u 2vj 2, u for any <u arbirary <x,y<. Proof. Le us fix x,. Using 22, i is easy o show ha p B, x, u, y =δ j+ y 2 p L, x δ j+ x,u, y. δ j+ y ¾ The formula now follows from 2. In his probabilisic derivaion of he closed-form soluion for he bond opion price in he lognormal model of Libor raes, Goldys [5] esablished he following resul which can also be used o value oher European coningen claims. Proposiion 4.4 Le X be a soluion of he following sochasic differenial equaion dx = X X λ dw, X = x, 24 where W follows a sard Brownian moion under P, λ : R + R is a locally square inegrable funcion. Then for any nonnegaive Borel measurable funcion g : R R any u> we have E P gx u = xe Q gh ζ + xe Q gh2 ζ, 25 where ζ has, under Q, a Gaussian law wih zero mean value variance Var Q ζ =v 2,u= u λs 2 ds. Furhermore, he funcions h,h 2 : R R are given by he formula h,2 z =+exp z +ln x ± x 2 v2,u, y R.

15 UNSW, Repor No.S97-, Sepember 997 his version: June Goldys [5] proved also he following formula, which indeed is closely relaed o 24 see Appendix A for deails. For any any nonnegaive Borel measurable funcion g : R R any u>we preserve here he noaion of Proposiion 4.4 E P gx u = x x e 8 v2,u E Q {g e z+ζ/2 + e z+ζ/2}, 26 +e z+ζ where we wrie z =lnx/ x. Equaliy 26 was employed in Goldys [5] o obain he closed-form expression for he price of a European call or pu opion, wih expiraion dae T j, wrienona zero-coupon bond of mauriy T j+. Since he payoff of a pu opion equals X =K BT j,t j+ +, i is clear ha he opion s price a ime admis he following represenaion π X =B, T j E PTj K FB T j,t j+,t j + F. The condiional expecaion in he righ-h side can be explicily evaluaed using, for insance, equaliy 26. On he oher h, i is no hard o check ha a cap seled in arrears ha is, a porfolio of call opions wrien on a Libor rae is essenially equivalen o he porfolio of pu opions on a zero-coupon bond. This equivalence leads o he closed-form soluion for he arbirage price of a cap wihin he framework of he lognormal model of forward Libor raes see [2], [8]. A similar approach o he pricing of caps wihin he lognormal model of forward Libor raes was previously adoped by Milersen e al. [8] who used, however, a slighly differen echnique in he las sep. Following Rady Smann [2], hey focused on he parial differenial equaion, which is saisfied by he funcion v = v, x ha expresses he forward price of a bond opion in erms of he forward price of he underlying bond. I can be shown ha v solves he following PDE v + 2 λ, T j 2 x 2 x 2 2 v x 2 = 27 wih he erminal condiion vt j,x=k x +. PDE 27 was explicily solved by Rady Smann [2], who obained in his way a closed-form soluion for he price of a bond opion hey worked, however, wihin a oally differen framework; namely, he Bühler Käsler [3] erm srucure model. In his conex, i is worhwhile o menion he paper by Rady [], in which a simple derivaion of he bond opion price based on he change of numeraire echnique is presened see Appendix B below. Le us finally observe ha from Brace e al. [2], i is clear ha he valuaion of a cap can also be easily done using he sard by now forward measure echnique i is enough o evaluae he expeced payoff of each caple under he corresponding forward measure P Tj. Neverheless equaliies are sill useful since hey allow us o value any European coningen claim of he form gbt j,t j+ or 8 which seles a ime T j. Before we end his secion, we shall check ha formula 22 can be rederived using formula 25. According o 25, we have E PTj gx u = I + I 2, where I + I 2 = x 2πv,u gh z e z2 /2v 2,u dz + x 2πv,u gh 2 z e z2 /2v 2,u dz. Firs, le us subsiue y = h z ini, so ha dy = y ydz. Then we obain 2 x gy ln x y I = 2πv,u y y exp y x 2 v2,u 2v 2,u dy, or equivalenly, I = x 2πv,u gy x y y y y x exp ln x y y x + 2 v2,u 2v 2,u 2 dy.

16 6 M.Rukowski: Spo, Forward, Fuures Libor Raes Similarly, subsiuing y = h 2 z ini 2 so ha once again dy = y y dz, we ge 2 x gy ln x y I 2 = 2πv,u y y exp y x + 2 v2,u 2v 2,u dy. By simple algebra, we find ha E PTj gx u = 2πv,u xgy y 2 y exp ln x y y x + 2 v2,u 2v 2,u 2 dy. The formula above easily generalizes o he case <uo his end, i is enough o consider he SDE 24 wih he iniial condiion X = x 2 xgy ln x y E PTj gx u X = x = 2πv, u y 2 y exp y x + 2 v2, u 2v 2, u dy. Applying he las formula o he process X = F B, T j+,t j, we obain an alernaive proof of Corollary Modelling of Fuures Libor Raes As already menioned, he properies of he fuures Libor raes in he lognormal model of forward Libor raes depend essenially on he choice of he spo maringale measure. For insance, Brace e al. [2] place hemselves in he HJM framework, herefore heir consrucion hinges on a prespecified spo maringale measure. They specify heir model by imposing he following condiion on he bond price volailiy: b, T =, for [T δ, T ], for any mauriy dae T. They assume ha he lengh of he accrual period equals a posiive consan δ in a discree-enor framework, his corresponds o he specificaion T j = T + jδ. Using he condiion above, wih T = T = δ, we find ha b, T 2 = δl, T +δl, T λ, T, [,T, 28, of course, b, T 2 = for every [T,T 2 ]. In he general HJM framework for any mauriy dae T he Radon-Nikodým densiy of he spo probabiliy measure P wih respec o he forward measure P T is known o saisfysee, for insance, Chaper 3 in Musiela Rukowski [] dp = E T bu, T dwu T 29 dp T on Ω, F T. In view of he las formula, i is clear ha P = P T on Ω, F T since he price volailiy of he T -mauriy bond vanishes idenically. Consequenly, he fuures Libor rae L f,t coincides wih he adjused forward Libor rae L,T. Le us now consider he nex dae, T 2. Using 29 reasoning along he similar lines, we conclude ha he spo maringale measure P is uniquely defined on Ω, F T2 hrough he formula dp = E T bu, T 2 dwu T2, 3 dp T2 wih he funcion b,t 2 given by 28. The fuures Libor rae for he dae T 2 can hus be deermined, a leas in principle. Indeed, we have cf. 7 L f, T 2 =E PT2 {LT 2 exp T where we have used he Bayes rule. Recall ha bu, T 2 dw T2 u 2 T } bu, T 2 2 du F,

17 UNSW, Repor No.S97-, Sepember 997 his version: June T 2 LT 2 =L, T 2 exp λu, T 2 dwu T3 T2 λu, T 2 2 du 2 dw T3 = dw T2 + δl, T 2 +δl, T 2 λ, T 2 d. I is hus apparen ha an explici expression for he condiional expecaion 4.5 is no easily available under presen assumpions hough i is possible o esablish an approximae formula. If condiion: b, T =, for [T δ, T ], is relaxed, he fuures Libor rae L f,t 2 saisfies L f, T 2 =E PT2 {LT 2 exp T2 bu, T 2 dw T2 u 2 T2 } bu, T 2 2 du F. Le us sress ha he specificaion of he bond price volailiy b,t 2 compaible wih he lognormal model of forward Libor raes is a raher nonrivial problem. A more promising approach would be perhaps o focus direcly on fuures Libor raes for a prespecified se of daes, wih he aim o produce a erm srucure model yielding closed-form soluions for prices of Eurodollar fuures opions wih differen expiraion daes. For a furher discussion of his poin, we refer o Rukowski [3]. References [] Amin, K., Ng, V.K.: Inferring fuure volailiy from he informaion in implied volailiy in Eurodollar opions: A new approach. Rev. Finan. Sud. 997, [2] Brace, A., G aarek, D., Musiela, M.: The marke model of ineres rae dynamics. Mah. Finance 7 997, [3] Bühler, W., Käsler, J.: Konsisene Anleihenpreise und Opionen auf Anleihen. Working paper, Universiy of Dormund, 989. [4] Flesaker, B.: Arbirage free pricing of ineres rae fuures forward conracs. J. Fuures Markes 3 993, [5] Goldys, B.: A noe on pricing ineres rae derivaives when LIBOR raes are lognormal. Finance Sochas. 997, [6] Heah, D., Jarrow, R., Moron, A.: Bond pricing he erm srucure of ineres raes: A new mehodology for coningen claim valuaion. Economerica 6 992, [7] Jamshidian, F.: LIBOR swap marke models measures. Finance Sochas. 997, [8] Milersen, K., Smann, K., Sondermann, D.: Closed form soluions for erm srucure derivaives wih log-normal ineres raes. J. Finance , [9] Musiela, M., Rukowski, M.: Coninuous-ime erm srucure models: Forward measure approach. Finance Sochas. 997, [] Musiela, M., Rukowski, M.: Maringale Mehods in Financial Modelling. Springer, Berlin Heidelberg New York, 997. [] Rady, S.: Opion pricing in he presence of naural boundaries a quadraic diffusion erm. Finance Sochas. 997, [2] Rady, S., Smann, K.: The direc approach o deb opion pricing. Rev. Fuures Markes 3 994, [3] Rukowski, M.: Models of forward Libor swap raes. Preprin, Universiy of New Souh Wales, 997 submied o Appl. Mah. Finance. [4] Rukowski, M.: Self-financing rading sraegies for sliding, rolling-horizon, consol bonds. Preprin, Universiy of New Souh Wales, 997 submied o Mah. Finance.

18 8 M.Rukowski: Spo, Forward, Fuures Libor Raes 5 Appendix A The resuls of his appendix are due o Goldys [5]. Le X be a soluion o he sochasic differenial equaion { dx = X X λ dw, 3 X = x, where we assume ha he funcion λ :[,T ] R d is bounded measurable, as usual, W is a sard Brownian moion defined on a filered probabiliy space Ω, F, P. For any x,, he exisence of a unique global soluion o 3 can be easily deduced from he general heory of sochasic differenial equaions. However, in Lemma 5. below we provide a direc proof by means of a simple ransformaion which is also crucial for he furher calculaions. Consider he following sochasic differenial equaion dz = e Z 2 +e λ 2 d λ dw 32 Z wih Z = z. Since he drif erm in his equaion is represened by a bounded globally Lipschiz funcion, equaion 32 is known 2 o have a unique srong non-exploding soluion for any iniial condiion z R. Lemma 5. For any x, he process x X = +e Z, [,T ], 33 where Z = z =ln x is a unique srong non-exploding soluion o equaion 3. Moreover, <X < for every [,T ]. Proof. I is easy o see ha equaion 32 can be rewrien in he form dz = X 2 λ 2 d λ dw. Hence, applying he Iô formula o he process X given by formula 33, we find ha dx = X X λ dw. Therefore, he process X is indeed a soluion o 3. Conversely, if X is any local weak soluion o 3 hen i is in fac a srong soluion because he diffusion coefficien in 3 is locally Lipschiz. Moreover, using again he Iô formula, one can check ha he process Z =lnx ln X, [,T ], 34 is a srong soluion of 32. Hence i can be coninued o a global one which is unique. The las par of he lemma follows from he definiion of he process X uniqueness of soluions o 3. ¾ Theorem 5. Le g : R R be a nonnegaive Borel funcion. Then for every x, > he expeced value E P gx is given by he formula E P gx = x x e 8 v2, E Q {g e z+ζ/2 + e z+ζ/2}, +e z+ζ where z =ln x x, he rom variable ζ has, under Q, a Gaussian law wih zero mean value variance v 2,= λu 2 du. 2 See, for insance, Theorem in: I.Karazas S.Shreve, Brownian Moion Sochasic Calculus. Springer, Berlin Heidelberg New York, 988.

19 UNSW, Repor No.S97-, Sepember 997 his version: June Proof. The proof borrowed from Goldys [5] is based on a simple idea ha for any rom variable U say we have E P U = E P X U+E P Y U, 35 where Y = X. I is essenial o observe ha X = x E Y u λu dw u 36 while Furhermore, from 32 i follows ha Y = x E X u λu dw u. 37 dz = λ dw X 2 λ d, or equivalenly, dz = λ dw Y + 2 λ d. Le us inroduce probabiliy measures P ˆP on Ω, F by seing Noice ha where he processes d P dp = E d ˆP dp = E Z = z W u = W u Ŵ u = W u Xu 2 λu dwu def = η 38 Yu + 2 λu dwu def = ˆη. 39 λu d W u = z u u Xv 2 λv dv, u [,], Yv + 2 λv dv, u [,], λu dŵu, 4 are known o follow sard Brownian moions under P ˆP, respecively. Simple manipulaions show ha η = E X u λu dw u exp λu d 2 W u + λu 2 du 8 so ha, using 37, we obain Y = x e 8 v2, η exp 2 Similarly, we have hus, in view of 36, ˆη = E Y u λu dw u exp 2 X = xe 8 v2, ˆη exp 2 λu dŵu + 8 λu d W u. 4 λu dŵu λu 2 du,. 42

20 2 M.Rukowski: Spo, Forward, Fuures Libor Raes Equaliy 42 yields while 4 gives E P X gx = xe 8 v2, E ˆP E P Y gx = x e 8 v2, E P Using 33 35, we conclude ha E P gx = e 8 v2, E Q {g { gx exp 2 { gx exp 2 +e z+ζ λu dŵu }, λu d W u }. xe 2 ζ + xe 2 ζ}, where ζ is, under Q, a Gaussian rom variable wih zero mean value variance v 2,. Equivalenly, E P gx = x x e 8 v2, E Q {g e z+ζ/2 + e z+ζ/2}. +e z+ζ This ends he proof of he heorem. ¾ Remark. To esablish direcly formula 25, a slighly differen changes of he underlying probabiliy measure P appear o be convenien. Namely, we pu on Ω, F d P dp = E X u λu dw u Now, under P ˆP we have respecively, where W u = W u u d ˆP dp = E Y u λu dw u. dz u = 2 λu du λu d W u dz u = 2λu d λu dŵu, X v λv dv, Ŵ u = W u u Y v λv dv, u [,], are sard Brownian moions under he corresponding probabiliies. Equaliy 25 can hus be easily derived from Corollary 5. Le g : R R be a nonnegaive Borel funcion. Then for every x, any <<T he condiional expecaion E P gx T F equals E P gx T F =kx, where he funcion k :, R is given by he formula kx = x x e 8 v2,t E Q {g e z+ζ/2 + e z+ζ/2}, +e z+ζ z =lnx/ x, he rom variable ζ has, under Q, a Gaussian law wih zero mean value variance T v 2, T = λu 2 du.

21 UNSW, Repor No.S97-, Sepember 997 his version: June In he nex proposiion he bond opion valuaion formula which was previously obained hrough he PDE approach by Rady Smann [2] is rederived. For sake of noaional simpliciy, we shall someimes wrie T j = T T j+ = T + δ. Proposiion 5. The price C a ime T of a European call opion wrien on a zero-coupon bond mauring a T + δ, wih expiraion dae T srike price <K<, equals C = KB, T + δnl + 2 v K B, T B, T + δ Nl 2 v, where Proof. l = l, T = v 2 = v 2, T = In view of Corollary 5., i is clear ha T λu, T 2 du v, T ln KB, T + δ K B, T B, T + δ. C = B, T E PT FB T,T + δ, T K + F = B, T kx, 43 where x = F B, T + δ, T gy =y K +. Using he noaion v 2 = v 2, T l = l, T = v ln Kx xk, we obain kx = x x e 8 v2 l +e K e z+vy 2 z+vy + e z+vy 2 ny dy, where n ss for he sard normal densiy. Le us denoe hy =e 2 z+vy + e 2 z+vy. Then kx = x x e hyny 8 v2 dy K l +e z+vy Equivalenly, kx = x x e 8 v2 I KI 2, where I = Consequenly, we find ha l e 2 z+vy ny dy = e 8 v2 + 2 z N l 2 v I 2 = I + e 8 v2 2 z N l + 2 v. l hyny dy. kx = x x Ke 2 z N l 2 v K x xe 2 z N l + 2 v, or, afer simplificaion, kx =x KNl + 2 v K xnl 2v. 44 Since x = F B, T + δ, T =B, T + δ/b, T, o end he proof i is enough o combine 43 wih 44. ¾

22 22 M.Rukowski: Spo, Forward, Fuures Libor Raes Using he pu-call pariy relaionship C P = B, T j KB, T j, i is easy o check ha he price of he corresponding pu opion equals P =K B, T j N l 2 v K B, T j B, T j N l + 2 v. 45 I is well known easy o check ha a caple is equivalen o he pu opion wrien on a zero-coupon bond, wih he srike price K = d j =+κd j more precisely, muliplied by a consan δ j =+κδ j. Therefore, using 45, we obain Cpl = B, T j B, T j N l + 2 v κδ j B, T j N l 2 v, since K = κδ δ j j. This ends a probabilisic derivaion of he cap valuaion formula wihin he lognormal model of forward LIBOR raes as already menioned, anoher approach o he bond opion valuaion relies on solving he parial differenial equaion Appendix B A more direc however, slighly less general probabilisic approach o he valuaion of a bond opion wihin he framework of he lognormal model of forward LIBOR raes was proposed by Rady []. We provide below a shor accoun of his resuls. We are given an arbirage-free model of securiy marke, le Z Z 2 be wo sricly posiive price processes of primary securiies. Our goal is o price an opion o exchange one asse for anoher, more specifically, we consider a European coningen claim Y ha seles a ime T Y =Z T KZ2 T + = Z T I A KZ 2 T I A, 46 where A = {ZT >KZ2 T } is he exercise se. I appears ha under specific assumpions on he dynamics of he relaive price process X = Z /Z 2, an judicious choice of numeraire asses leads o a closed-form soluion. We posulae ha here exiss consans a, b, c, d such ha ad bc 3 dx =ax + bcx + dλ dw, 47 where λ :[,T] R d is a coninuous funcion. Moreover, W follows a sard Brownian moion, under he maringale measure P ha corresponds o he choice of securiy Z 2 as a numeraire asse. Le us assume ha ax + b cx + d. Then Ṽ def = ax + b, Ũ def = cx + d, 48 for every [,T] his can be deduced from he non-confluen propery of soluions of sochasic differenial equaions wih Lipschiz coninuous coefficiens. I is useful o noe ha dx = ṼŨλ dw, Le us denoe def V = az + bz 2 = ṼZ 2 def, U = cz + dz 2 = ŨZ 2. I is eviden ha V U represen wealh processes of wo self-financing rading sraegies, hus hese processes may be aken as numeraire asses noe ha hey never vanish. I is crucial o observe ha Y admis he following represenaion Y = Kc + dv T I A Ka + bu T I A ad bc = αv T I A + βu T I A, 3 Exisence uniqueness of a non-exploding srong soluion o SDE 47 can be jusified using he sard resuls of sochasic calculus see, in paricular, Theorem in: I.Karazas S.Shreve, Brownian Moion Sochasic Calculus. Springer, Berlin Heidelberg New York, 988. Le us only indicae ha if he iniial condiion lies beween he zeros of he funcion fx =ax + bcx+ d, hen he soluion says in his inerval forever.

23 UNSW, Repor No.S97-, Sepember 997 his version: June where α β are consans. Moreover, he exercise se A saisfies A = {αv T >βu T }. If, for insance V T >, U T >, ad bc >, Ka+ b>kc + d>, hen we have also A = {V T /U T >D} = {U T /V T <D }, 49 where we wrie D =Ka + bkc + d. From he general heory i is clear ha if he claim Y is aainable, hen is arbirage price π Y equals π Y =αv P{A F } + βu ˆP{A F }, 5 where P ˆP, respecively is he maringale measure which corresponds o he choice of he process V U, respecively as a numeraire asse. To find explici formulae for he condiional probabiliies in he righ-h side, we shall use he following resul which follows easily from Iô s formula. Lemma 6. Suppose ha X saisfies 47 processes Ṽ,Ũ are given by 48. Then where λ =ad bcλ. dṽ/ũ =Ṽ/Ũ λ dw cṽλ d, From Lemma 6. i is clear ha du/v =dũ/ṽ =U/V λ d W dv/u=dṽ/ũ =V/U λ dŵ, where W Ŵ, respecively follows a sard Brownian moion under P ˆP, respecively. This observaion combined wih equaliy 5 leads o he following valuaion resul, whose proof is sard, hus is omied. Proposiion 6. The arbirage price a ime of a claim Y given by 46 equals π Y =αv N d Z,Z2,,T + βu N d 2 Z,Z2,,T, where d,2 Z,Z 2,,T= lnv /U ln D ± σ 2, T σ, T D =Ka + bkc + d, σ 2, T = T T λu 2 du =ad bc 2 λu 2 du. Le us reurn o he case of a European call opion wrien on a zero-coupon bond mauring a T + δ, wih expiraion dae T srike price <K<. We have C T = BT,T + δ K + =Z T KZ 2 T +,, where Z = B, T + δ Z2 = B, T. Furhermore, he forward price of a zero-coupon bond X = B, T + δ/b, T =Z /Z 2 saisfies see 23 dx = X X λ, T dw T, 5

24 24 M.Rukowski: Spo, Forward, Fuures Libor Raes where W T follows a sard Brownian moion under he forward measure P T. To apply he change of numeraire mehod, i is hus enough o se V = Z = B, T + δ >, U = Z 2 Z = B, T B, T + δ >, ha is, we may ake a =,b=,c= d =. I is now sraighforward o check ha α = K, β = K, D = K/ K, represenaion 49 of he opion s exercise se is valid. Therefore, using Proposiion 6., we may find he price of a call bond opion, namely, C = KB, T + δn d Z,Z2,,T K B, T B, T + δ N d 2 Z,Z2,,T. Since see Proposiion 5. d,2 Z,Z2,,T=l, T ± 2 v, T σ 2, T =v 2, T, we conclude ha he las formula coincides wih he valuaion formula obained in Proposiion 5.. This complees Rady s derivaion of he bond opion valuaion formula for he lognormal model of forward Libor raes. I is worhwhile o poin ou ha since he formula esablished in Proposiion 6. applies only o opions o exchange one asse for anoher, i is less general hen any of represenaions obained by Goldys [5]. In principle, he laer may serve o value any coningen claim, eiher hrough numerical inegraion or hrough he Mone Carlo procedure. We end his noe by presening a minor modificaion of Rady s approach o he valuaion he bond opion wihin he framework of he lognormal model of forward Libor raes. Noe firs ha C T = BT,T + δ K + = KBT,T + δ K BT,T BT,T + δ + so ha as before, we denoe D = K K C T = K Z T D Z 2 T / Z T +, where Z = B, T + δ Z 2 = B, T B, T + δ. Le us denoe Z = Z 2 / Z = B, T B, T + δ B, T + δ = X. Using Iô s formula combined wih 5, we obain d Z = dx = Z λ, T dw T + A d, where A is an adaped process. Consequenly, under he maringale measure P which corresponds o he choice of he process B,T + δ as a numeraire i.e., under he forward measure for he dae T + δ wehave d Z = Z λ, T d W, where W is a P-Brownian moion. Consequenly Z T = Z T exp λu, T d W u T λu, T 2 du. 2 Since he opion s price a ime equals C = K Z E P D ZT + F, he formula of Proposiion 5. follows easily by sard argumens.