Continuous-time term structure models: Forward measure approach

Transcription

1 Finance Sochas. 1, (1997 c Springer-Verlag 1997 Coninuous-ime erm srucure models: Forward measure approach Marek Musiela 1, Marek Rukowski 2 1 School of Mahemaics, Universiy of New Souh Wales, Sydney 252, NSW, Ausralia ( musiela@soluion.mahs.unsw.edu.au 2 Insiue of Mahemaics, Poliechnika Warszawska, PL--661 Warszawa, Poland ( markru@alpha.im.pw.edu.pl Absrac. The problem of erm srucure of ineres raes modelling is considered in a coninuous-ime framework. The emphasis is on he bond prices, forward bond prices and so-called LIBOR raes, raher han on he insananeous coninuously compounded raes as in mos radiional models. Forward and spo probabiliy measures are inroduced in his general se-up. Two condiions of noarbirage beween bonds and cash are examined. A process of savings accoun implied by an arbirage-free family of bond prices is idenified by means of a muliplicaive decomposiion of semimaringales. The uniqueness of an implied savings accoun is esablished under fairly general condiions. The noion of a family of forward processes is inroduced, and he exisence of an associaed arbirage-free family of bond prices is examined. A sraighforward consrucion of a lognormal model of forward LIBOR raes, based on he backward inducion, is presened. Key words: Term srucure of ineres raes, forward measure, maringale measure, LIBOR rae JEL classificaion: E43, E44 Mahemaics Subjec Classificaion (1991: 6G44, 6H3, 9A9 Par of his work was carried ou while visiing he Universiy of Bonn. We would like o acknowledge financial suppor from he Ausralian Research Council and from he Deusche Forschungsgemeinschaf, Sonderforschungsbereich 33. We hank an anonymous referee for his valuable commens. Manuscrip received: July 1996; final version received: Ocober 1996

2 262 M. Musiela, M. Rukowski 1. Inroducion The Heah-Jarrow-Moron erm srucure mehodology (see Heah e al. [11] is based on an arbirage-free dynamics of he insananeous coninuously compounded forward raes. The exisence of such raes, however, requires a cerain degree of smoohness wih respec o he enor of he bond prices and heir volailiies, and hus working wih such models may be inconvenien. 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 [18], who focused on he effecive annual ineres rae. This approach was furher developed in he following series of papers: Sandmann e al. [19], Goldys e al. [9], Musiela [16], Milersen e al. [15]. I is ineresing o observe ha Brace e al. [4] paramerize heir version of he lognormal forward LIBOR model inroduced by Milersen, Sandmann and Sondermann in [15] wih a piecewise consan volailiy funcion, however, hey need o consider smooh volailiy funcions in order o analyse he model in he Heah-Jarrow-Moron framework. In he presen paper, he problem of coninuous-ime modelling of erm srucure of ineres raes is considered in a general manner. We describe cerain properies which are valid for wide classes of erm srucure models, so ha a basis for he discussion of any specific model is developed. Three such special sysems are pu forward, and heir properies are discussed (we refer o hem as o he se-ups (BP, (FP and (LR in wha follows. The paper proceeds as follows. In Secion 2, we deal wih he quesion of exisence and uniqueness of a savings accoun implied by a given (weakly arbirage-free coninuous-ime family of bond prices. The nex secion is devoed o he problem of a consrucion of an arbirage-free family of bond prices given a family of sochasic volailiies of forward processes and an iniial erm srucure. Finally, in Secion 4 a consrucion of he lognormal model of forward LIBOR raes is presened. Le us commen briefly on he exisence of a shor-erm rae of ineres. In he radiional models, in which he insananeous coninuously compounded shor-erm rae r is well defined, he savings accoun process, B say, saisfies B ( = exp r u du so ha i represens he amoun generaed a ime by coninuously reinvesing $1 in he shor-erm rae r (in such a framework, he absence of arbirage is relaed o he non-negaiviy of he shor-erm rae. Though we will deal someimes wih an (implied savings accoun, we will no assume ha is sample pahs are absoluely coninuous wih respec o he Lebesgue measure, herefore, erm srucure models in which he insananeous rae of ineres is no well-defined will be also covered by our subsequen analysis. In his more general seing, he absence of arbirage beween all bonds wih differen mauriies implies he exisence of a savings accoun which follows a process of finie variaion. If, (1

3 Coninuous-ime erm srucure models 263 in addiion, he absence of arbirage beween bonds and cash is assumed, he savings accoun is shown o follow an increasing process. 2. Bond price models We assume ha we are given a probabiliy space (Ω,(F [,T ], P equipped wih a filraion (F [,T ] which saisfies he usual condiions. For he ease of exposiion, we make he following sanding assumpion (see Björk e al. [2] [3] for more general erm srucure models. Assumpion (A. The process W is a d-dimensional sandard Wiener process defined on a filered probabiliy space (Ω,(F [,T ], P. The underlying filraion (F [,T ] coincides wih he usual P-augmenaion of he naural filraion of W. We wrie M loc (P and M(P o denoe he class of all real-valued local maringales and he class of all real-valued maringales, respecively. The subscrip c will indicae ha we consider processes wih coninuous sample pahs, and he superscrip + will denoe he collecion of all sricly posiive processes which belong o a given class of processes. For insance, M c + (P sands for he class of sricly posiive P-maringales wih coninuous sample pahs. We denoe by V (by A, respecively he class of all real-valued adaped (predicable, respecively processes of finie variaion. We wrie S p (P o denoe he class of all real-valued special semimaringales, i.e., hose processes X which admi a decomposiion X = X + M + A, where M M loc (P and A A. Abusing slighly our convenion, we denoe by S p + (P he class of hose special semimaringales X S p (P which are sricly posiive, and such ha, in addiion, he process of lef hand limis, X, is also sricly posiive (all he processes considered here are assumed o be càdlàg ha is, wih almos all sample pahs being righ-coninuous funcions, wih finie lef-hand limis. Noice ha he class S p (P (as well as S p + (P is invarian wih respec o an equivalen change of he underlying probabiliy measure. More precisely, S p (Q =S p (P and S p + (Q =S p + (PifQand P are muually equivalen probabiliy measures on (Ω,F T such ha he Radon-Nikodým densiy Λ = dq dp F, [, T ], follows a locally bounded process (see [5], p.258. For breviy, we wrie Q P o denoe ha wo probabiliy measures Q and P are muually equivalen. Since he filraion (F [,T ] is generaed by a Wiener process, he Radon-Nikodým densiy Λ will always follow a coninuous exponenial maringale, hence a locally bounded process. Therefore, we may and do wrie simply S p and S p + in wha follows.

4 264 M. Musiela, M. Rukowski 2.1. Family of bond prices Le us fix a sricly posiive horizon dae T >, and le B(, T sand for he price a ime T of a zero-coupon bond which maures a ime T T. By a family of bond prices we mean an arbirary family of sricly posiive real-valued adaped processes B(, T, [, T ], wih B(T, T = 1 for every T [, T ]. Noice ha, for convenience, he assumpion ha he bond price B(, T follows a semimaringale is no imposed in he definiion of a family of bond prices. In his secion, a family B(, T of bond prices is assumed o be given ha is, already consruced by means of a cerain procedure. We shall usually make he following assumpions. (BP.1 For any dae T [, T ], he bond price B(, T, [, T ], belongs o he class S p +. (BP.2 For any fixed T [, T ], he forward process F B (, T, T def = B(, T B(, T, [, T ], follows a maringale under P, or equivalenly, ( B(,T B(, T =E P B(T,T F, [, T ]. (2 In view of (BP.1 (BP.2, he process F B (, T, T, [, T ], follows under P a sricly posiive coninuous maringale wih respec o he filraion of a Wiener process, so ha F B (, T, T isinm c(p. + Consequenly, for any fixed T [, T ] here exiss a R d -valued predicable process γ(, T, T, [, T ], inegrable wih respec o he Wiener process W, and such ha ( F B (, T, T =F B (, T, T E γ(u, T, T dw u, or more explicily, ( F B (, T, T =F B (, T, T exp γ(u, T, T dw u 1 2 γ(u, T, T 2 du. Pu anoher way, for any fixed mauriy T [, T ] he process F B (, T, T saisfies df B (, T, T =F B (,T,T γ(,t,t dw. (3 Le us now consider any wo mauriies T, U [, T ]. We define he forward process F B (, T, U by seing F B (, T, U def = F B(, T, T F B (, U, T = B(, T, [, T U ]. (4 B(, U Suppose firs ha U > T ; hen he amoun

5 Coninuous-ime erm srucure models 265 f s (, T, U =(U T 1 (F B (,T,U 1 (5 is he add-on (annualized forward rae over he fuure ime inerval [T, U ] prevailing a ime, and f (, T, U = ln F B(, T, U. U T is he (coninuously compounded forward rae a ime over his inerval. On he oher hand, if U < T hen F B (, T, U represens he value a ime of he forward price of a T -mauriy bond for he forward conrac which seles a ime U. The following lemma is a sraighforward consequence of Iô s formula. Lemma 2.1 For any mauriies T, U [, T ], he dynamics under P of he forward process are given by he following expression where df B (, T, U =F B (,T,Uγ(,T,U (dw γ(, U, T d, (6 for every [, U T ] γ(, T, U =γ(,t,t γ(,u,t (7 Combining Lemma 2.1 wih Girsanov s heorem, we obain where for every [, U ] df B (, T, U =F B (,T,Uγ(,T,U dw U, (8 W U = W γ(u, U, T du. (9 The process W U is a sandard Wiener process on he filered probabiliy space (Ω,(F [,U ], P U, where he probabiliy measure P U P is defined on (Ω,F U by means of is Radon-Nikodým derivaive wih respec o he underlying probabiliy measure P dp ( U dp = E U γ(u, U, T dw u, P-a.s. (1 I is apparen ha he forward process F B (, T, U follows an exponenial local maringale under he forward probabiliy measure P U, since equaliy (8 yields ( F B (, T, U =F B (, T, U E γ(u, T, U dwu U for [, U T ]. Observe also ha we have P T = P and W T = W. We hus recognize he underlying probabiliy measure P as a forward maringale measure associaed wih he horizon dae T.

6 266 M. Musiela, M. Rukowski 2.2. Spo and forward maringale measures The concep of a forward maringale measure is inroduced in erms of he behaviour of relaive bond prices. In he presen conex, we find i convenien o make use of he noion of a forward process. Definiion 2.1 Le U be a fixed mauriy dae. A probabiliy measure Q U P on (Ω,F U is called a forward maringale measure for he dae U if for any mauriy T [, T ] he forward process F B (, T, U, [, T U ], follows a local maringale under Q U. I follows immediaely from assumpion (BP.2, ha he underlying probabiliy measure P is indeed a forward probabiliy measure for he dae T, in he sense of Definiion 2.1. Le us now inroduce he noion of a spo maringale measure wihin he presen framework. Inuiively speaking, a spo measure is a forward measure associaed wih he iniial dae T =. Is formal definiion relaes o a very specific kind of discouning, however. I should be sressed ha neiher a forward measure for he dae T, nor a spo measure, are uniquely defined, in general. Definiion 2.2 A spo maringale measure for he se-up (BP.1 (BP.2 is any probabiliy measure P P on (Ω,F T for which here exiss a process B A +, wih B =1,and such ha for any mauriy T (, T ] he bond price B(, T saisfies B(, T =E P (B /B T F, [, T ]. ( Arbirage-free properies We shall sudy wo forms of absence of arbirage. The firs, weaker noion refers o a pure bond marke. The second form assumes, in addiion, ha cash is also presen. Noe ha by cash we mean here money which can be carried over a no cos, raher hen a savings accoun yielding a posiive ineres. We shall formulae now a sufficien condiion for he absence of arbirage beween bonds wih differen mauriies (as well as beween bonds and cash. Definiion 2.3 A family B(, T of bond prices is said o saisfy he weak noarbirage condiion if and only if here exiss a probabiliy measure Q P on (Ω,F T such ha for any mauriy T < T he forward process F B (, T, T = B(,T/B(,T belongs o M loc (Q. We say ha he family B(, T saisfies he no-arbirage condiion if, in addiion, inequaliy B(T, U 1 holds for any mauriies T, U [, T ] such ha T U. Assumpion (BP.2 is manifesly sufficien for he family B(, T o saisfy a weak no-arbirage condiion as we may ake Q = P. As menioned, if a family B(, T saisfies a weak no-arbirage condiion hen i is possible o consruc a model of he securiies marke wih he absence of arbirage across bonds wih

7 Coninuous-ime erm srucure models 267 differen mauriies (le us sress ha he weak no-arbirage condiion makes no explici reference o he presence of cash or a savings accoun. We shall now focus on he absence of arbirage beween all bonds and cash. Under (BP.2, inequaliy B(T, U 1, which is equivalen o F B (T, U, T 1, gives immediaely F B (, U, T =E P (F B (T,U,T F 1 (12 for every [, T ]. Since almos all sample pahs of he forward process F B (, U, T are coninuous funcions, we may reformulae his condiion in he following way. (BP.3 For any wo mauriies T U, he following inequaliy holds wih probabiliy 1 B(, U B(, T, [, T ]. (13 Suppose, on he conrary, ha B(, U > B(, T for cerain mauriies U > T. In such a case, by issuing a ime a bond of mauriy U, and purchasing a T -mauriy bond, one could lock in in a risk-free profi if, in addiion, cash were presen in he marke. Indeed, o mee he liabiliy a ime U i would be enough o carry over he period [T, U ] (a no cos one uni of cash received a ime T. A he inuiive level, he following hree condiions are equivalen: (i he bond price B(, T is a non-increasing funcion of mauriy T ; (ii he forward process F B (, T, U is never less han one; and (iii he bond price B(, T is never sricly greaer han 1 (cf., Corollary 2.3 below. No surprisingly, he absence of arbirage beween bonds and cash appears o be closely relaed o he quesion of exisence of an increasing savings accoun implied by he family B(, T. We adop he following definiion. Definiion 2.4 A savings accoun implied by he family B(, T of bond prices is an arbirary process B which belongs o A +, wih B =1,and such ha here exiss a probabiliy measure P P on (Ω,F T such ha he relaive bond price Z (, T def = B(, T /B, [, T ], (14 is a P -maringale for any mauriy T [, T ]. I is clear ha Z (, T isap -maringale if for any mauriy T he bond price B(, T saisfies B(, T =E P (B /BT F, [, T ]. (15 In paricular, we have B(, T =E P (1/B T, [, T ], (16 so ha an implied savings accoun B maches also he iniial erm srucure B(, T, T [, T ]. I is also clear ha he probabiliy measure P of Definiion 2.4 is a spo maringale measure for he family B(, T, in he sense of Definiion 2.2. One migh wonder if he normalized bond price process

8 268 M. Musiela, M. Rukowski B def = B(, T /B(, T would be a plausible choice of an implied savings accoun (corresponding o P = P. In view of (BP.2, we have E P (B /B T F =E P (B(,T /B(T,T F =B(,T, [, T ], (17 for any mauriy dae T. In a ypical coninuous-ime model of he erm srucure, he sample pahs of he bond price process B(, T are of infinie variaion, however. In such a case, a normalized bond price canno simulaneously play he role of a savings accoun. Consequenly, he spo and forward maringale measures are usually disinc Implied savings accoun In his secion, we ake up he issue of exisence of an implied savings accoun. We shall show ha, under (BP.1 (BP.3, here exis an increasing process B which represens an implied savings accoun for he family B(, T. We sar wih an auxiliary resul which deals wih he behaviour of he erminal discoun facor D = B 1 (, T, [, T ]. Noe ha he process D belongs o he class S + p since B(, T does. Lemma 2.2 Under he assumpions (BP.1 (BP.3, he erminal discoun facor D follows a sricly posiive supermaringale under he forward maringale measure P. Proof. Combining (2 wih (13, we obain B(, U =E P ( B(,T B(U,T F E P ( B(,T B(T,T F =B(,T, so ha E P (D U F E P (D T F for T U T. Seing = T in he las inequaliy, we find ha E P (D U F T E P (D T F T =D T for every T U T, so ha D is a P-supermaringale. To show he exisence of an implied savings accoun, we shall make use of he following sandard resul of Iô sochasic calculus (see, for insance, Theorem 6.19 in [12]. Proposiion 2.1 Suppose ha X belongs o he class S + p, wih X =1.There exiss a unique pair (M, A of sochasic processes such ha X = MA, he process M belongs o M + loc (P, wih M =1,and A belongs o A +, wih A =1. If, in addiion, X is a supermaringale hen A is a decreasing process.

9 Coninuous-ime erm srucure models 269 I is well known ha if a sricly posiive special semimaringale X follows a supermaringale, hen he process of lef hand limis X is also sricly posiive (see Proposiion 6.2 in [12], hence X belongs o he class S p +. Assume ha he process M in he decomposiion above has coninuous sample pahs his holds in our case, since he underlying filraion is generaed by a Wiener process. Then he process A is easily seen o belong o S p +. We find i convenien o idenify he implied savings accoun using a muliplicaive decomposiion of he erminal discoun facor D. To his end, le us formulae a corollary o Proposiion 2.1. Corollary 2.1 Under (BP.1 (BP.2, here exiss a predicable process ξ inegrable wih respec o he Wiener process W, and such ha he erminal discoun facor D admis he unique decomposiion ( D = D à M = D à E ξ u dw u, [, T ], (18 where M isinm + c,loc (P and à belongs o A+, wih à = M =1. If, in addiion, condiion (BP.3 is me hen à is a decreasing process. Proof. All asserions are immediae consequences of Lemma 2.2, combined wih Proposiion 2.1 and he represenaion heorem for sricly posiive maringales wih respec o he naural filraion of a Wiener process. Noe ha if here exiss a savings accoun B hen N = B /B(, T should follow a P-local maringale. Consequenly, D =1/B(,T =(B 1 N,which is essenially he muliplicaive decomposiion of D. We find i convenien o rewrie (18 as follows B def = 1/à = B(,T ( B(, T E ξ u dw u, [, T ]. (19 To show he exisence of an implied savings accoun, i is enough o check ha he process B given by (19 saisfies Definiion 2.4. We formulae he nex resul under he hypoheses (BP.1 (BP.3. Under (BP.1 (BP.2, all claims of Proposiion 2.2 remain valid, excep ha B is no necessarily an increasing process if he assumpion (BP.3 is relaxed. Proposiion 2.2 Le he family B(, T of bond prices saisfy (BP.1 (BP.3. Assume, in addiion, ha he process M, defined by he muliplicaive decomposiion (18 of he erminal discoun facor D is a maringale, and no only a local maringale under P. Le B =1/Ãbe an increasing predicable process uniquely deermined by (18. Then (i B represens a savings accoun implied by he family B(, T. (ii B is associaed wih he spo maringale measure P which equals dp dp def = M T = B T B(, T, P-a.s. (2 (iii The relaive price process B /B(, T follows a maringale under he forward maringale measure P for he dae T.

10 27 M. Musiela, M. Rukowski Proof. Le P be an arbirary probabiliy measure on (Ω,F T equivalen o P. Then he Radon-Nikodým densiy of P wih respec o P resriced o he σ-field F equals dp ( = E ξ u dw u, P -a.s., (21 dp F for some predicable process ξ. We sar by considering a zero-coupon bond of mauriy T. In view of (18, he relaive bond price Z (, T =B(,T /B saisfies under P ( Z (, T =B(, T /M = B(, T E 1 ξ u dw u (22 for [, T ]. Consequenly, under P we have ( Z (, T =B(, T exp ξ u dw u 1 2 ξ u (2 ξ u ξ u du, where W = W ξ u du follows a Wiener process under P. I is hus eviden ha he relaive bond price Z (, T is a local maringale under P provided ha ξ = ξ. Under his assumpion we have Z (, T =B(, T E ( ξ u dwu. (23 We are in a posiion o define a candidae for a spo probabiliy measure by seing ξ = ξ in (21. In view of Definiion 2.4, we have o check ha for any mauriy T < T he relaive bond price Z (, T follows a maringale under P. I is enough o show ha for any mauriy T < T we have B(, T =E P ( B B T F, [, T ]. (24 For his purpose, observe firs ha equaliy ξ = ξ, combined wih (21 (22, gives η = dp = B(, T dp F Z (, T = B B(, T B(, T, [, T ]. (25 Consequenly, using he absrac Bayes rule we obain ( def B ( I = E B P BT F = E η ( T B(, T P BT η F = E P B(T, T F, where he las equaliy follows from (25. Using he assumed equaliy (2, we find ha I = B(, T, as required. We have hus shown ha he process B =1/Ã saisfies all condiions of he definiion of an implied savings accoun, and P is he associaed spo maringale measure (his follows immediaely from (25. Noice ha he probabiliy measure P given by (2 is he spo maringale measure associaed wih he forward maringale measure P for he dae T. More generally, he forward measure P and he associaed spo measure P are relaed o each oher hrough he formula

11 Coninuous-ime erm srucure models 271 dp dp = B T B(, T, P-a.s. (26 Therefore, boh measures coincide if and only if he random variable BT is consan. As menioned earlier, he uniqueness of a spo and forward measure is no a universal propery. Summarizing, for any forward measure Q for he dae T, he probabiliy measure Q which is defined on (Ω,F T by he formula dq dq = B T B(, T, Q-a.s., (27 is a spo measure for he family B(, T. Conversely, if Q is a spo measure, hen he probabiliy Q given by (27 is a forward measure for he dae T Uniqueness of an implied savings accoun The aim of his secion is o esablish uniqueness of an implied savings accoun. We sar by an auxiliary resul. Proposiion 2.3 Le B and ˆB be wo processes from A + such ha for every T [, T ] E P ( B / B T F =EˆP (ˆB /ˆB T F, [, T ], (28 where P ˆP are wo probabiliy measures on (Ω,F T. If B = ˆB hen B = ˆB. Before we proceed o he proof of Proposiion 2.3, le us quoe he following resul from Dellacherie and Meyer [5] (p.231. Lemma 2.3 Le A be an increasing process, defined on a filered probabiliy space (Ω,(F [,T ], P which saisfies he usual condiions, and such ha he random variable A T is P-inegrable. Denoe by A p he dual predicable projecion of A. Then A p 2 n 1 = lim E P (A (k+12 n n A k2 n F k2 n, [, T ], k= where he convergence is in he sense of he weak L 1 norm. If A has no predicable jumps hen he convergence is in he sense of (srong L 1 norm. Moreover, for any bounded predicable process H we have 2 n 1 H u da p u = lim H k2 n n E P (A (k+12 n A k2 n F k2 n, [, T ]. k= Proof of Proposiion 2.3. We inroduce predicable processes of finie variaion à =1/ B and  =1/ˆB.Assume firs ha P = ˆP, so ha we have Y def =  E P (à T F =à E P ( T F, [, T ].

12 272 M. Musiela, M. Rukowski Equaliy,  = à follows immediaely from he uniqueness of a muliplicaive decomposiion of sricly posiive semimaringale Y (i is clear ha Y belongs o. We now consider he general case. Since P ˆP, he process Λ defined by S + p Λ = d ˆP d P F = E P (Λ T F, [, T ], follows a sricly posiive coninuous (hence predicable maringale under P. Equaliy (28 combined wih he Bayes rule yields E P (à T /à F =EˆP ( T / F =E P (Λ TÂT/(Λ Â F for every T [, T ], and hus E P (Λ (à T à /à F = E P (Λ T ( T  / F (29 for every [, T ]. We wish o show ha processes à and  admi he same dual predicable projecion, and hus coincide. Le us fix an arbirary [, T ]. I follows from (29, ha E P ( Λ n (à k n à k+1 n /à k n k F n k ( = E P Λ n ( k+1 n  k+1 n / k n F n k k, where for every naural n and every k =,...,2 n 1, we se k n = k2 n. By virue of Lemma 2.3, for he lef-hand side of he las equaliy we ge 2 n 1 lim n k= Λ n k à 1 E n k P (à n à k+1 n F n = Λ k k u à 1 u dã u, since he process H = Λ /à is predicable, and manifesly à p = Ã. To show ha 2 n 1 ( lim  1 n E n k P Λ n ( k+1 n  k+1 n F n = Λ k k u  1 u dâ u, [, T ], k= i is enough o verify ha 2 n 1 ( lim  1 n E n k P (Λ n Λ n ( k+1 k n  k+1 n F n =  1 k k u d Λ,  u =. k= The las equaliy follows from he fac ha he predicable quadraic covariaion Λ,  vanishes ( being a predicable process of finie variaion has null coninuous maringale componen. The following corollary o Proposiion 2.3 esablishes he uniqueness of an implied savings accoun. Corollary 2.2 Under (BP.1 (BP.2, he uniqueness of an implied savings accoun holds.

13 Coninuous-ime erm srucure models 273 Proof. Le B and ˆB be wo arbirary savings accouns implied by he family B(, T. Definiion 2.4 yields for every T [, T ] E P ( B / B T F =B(,T=EˆP (ˆB /ˆB T F, [, T ], where P and ˆP are muually equivalen probabiliy measures on (Ω,F T. Also, B and ˆB are predicable processes of finie variaion, hence, equaliy B = ˆB is a sraighforward consequence of Proposiion 2.3. The nex resul examines a relaionship beween spo and forward measures (for he proof, see Musiela and Rukowski [17]. Proposiion 2.4 Under he hypoheses (BP.1 (BP.2, he class of forward measures for he dae T and he class of spo measures admi a common elemen if and only if he implied savings accoun saisfies B T = E P ( B T, ha is, if he random variable B T is degenerae. In he nex corollary we deal wih he equivalence of various forms of noarbirage wih cash condiion. I should be noiced ha in he proof he implicaion (iii (iv we make use, in paricular, of Assumpion (A. Corollary 2.3 Under (BP.1 (BP.2, he following are equivalen. (i The bond price B(, T is a non-increasing funcion of mauriy dae T. (ii The forward process F B (, T, U, T U, is never sricly less han one. (iii The bond price B(, T is never sricly greaer han 1. (iv The implied savings accoun follows an increasing process. Proof. Equivalence of (i, (ii and (iv is rivial. Also i is obvious ha (iv implies (iii. I remains check ha (iv follows from (iii. Le B be he unique savings accoun associaed wih he family B(, T. Condiion B(, T 1 implies immediaely ha he process 1/B follows a sricly posiive supermaringale under he spo measure P. Since i is a process of finie variaion, is maringale par, being coninuous maringale by virue of Assumpion (A, vanishes idenically. Therefore, B is an increasing process Bond price volailiy Throughou his secion, we assume ha a family B(, T of bond prices saisfies (BP.1 (BP.3. In he presen se-up, we find convenien o inroduce he noion of a bond price volailiy by means of he following definiion. Definiion 2.5 An R d -valued adaped process b(, T is called a bond price volailiy for mauriy T if he bond price B(, T admis he represenaion db(, T =B(,Tb(,T dw + dc T, (3 where C T is a predicable process of finie variaion.

14 274 M. Musiela, M. Rukowski Under (BP.1 (BP.2, he exisence and uniqueness of bond price volailiy b(, T for any mauriy T is a simple consequence of he canonical decomposiion of he special semimaringale B(, T S p +, combined wih he predicable represenaion heorem. Also, i is no hard o check ha he bond price volailiy, as defined above, is invarian wih respec o an equivalen change of probabiliy measure. More precisely, if (3 holds, hen under any probabiliy measure P P we have db(, T =B(,Tb(,T d W +d C T (31 for some predicable process of finie variaion C T, where W follows a Wiener process under P. Since we have assumed ha condiions (BP.1 (BP.3 are saisfied, here exiss a unique savings accoun B associaed wih a spo probabiliy measure P. For any mauriy T, he relaive bond price Z (, T =B(,T/B follows a local maringale under P so ha ( Z (, T =B(, T E b(, T dwu. (32 By comparing he las equaliy wih (23, we find ha b(, T = ξ,i.e., he volailiy of a T -mauriy bond is deermined by he muliplicaive decomposiion (18. Upon seing T = in (32, we obain he following represenaion for a savings accoun B in erms of bond price volailiies B for every [, T ]. ( = B 1 (, exp b(u, dw u b(u, 2 du Remark 2.1 Observe ha for any mauriies T, U [, T ] we have (33 γ(, T, U =b(,t b(,u, [, T U ], (34 where γ(, T, U is he volailiy of he forward process F B (, T, U. Therefore, he forward volailiies γ(, T, U are uniquely specified by he bond price volailiies b(, T. I is hus naural o ask if he converse implicaion holds; ha is, wheher he bond price volailiies are uniquely deermined by he forward volailiies. Example 2.1 Le us focus on a special case when processes C T are absoluely coninuous; ha is, when for any mauriy T T we have db(, T B(, T = a(, T d + b(, T dw (35 for some adaped processes a(, T and b(, T. We assume, for simpliciy, ha a and b are uniformly bounded ha is, a(, T + b(, T K, for some consan K. Our goal is o show ha (35, combined wih he weak no-arbirage condiion, implies he exisence of an absoluely coninuous savings accoun. I leads also, under mild addiional assumpions, o he exisence of coninuously

15 Coninuous-ime erm srucure models 275 compounded forward raes. Noe ha forward process F B (, T, T follows under P ( (c(,t c(,t df B (, T, T =F B (,T,T d + γ(, T, T dw, where c(, T def = a(, T b(, T b(, T, [, T ]. Suppose firs ha a family B(, T saisfies he weak no-arbirage condiion. More specifically, assume ha all forward processes F B (, T, T follow maringales under a probabiliy measure Q equivalen o P (noice ha he underlying probabiliy measure P is no assumed o be a forward maringale measure. In paricular, he expeced value E Q (B 1 (T, T is finie for every T T. Then here exiss an adaped process, h say, such ha dq ( dp = E T h u dw u, P-a.s., and for every T T c(, T c(, T +h (b(,t b(,t =, [, T ]. This implies ha he quaniy N (, T def = c(, T +h b(,t, [, T ], is in fac independen of variable T, meaning ha for any mauriy T T we have r def = c(, T +h b(,t =c(,t+h b(,t, [, T ]. In he formula above, he process r is adaped o he filraion (F [,T ], wih almos all sample pahs inegrable on [, T ]. Furhermore, he bond price saisfies under Q db(, T B(, T = ( r + b(, T b(, T d + b(, T dŵ, where Ŵ = W h u du. Le us pu η = E ( b(, T dŵ, [, T ], and le us assume ha E Q (η T =1. Also, le B be an adaped coninuous process of finie variaion given by he righ-hand side of (1. I is easily seen ha he process Y = B /B(, T also follows a maringale under P, since Y saisfies he SDE dy = Y b(, T dŵ wih Y =1/B(, T. We deduce easily ha Y = η /B(, T for [, T ]. I is also useful o observe ha we have

16 276 M. Musiela, M. Rukowski η = B B 1 (, T B(, T, [, T ]. (36 Le us define a probabiliy measure P Q by seing dp = η T dq. In view of (36, we obain ( 1 ( η ( B E P B(T, T F = E P η T B(T, T F = E P B(, T BT F. By combining his wih he maringale propery of F B (, T, T under P, we find ha ( B B(, T =B(,T E P (F B (T,T,T F =B E P BT F. (37 I is now clear ha for any mauriy T he discouned process Z (, T = B(,T/B is a maringale under P. We conclude ha B is he unique savings accoun implied by he family B(, T. To show he exisence of insananeous forward raes f (, T we shall follow [1]. We assume, in addiion, ha ( T E P r B 1 d <, and we denoe by G(, u he joinly measurable version of he maringale (we refer o [1] for he exisence of such a version of G(, u G(, u =E P (r u B 1 u F, [, u]. The condiional version of Fubini s heorem yields T ( T G(, u du = E P r u Bu 1 du F =1 E P (B 1 T F (38 since db 1 = r B 1 d. By combining (37 wih (38, we obain B(, T =B (1 T G(,udu. (39 I follows immediaely from (39 ha B(, T is differeniable in T. Furhermore, for any fixed T T he implied insananeous forward ineres rae f (, T equals ln B(, T f (, T = = B B 1 (, T G(, T, (4 T or equivalenly, f (, T =B B 1 (,TE P (r T B 1 T F. I is now easy o check ha f (, T =E P (r T F, [, T ], so ha he forward rae f (, T is a maringale under he forward measure P. I follows ha for any fixed mauriy T, he process f (, T follows a coninuous

17 Coninuous-ime erm srucure models 277 semimaringale wih an absoluely coninuous componen of finie variaion. More explicily, we have f (, T =f(, T + α(u,tdu + σ(u, T dwu (41 for some adaped processes α and σ, where W = Ŵ b(u, T du is a Wiener process under P. Moreover, for any T T we have ln b(, T σ(, T =, α(,t= σ(,t b(,t. (42 T To check he firs equaliy in (42 i is enough o show ha he bond price volailiies are absoluely coninuous wih respec o T, or more specifically, ha for any mauriy T he bond price volailiy b(, T saisfies b(, T = T σ(,udu, [, T ]. This can be done by a sraighforward applicaion of Fubini s heorem o he formula ( T B(, T = exp f (, u du, [, T ]. The second equaliy in (42 now follows from Girsanov s heorem. For T = T, i is enough o examine firs he maringale f (, T under he forward measure P, and hen o derive he dynamics of f (, T under he spo measure P. 3. Forward processes In his secion, we examine a mehod of bond price modelling based on he exogenous specificaion of forward volailiies ha is, he volailiies of forward processes. I should be sressed ha we no longer assume ha we are given a family of bond prices. We make insead he following assumpions. (FP.1 For any T [, T we are given an adaped R d -valued process γ(, T, T, [, T ], such ha ( T P γ(u, T, T 2 du < + =1. (43 By convenion, γ(, T, T = R d for every [, T ]. (FP.2 We are given a deerminisic funcion P(, T, T [, T ], wih P(, = 1, which represens an iniial erm srucure of ineres raes. Noice ha P(, T is an exogenously given iniial erm srucure, which should be mached by a family of bond prices, which we are going o consruc. Le us inroduce he noion of a family of forward processes implied by he se-up (FP.1 (FP.2.

18 278 M. Musiela, M. Rukowski Definiion 3.1 Given he se-up (FP.1 (FP.2, for any mauriy T [, T ] we define he forward process F(,T,T, [, T ], by specifying is dynamics under P df(, T, T =F(,T,T γ(,t,t dw, (44 and he iniial condiion F(, T, T = P(, T P(, T, T [, T ]. (45 For any T T, he unique soluion of (44 is given by he sandard exponenial formula F(, T, T = P(, T ( P(, T E γ(u, T, T dw u, (46 where [, T ]. We posulae ha he process F(, T, T has a financial inerpreaion as he raio of bond prices, more exacly, we require ha F(, T, T = B(,T B(,T, [, T ], (47 where bond prices B(, T remain ye unspecified. Indeed, our goal is o consruc a family B(, T which would be consisen wih he dynamics (46 of forward processes, and would mach he iniial erm srucure P(, T ; ha is, B(, T = P(, T for any mauriy T T. Noe ha in his secion he bond price is no required a priori o be a semimaringale. Neverheless, in some circumsances we shall make reference o he volailiy of a bond price, which is defined only for he bond price which follows a semimaringale. In view of assumpions (FP.1 (FP.2 and (47, o find a family B(, T, i is sufficien o specify he price of T -mauriy bond. When searching for a candidae for he process B(, T, we need o ake ino accoun he erminal condiion B(T, T = 1 and he iniial condiion B(, T =P(, T. A family B(, T is hen defined by seing B(, T def = F(, T, T B(, T for every T T. Such a family is easily seen o mach a prespecified iniial erm srucure, he erminal condiion B(T, T = 1 is no necessarily saisfied, however, unless a judicious choice of he process B(, T is made. Le us inroduce a counerpar of condiion (BP.3. We find i convenien o inroduce he family of processes F(, T, U by seing F(, T, U def = F(, T, T F(, U, T, [, T U ]. (48 (FP.3 For any mauriies T, U [, T ] such ha T U we have F(, T, U 1, [, T ]. (49 Noice ha (FP.3 implies, in paricular, ha P(, U P(, T for T U. A family of bond prices associaed wih he se-up (FP.1 (FP.2 is defined as follows.

19 Coninuous-ime erm srucure models 279 Definiion 3.2 We say ha a family B(, T of bond prices is associaed wih (FP.1 (FP.2 if he following holds. (a Processes F(, T, T given by (46 coincide wih processes F B (, T, T which are given by he formula F B (, T, T def = B(, T B(, T, [, T ]. (5 (b Equaliy B(, T =P(, T is saisfied for every T [, T ]. To show ha any family of forward processes F(, T, T admis an associaed family of bond prices, we shall use he noion of a savings accoun implied by he se-up (FP.1 (FP.2. Formally, a savings accoun implied by (FP.1 (FP.2 is any process which represens an implied savings accoun for some family of bond prices associaed wih (FP.1 (FP.2. I is clear ha we may represen he volailiy γ(, T, T as follows ˆb(, T =γ(,t,t +ˆb(,T, [, T ]. (51 for some family of processes ˆb(, T, T T. Given a family of forward volailiies γ(, T, T, in order o deermine uniquely all processes ˆb(, T i suffices o specify he process ˆb(, T. The bond price volailiies b(, T ofany associaed family B(, T, if well-defined, necessarily saisfy relaionship (51; ha is, for any mauriy T T we have b(, T =γ(,t,t +b(,t, [, T ]. (52 This does no mean, of course, ha arbirary processes ˆb(, T which saisfy (51 are indeed price volailiies of some family B(, T of bond prices associaed wih (FP.1 (FP.2. On he oher hand, i follows immediaely from (51 (52 ha for an arbirary choice of he process ˆb(, T, here exiss a unique process ψ such ha he rue bond price volailiy b(, T saisfies b(, T =ˆb(,T+ψ, [, T ], for any mauriy T T. Indeed, i is enough o se ψ = b(, T ˆb(, T for every [, T ]. For he sake of exposiional simpliciy, we shall assume from now on ha he forward volailiies γ(, T, T are bounded. Our goal is o find explicily a family of bond prices associaed wih (FP.1 (FP.2. Firs, we ake an arbirary bounded adaped R d -valued process ˆb(, T, and we define he probabiliy measure ˆP P on (Ω,F T by seing d ˆP ( dp = E T ˆb(u, T dw u, P-a.s. (53 The process Ŵ given by he formula Ŵ = W + ˆb(u, T du is a Wiener process under ˆP. In he second sep, we inroduce a candidae for he savings accoun process ˆB

20 28 M. Musiela, M. Rukowski ( ˆB = P 1 (, exp ˆb(u, dŵ u + 1 ˆb(u, 2 du, (54 2 where ˆb(, T is defined by (51. Remark 3.1 I is no known a priori wheher he process ˆB is of finie variaion (or even if i follows a semimaringale. I appears ha ˆB is of finie variaion if and only if i represens an implied savings accoun for a family B(, T of bond prices defined by formula (55 below. In he opposie case, neiher he process ˆB, nor he bond prices are semimaringales. We are in a posiion o inroduce a family B(, T by seing ( B(, T =P(, T ˆB E ˆb(u, T dŵ u, [, T ], (55 for any mauriy T [, T ]. We claim ha B(, T is a family of bond prices associaed wih (FP.1 (FP.2. To check his, we analyse he forward process F B (, T, T associaed wih he family B(, T. I is clear ha F B (, T, T = P(, T ( P(, T exp γ(u, T, T dŵ u 1 2 where δ u (T, T = ˆb(u,T 2 ˆb(u,T 2 δ u (T, T du, for every u [, T ]. Le us check ha he condiion (a of Definiion 3.2 is saisfied. To his end, noice ha making using of (51 and (55, we ge afer simple manipulaions F B (, T, T = P(, T P(, T E ( γ(u, T, T dwu. Condiion (b of Definiion 3.2 is an immediae consequence of (54 (55. Family B(, T of bond prices inroduced above manifesly saisfies he weak no-arbirage condiion. Furhermore, if assumpion (FP.3 is me, family B(, T saisfies he no-arbirage condiion. Recall ha we assume hroughou, for simpliciy of exposiion, ha he volailiies γ(, T, T of forward processes are bounded. Proposiion 3.1 For any bounded adaped process ˆb(, T, processes B(, T given by (53 (55 represen a family of bond prices associaed wih (FP.1 (FP.2. This family saisfies he weak no-arbirage condiion (i saisfies he noarbirage condiion if (FP.3 holds. The process ˆB given by (54 represens a savings accoun implied by he family B(, T if and only if i follows a predicable process of finie variaion. Proof. In view of previous consideraions, only he las claim is no obvious. The only if clause follows direcly from he definiion of a savings accoun. The if clause is a consequence of resuls of he previous secion. In fac, for any mauriy T he relaive process

21 Coninuous-ime erm srucure models 281 Ẑ (, T def = B(, T / ˆB = P(, T E ( ˆb(u, T dŵ u, [, T ], is evidenly in M loc ( ˆP. If he volailiy of a T -mauriy bond equals ˆb(u, T hen, of course, he process ˆb(, T given by (51 is he bond price volailiy for mauriy T. To conclude, i is enough o compare (54 wih (33. Example 3.1 Le us now consider a simple example (we ake d =1,for convenience. Assume ha he forward volailiies γ(, T, T are consan, more precisely, here exiss a non-zero real γ such ha γ(, T, T = γ for every T [, T and [, T ]. Furhermore, we have as usual γ(, T, T =for every [, T ]. This implies ha for any mauriy T [, T F(, T, T = P(, T ( P(, T exp γw 1 2 γ2, [, T ]. (56 On he oher hand, we assume ha he deerminisic funcion P(, represening he iniial erm srucure belongs o S p +. Le us choose ˆb(, T = for every [, T ] so ha for any mauriy T [, T we have (cf. (51 ˆb(, T =γ(,t,t =γ, [, T ]. (57 Noice also ha he probabiliy measure ˆP defined by (53 saisfies ˆP = P, so ha Ŵ = W. The process ˆB, given by (54, hus equals 1 ( ˆB = P 1 (, exp γw γ2, [, T, (58 and ˆB T = P 1 (, T. Le us firs find he bond price B(, T. By virue of (55, i is clear ha ˆB(, T =P(, T ˆB for every [, T ]. More explicily, B(, T = P(, T P(, ( exp γw γ2, [, T, and B(T, T =1.Le us now consider a bond of mauriy T < T. In view of (55, we have ( B(, T =P(, T ˆB exp ˆb(u, T dw u 1 2 ˆb 2 (u, T du, [, T ]. Combining (57 wih (58, we find ha for any mauriy T < T we have B(, T =P(, T /P(, for every [, T ]. This complees he consrucion of a family B(, T associaed wih (FP.1 (FP.2. Le us now invesigae basic properies of his family. Firs, observe ha for any mauriy T < T we have F B (, T, T def = B(, T B(, T = P(, T ( P(, T exp γw 1 2 γ2, [, T ], 1 Noice ha ˆB is predicable, since any opional process wih respec o a filraion of a Wiener process is predicable. On he oher hand, ˆB, being obviously a semimaringale, does no follow a process of finie variaion as i admis a non-zero coninuous maringale componen.

22 282 M. Musiela, M. Rukowski so ha he forward processes F B (, T, T and F(, T, T coincide. We shall now check ha he process B which equals B = P 1 (, for [, T, and = P 1 (, T exp (γw T 1 2 γ2 T B T is he unique implied savings accoun for he family B(, T. I is clear ha B belongs o A + ; i is hus enough o check ha all relaive bond prices Z (, T =B(,T/B follow local maringales under some probabiliy measure P P. For any mauriy T < T we have Z (, T = P(, T for every [, T ], hence Z (, T follows rivially a maringale under any probabiliy measure equivalen o P. For T we have ( Z (, T =P(, T exp γ (W γ 1 2 γ2, [, T ], so ha Z (, T is a maringale under he probabiliy measure P P which is given by he formula dp ( dp = exp γw T 1 2 γ2 T, P-a.s. Observe ha due o he jump a ime T, he savings accoun B is no increasing, even if he iniial erm srucure P(, is a sricly decreasing funcion. Le us now deermine he bond price volailiies. I is apparen ha b(, T = for any mauriy T < T, while b(, T = γ(i seems ineresing o compare his wih he iniial guess: ˆb(, T =γfor T < T, and ˆb(, T =. This example, hough raher simplisic, provides some insigh ino he feaures of he proposed procedure. Firs, he process ˆB given by (54 does no necessarily represen he savings accoun implied by he family B(, T. Second, he implied savings accoun may follow a disconinuous process; in our example, his feaure is relaed o he fac ha he forward volailiies γ(, T, U are disconinuous in U. Le us now examine he problem of uniqueness of a family of bond prices associaed wih a given collecion of forward processes. Since any family B(, T of bond prices associaed wih he se-up (FP.1 (FP.2 saisfies F(, T, T =F B (,T,T = B(,T B(,T, [, T ], we have (see formula (48 B(, T = B(,T B(, =F(,T,T F(,,T =F(,T,, [, T ]. (59 Therefore, a family of bond prices associaed wih a given collecion F(, T, T of forward processes is uniquely deermined; his implies in urn he uniqueness of he savings accoun implied by forward processes. Noice, however, ha formula (59 is no very useful in pracice, since he dynamics of F(, T, are no easily available.

23 Coninuous-ime erm srucure models Models of forward LIBOR raes To inroduce he noion of a forward LIBOR rae, we place ourselves wihin he se-up (BP.1 (BP.2. This means ha we are given a family B(, T of bond prices, and hus also he collecion F B (, T, U of forward processes. A sricly posiive real number δ<t is fixed hroughou. By he definiion, he forward δ-libor rae 2 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 ]. (6 Comparing (6 wih (5 we find ha L(, T =f s (,T,T+δ so ha he forward LIBOR rae L(, T represens in fac 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 B(,T 1+δL(,T=, [, T ]. (61 B(,T+δ In paricular, he iniial erm srucure of forward LIBOR raes saisfies L(, T =f s (, T, T + δ =δ 1( B(, T B(, T + δ 1. (62 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+δγ(,T,T+δ dw T +δ, where W T +δ and P T +δ are defined by (9 and (1, respecively. This means ha L(, T solves he equaion dl(, T =δ 1 (1 + δl(, T γ(, T, T + δ dw T +δ (63 subjec o he iniial condiion (62. Suppose ha forward LIBOR raes L(, T are sricly posiive. Then formula (63 can be rewrien as follows where for any [, T ] dl(, T =L(,Tλ(,T dw T +δ, (64 λ(, T = 1+δL(,T γ(,t,t +δ. (65 δl(,t This shows ha he collecion of forward processes specifies uniquely he family of forward LIBOR raes. The consrucion of a model of forward LIBOR raes relies on he following assumpions. 2 In pracice, several ypes of LIBOR raes occur, e.g., 3-monh LIBOR and 6-monh LIBOR. For he ease of exposiion, we consider a fixed mauriy δ.

24 284 M. Musiela, M. Rukowski (LR.1 For any mauriy T T δ, we are given a R d -valued bounded deerminisic funcion 3 λ(, T which represens he volailiy of he forward LIBOR rae process L(, T. (LR.2 We assume a sricly decreasing and sricly posiive iniial erm srucure P(, T, T [, T ], and hus an iniial erm srucure L(, T of forward LIBOR raes L(, T =δ 1( P(, T P(, T + δ 1, T [, T δ]. ( Discree-enor case We sar by sudying a discree-enor version of a lognormal model of forward LIBOR raes. I should be sressed ha a so-called discree-enor model sill possesses cerain coninuous-ime feaures, for insance, he forward LIBOR raes follow coninuous-ime processes. For he ease of noaion, we shall assume ha he horizon dae T is a muliple of δ, say, T = M δ for a naural M. We shall focus on a finie number of daes, Tmδ = T mδ for m =1,...,M 1. The consrucion is based on backward inducion, herefore, we sar by defining he forward LIBOR rae wih he longes mauriy, L(, Tδ. We posulae ha he rae L(, Tδ is governed under he probabiliy measure P by he following SDE (cf. (64 dl(, Tδ =L(,Tδλ(,T δ dw (67 wih he iniial condiion L(, Tδ =δ 1( P(, Tδ P(, T 1. (68 Pu anoher way, we posulae ha for every [, Tδ ] L(, Tδ =δ 1( P(, Tδ ( P(, T 1 E λ(u, Tδ dw u. (69 Since P(, Tδ > P(, T i is clear ha L(, Tδ isinm+ c(p.also, for any fixed T δ he random variable L(, Tδ has a lognormal probabiliy law under P. The nex sep is o define he forward LIBOR rae for he dae T2δ, using relaionship (65 wih T = Tδ, ha is, γ(, T δ, T = δl(,t δ 1+δL(,T δ λ(,t δ, [, T δ]. (7 Given ha he volailiy γ(, T δ, T is deermined by (7, he forward process F B (, T δ, T is known o solve under P df B (, T δ, T =F B (,T δ,t γ(,t δ,t dw (71 3 Volailiy λ could follow a sochasic process; we deliberaely focus here on a lognormal model of forward LIBOR raes in which λ is deerminisic.

25 Coninuous-ime erm srucure models 285 and iniial condiion is F B (, Tδ, T =P(, Tδ /P(, T. The forward process F B (, Tδ, T belongs o M c + (P, since he volailiy γ(, Tδ, T follows a bounded process. We inroduce a d-dimensional Wiener process W T δ, which corresponds o he dae Tδ, by seing W T δ = W γ(u, Tδ, T du, [, Tδ ]. (72 Due o he boundedness of he process γ(, Tδ, T, he exisence of he process W T δ and of he associaed probabiliy measure PT P, which is given by he δ formula dp T ( δ dp = E T γ(u, T δ δ, T dw u, P-a.s. (73 is rivial. The process W T δ may be inerpreed as he forward Wiener process for he dae Tδ. We are in a posiion o specify he dynamics of he forward LIBOR rae for he dae T2δ under he forward probabiliy measure P T. Analogously o δ (67 we se dl(, T2δ =L(,T2δλ(,T 2δ dw T δ, (74 wih he iniial condiion L(, T2δ =δ 1( P(, T2δ P(, Tδ 1. (75 Solving (74 and comparing wih (65 for T = T2δ, we obain γ(, T 2δ, T δ = δl(,t 2δ 1+δL(,T 2δ λ(,t 2δ, [, T 2δ]. (76 To find γ(, T 2δ, T we make use of he relaionship (cf. (7 γ(, T 2δ, T δ =γ(,t 2δ,T γ(,t δ,t, [, T 2δ]. (77 Given he process γ(, T2δ, T δ, we can define he pair (W T 2δ, PT corresponding 2δ o he dae T2δ and so forh. By working backward o he firs relevan dae T(M 1δ = δ, we consruc a family of forward LIBOR raes L(, T mδ, m = 1,...,M 1. Noice ha he lognormal probabiliy law of every process L(, Tmδ under he corresponding forward probabiliy measure P T is ensured. Indeed, (m 1δ for any m =1,...,M 1 we have where W T (m 1δ dl(, T mδ =L(,T mδλ(,t mδ dw T (m 1δ, (78 is a sandard Wiener process under PT. This complees he (m 1δ derivaion of he lognormal model of forward LIBOR raes in a discree-enor framework. Noe ha we have consruced simulaneously a family of forward LIBOR raes and a family of associaed forward processes.

26 286 M. Musiela, M. Rukowski 4.2. Implied savings accoun We shall now examine he exisence and uniqueness of he implied saving accoun, in a discree-ime seing. The implied savings accoun is hus seen as a discree-ime process, B, =,δ,...,t = Mδ. Inuiively, he value B of a savings accoun a ime can be inerpreed as cash amoun accumulaed up o ime by rolling over a series of zero-coupon bonds wih he shores mauriies available. In a discree-enor framework, o find he process B, we do no have o specify explicily all bond prices; he knowledge of forward bond prices is sufficien. Indeed, from (4 we ge F B (, T j, T j +1 = F B(,T j,t F B (,T j+1, T = B(, T j B(, T j +1, where we wrie T j = j δ. This in urn yields, upon seing = T j, F B (T j, T j, T j +1 =1/B(T j,t j+1, (79 so ha he price B(T j, T j +1 of a one-period bond is uniquely specified for every j. Though he bond which maures a ime T j does no physically exis afer his dae, i seems jusified 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. Pu anoher way, he spo value a ime T j +1 of one cash uni received a ime T j equals B 1 (T j, T j +1. The discree-ime savings accoun B hus equals B T k = k ( ( k F B Tj 1, T j 1, T j = B ( 1 T j 1, T j j =1 for k =,...,M 1, since by convenion B =1.Noe ha ( F B Tj, T j, T j +1 =1+δL(Tj,T j+1 > 1 for j =1,...,M 1, and since BT j +1 = F B (T j, T j, T j +1 BT j, we find ha BT j +1 > BT j for every j =,...,M 1. We conclude ha he implied savings accoun B follows a sricly increasing discree-ime process. We define he probabiliy measure P P on (Ω,F T by he formula (cf., (26 dp dp = B T P(, T, P-a.s. (8 The probabiliy measure P appears o be a plausible candidae for a spo maringale measure. Indeed, if we se B(T l, T k =E P (BT l /BT k F Tl (81 for every l k M, hen in he case of l = k 1, equaliy (81 coincides wih (79. I should be sressed 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 knowledge of forward LIBOR raes for all mauriies is necessary for his. j =1