2 Models of Forward LIBOR and Swap Raes 2 Preliminaries Le T > be a horizon dae for our model of economy. In his preliminary secion, we focus on LIBOR

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1 Models of Forward LIBOR and Swap Raes MAREK RUTKOWSKI Insyu Maemayki, Poliechnika Warszawska, -66 Warszawa, Poland Absrac The backward inducion approach is sysemaically used o produce various models of forward marke raes. These include he lognormal model of forward LIBOR raes examined in Milersen e al. (997) and Brace e al. (997), as well as he lognormal model of (xed-mauriy) forward swap raes proposed by Jamshidian (996, 997). The valuaion formulae for European caps and swapions are given. In he las secion, he Eurodollar fuures conracs and opions are examined wihin he framework of he lognormal model of forward LIBOR raes. Keywords zero-coupon bond, LIBOR rae, swap rae, cap, swapion, Eurodollar fuures Inroducion The aim of his noe is o presen in a sysemaic way he backward inducion approach o he modelling of forward marke raes (a generic erm marke rae is used here o describe any kind of marke ineres rae, such as LIBOR rae or swap rae). In he conex of marke rae modelling, such an approach was iniiaed by Musiela and Rukowski (997), who employed his mehodology o consruc he lognormal model of forward LIBOR raes (wih xed accrual period). The idea of erm srucure modelling wih he direc use of forward measures was previously exploied by Hansen (994), who used he forward inducion o produce an arbirage-free diusion-ype model for prices of a nie family of zero-coupon bonds wih dieren mauriies. Le us focus on he mos imporan feaures of he backward inducion approach (also ermed he forward measure approach). Firs, in conras o a considerable number of published papers on erm srucure modelling, we are no ineresed (a leas, no primarily) in producing a muli- (or innie-) dimensional process which would describe he bond prices (no o menion he process of he shor-erm ineres rae), bu raher we focus direcly on a nie collecion of specic marke raes. The basic inpus of each model are hus he quaniies (ypically, volailiies) associaed wih hese raes, as well as he iniial erm srucure, reeced here by prices of a nie family of bonds wih prespecied mauriies. Second, he arbirage-free feaures of a model are embedded in assumpions which are imposed (in an implici way) on he dynamics of relaive bond price under he (generalized) forward measure. In his sense, he mehodology presened in his noe is also closely relaed o he change of numeraire echnique which is now commonly sandard in arbirage pricing (see, e.g., Geman e al. (995)). The rs feaure leads o he following naural and imporan quesion is i possible o nd an arbirage-free family of (eiher some or all) bond prices wih a paricular model of forward marke raes Due o second feaure of he backward inducion procedure employed in he modelling of marke raes, he answer o his quesion is posiive, in general. I is worhwhile o sress ha he underlying bond prices are no uniquely deermined, however. Therefore he problem of he righ choice of he underlying family of bond prices appears here in a naural way (for a pracical soluion o his problem, we refer o Sawa (997)). Presen address School of Mahemaics, Universiy of New Souh Wales, 252 Sydney, NSW, Ausralia. Hansen's work is indeed much closer in he spiri o he paper Ball and Torous (983), who modelled bond prices hrough Brownian bridges. In conras o heir raher crude approach, a clever choice of he drif and diusion coeciens guaranees he arbirage-free propery of Hansen's model.

2 2 Models of Forward LIBOR and Swap Raes 2 Preliminaries Le T > be a horizon dae for our model of economy. In his preliminary secion, we focus on LIBOR raes which correspond o a xed ime period, denoed by Le B(; T ) sand for he price a ime of a uni zero-coupon bond ha maures a ime T By deniion, he forward -LIBOR rae L(; T ) = L(; T; ) prevailing a ime for he fuure dae T T (or raher, over he fuure ime period [T; T + ]) is dened hrough he formula Pu more explicily, we have + L(; T ) = B(; T ) ; 8 2 [; T ] () B(; T + ) L(; T ) = B(; T ) B(; T + ) ; 8 2 [; T ] (2) B(; T + ) In paricular, he iniial erm srucure of forward LIBOR raes and he iniial erm srucure of bond prices are relaed o each oher hrough he following relaionship L(; T ) = B(; T ) B(; T + ) ; 8 T 2 [; T ] B(; T + ) Equivalenly, in erms of he iniial yield curve Y (; ) we have L(; T ) = e (T +)Y (;T +)T Y (;T ) since Y (; ) saises B(; T ) = e T Y (;T ) for any mauriy T To inroduce he uncerainy in our model, we assume hroughou ha we are given a d- dimensional sandard Brownian moion W; dened on an underlying lered probabiliy space (; F; P) The lraion F = (F ) 2[;T ] coincides wih he P-compleion of he naural lraion of he underlying noise process W 3 Model of Forward LIBOR Raes In his secion, we describe he backward inducion approach o he modelling of forward LIBOR raes. The consrucion presened below is a sligh modicaion and exension of ha given by Musiela and Rukowski (997a). Le us sar by inroducing some noaion. We are given a prespecied collecion of rese/selemen daes T = < T < < T M = T ; referred P o as he enor j srucure. Le us denoe j = T j T j for j = ; ; M Then obviously T j = i= i for every j = ; ; M We nd i convenien o denoe T m = T MX j=mm+ j = T Mm ; 8 m = ; ; M For any j = ; ; M ; we dene he forward LIBOR rae L(; T j ) by seing L(; T j ) = L(; T j ; j+ ) = B(; T j) B(; T j+ ) ; 8 2 [; T j ] j+ B(; T j+ ) Deniion 3. Le us x j = ; ; M A probabiliy measure P Tj on (; F Tj ); equivalen o P; is said o be he forward LIBOR measure for he dae T j if, for every k = ; ; M; he relaive bond price U Mj+ (; T k ) def = B(; T k) j B(; T j ) ; 8 2 [; T k ^ T j ]; follows a local maringale under P Tj

3 Marek Rukowski 3 I is clear ha he noion of forward LIBOR measure is in fac idenical wih ha of a forward probabiliy measure for a given dae (see, e.g., Geman e al. (995)). The sligh modicaion of he sandard erminology emphasizes our inenion o make a clear disincion beween various kinds of forward probabiliies, which we are going o sudy in he sequel. Also, i is rivial o observe ha he forward LIBOR rae L(; T j ) necessarily follows a local maringale under he forward LIBOR measure for he dae T j+ If, in addiion, i is a sricly posiive process, he exisence of he associaed volailiy process can be easily jusied. In our furher developmen, we shall go he oher way around; ha is, we will assume ha for any dae T j ; he volailiy (; T j ) of he forward LIBOR rae L(; T j ) is exogenously given. Basically, i can be a deerminisic R d -valued funcion of ime, a R d -valued funcion of he underlying forward LIBOR raes, or a d-dimensional adaped sochasic process. For simpliciy, we assume hroughou ha he volailiies of forward LIBOR raes are bounded. Our aim is o consruc a family L(; T j ); j = ; ; M of forward LIBOR raes, a collecion of muually equivalen probabiliy measures P Tj ; j = ; ; M; and a family W Tj ; j = ; ; M of processes in such a way ha (i) for any j = ; ; M he process W Tj follows a d-dimensional sandard Brownian moion under he probabiliy measure P Tj ; (ii) for any j = ; ; M ; he forward LIBOR rae L(; T j ) saises he SDE wih he iniial condiion dl(; T j ) = L(; T j ) (; T j ) dw Tj+ ; 8 2 [; T j ]; L(; T j ) = B(; T j) B(; T j+ ) j+ B(; T j+ ) As already menioned, he consrucion of he model is based on backward inducion, herefore we sar by dening he forward LIBOR rae wih he longes mauriy, i.e., T M We posulae ha L(; T M ) = L(; T ) is governed under he underlying probabiliy measure P by he following SDE (noe ha, for simpliciy, we have chosen he underlying probabiliy measure P o play he role of he forward LIBOR measure for he dae T ) wih he iniial condiion Pu anoher way, we have L(; T ) = B(; T ) B(; T ) M B(; T ) dl(; T ) = L(; T ) (; T ) dw ; L(; T ) = B(; T ) B(; T ) M B(; T ) exp Z (u; T ) dw u 2 Z j(u; T )j 2 du Since B(; T ) > B(; T ); i is clear ha he L(; T ) follows a sricly posiive maringale under P T = P The nex sep is o dene he forward LIBOR rae for he dae T 2 For his purpose, we need o inroduce rs he forward probabiliy measure for he dae T By deniion, i is a probabiliy measure Q; equivalen o P; and such ha processes U 2 (; T k ) = B(; T k ) M B(; T ) are Q-local maringales. I is imporan o observe ha he process U 2 (; Tk ) admis he following represenaion U 2 (; Tk ) = M M U (; Tk ) M L(; T ) + Le us formulae an auxiliary resul, which is a sraighforward consequence of I^o's rule.

4 4 Models of Forward LIBOR and Swap Raes Lemma 3. Le G and H be real-valued adaped processes, such ha dg = dw ; dh = dw Assume, in addiion, ha H > for every and denoe Y = ( + H ) Then I follows immediaely from Lemma 3. ha d(y G ) = Y Y G dw Y d du 2 (; T k ) = k dw ML(; T ) + M L(; T ) (; T ) d for a cerain process k Therefore i is enough o nd a probabiliy measure under which he process W T def = W Z M L(u; T ) + M L(u; T ) (u; T ) du = W Z (u; T ) du; 8 2 [; T ]; follows a sandard Brownian moion (he deniion of (; T ) is clear from he conex). This can be easily achieved using Girsanov's heorem, as we may pu dp T dp = exp Z T (u; T ) dw u 2 Z T j(u; T )j 2 du ; P-a.s. We are in a posiion o specify he dynamics of he forward LIBOR rae for he dae T 2 under P T ; namely we posulae ha dl(; T 2 ) = L(; T 2 ) (; T 2 ) dw T ; wih he iniial condiion L(; T 2 ) = B(; T 2 ) B(; T ) M B(; T ) Le us now assume ha we have found processes L(; T ); ; L(; T m) In paricular, he forward LIBOR measure P T m and he associaed Brownian moion W T m are already specied. Our aim is o deermine he forward LIBOR measure P T m I is easy o check ha U m+ (; Tk ) = Mm Mm U m (; Tk ) Mm L(; Tm) + Using Lemma 3., we obain he following relaionship W T m = W T m Z Mm L(u; T m) + Mm L(u; T m) (u; T m) du; 8 2 [; T m] The forward LIBOR measure P T m can hus be easily found using Girsanov's heorem. Finally, we dene he process L(; T m+) as he soluion o he SDE wih he iniial condiion dl(; T m+) = L(; T m+) (; T m+) dw T m ; L(; T m+) = B(; T m+) B(; T m) Mm B(; T m) Remarks. If he volailiy coecien (; T m ) [; T m ]! R d is deerminisic, hen, for each dae 2 [; T m ]; he random variable L(; T m ) has a lognormal probabiliy law under he forward probabiliy measure P Tm+ Such a model, 2 commonly referred o as he lognormal model of LIBOR raes, was examined hrough dieren means by Milersen e al. (997), Brace e al. (997), Musiela and Rukowski (997a), Jamshidian (997). Le us menion ha he fuures Libor rae and is derivaives are sudied in Secion 5 below. 2 In fac, some auhors examined a fully coninuous-ime version of he lognormal model of forward LIBOR raes, in which he raes L(; T ) are specied for all mauriies.

5 Marek Rukowski 5 3. Spo LIBOR Measure The backward inducion approach o modelling of forward LIBOR raes was re-examined and essenially generalized by Jamshidian (997). In his secion, we presen briey his alernaive approach o he modelling of forward LIBOR raes. As made apparen in he preceding secion, in he direc modelling of LIBOR raes, no explici reference is made o he bond price processes, which are used o formally dene a forward LIBOR rae hrough equaliy (2). Neverheless, o explain he idea ha underpins Jamshidian's approach, we shall emporarily assume ha we are given a family of bond prices B(; T j ) for he fuure daes T j ; j = ; ; M By deniion, he spo LIBOR measure is ha probabiliy measure equivalen o P; under which all relaive bond prices are local maringales, when he price process obained by rolling over one-period bonds, is aken as a numeraire. The exisence of such a measure can be eiher posulaed, or derived from oher condiions. 3 Le us pu Y m() G = B(; T m() ) B (T j ; T j ); 8 2 [; T ]; (3) m() = inf fk 2 N j j= kx i= i g = inf fk 2 N j T k g I is easily seen ha G represens he wealh a ime of a porfolio which sars a ime wih one uni of cash invesed in a zero-coupon bond of mauriy T ; and whose wealh is hen reinvesed a each dae T j ; j = ; ; M ; in zero-coupon bonds which maure a he nex dae; ha is, T j+ Deniion 3.2 A spo LIBOR measure P L is any probabiliy measure on (; F T ); which is equivalen o P; and such ha for any j = ; ; M he relaive bond price B(; T j )=G follows a local maringale under P L Noe ha B(; T k+ )=G = m() Y j= + j L(T j ; T j ) ky j=m()+ + j L(; T j ) ; so ha all relaive bond prices B(; T j )=G ; j = ; ; M are uniquely deermined by a collecion of forward LIBOR raes. In his sense, G is he correc choice of he reference price process in he presen seing. We shall now concenrae on he derivaion of he dynamics under P L of forward LIBOR raes L(; T j ); j = ; ; M Our aim is o show ha hese dynamics involve only he volailiies of forward LIBOR raes (as opposed o volailiies of bond prices or oher processes). Therefore, i is possible o dene he whole family of forward LIBOR raes simulaneously under one probabiliy measure (of course, his feaure can also be deduced from he preceding consrucion). To faciliae he derivaion of he dynamics of L(; T j ); we posulae emporarily ha bond prices B(; T j ) follow I^o processes under he underlying probabiliy measure P; more explicily db(; T j ) = B(; T j ) a(; T j ) d + b(; T j ) dw (4) for every j = ; ; M;, as before, W follows a d-dimensional sandard Brownian moion under an underlying probabiliy measure P (i should be sressed, however, ha we do no assume here ha P is a forward (or spo) maringale measure). Combining (3) wih (4), we obain dg = G a(; Tm() ) d + b(; T m() ) dw 3 One may assume, e.g., ha bond prices B(; T j ) saisfy he weak no-arbirage condiion, meaning ha here exiss a probabiliy measure ~P; equivalen o P; and such ha all processes B(; T k )=B(; T ) are ~P-local maringales.

6 6 Models of Forward LIBOR and Swap Raes Furhermore, by applying I^o's rule o equaliy we nd ha and (; T j ) = + j+ L(; T j ) = B(; T j) B(; T j+ ) ; (5) dl(; T j ) = (; T j ) d + (; T j ) dw ; B(; T j ) a(; Tj ) a(; T j+ ) (; T j )b(; T j+ ) j+ B(; T j+ ) (; T j ) = B(; T j ) b(; Tj ) b(; T j+ ) (6) j+ B(; T j+ ) Using (5) and he las formula, we arrive a he following relaionship b(; T m() ) b(; T j+ ) = jx k=m() k+ (; T k ) + k+ L(; T k ) (7) By deniion of a spo LIBOR measure P L ; each relaive price process B(; T j )=G follows a local maringale under P L Since, in addiion, P L is assumed o be equivalen o P; i is clear ha i is given by he Doleans exponenial, ha is dp L dp = exp Z T h u dw u 2 Z T jh u j 2 du ; P-a.s. for some adaped process h I i no hard o check, using I^o's rule, ha h need o saisfy a(; T j ) a(; T m() ) = b(; T m() ) h b(; Tj ) b(; T m() ) ; 8 2 [; T j ]; for every j = ; ; M Combining (6) wih he las formula, we obain and his in urn yields B(; T j ) j+ B(; T j+ ) dl(; T j ) = (; T j ) Using (7), we conclude ha process L(; T j ) saises dl(; T j ) = a(; Tj ) a(; T j+ ) = (; T j ) b(; T m() ) h ; jx k=m() b(; Tm() ) b(; T j+) h d + dw k+ (; T k ) (; T j ) + k+ L(; T k ) d + (; T j ) dw L ; R he process W L = W h u du follows a d-dimensional sandard Brownian moion under he spo LIBOR measure P L To furher specify he model, we assume ha processes (; T j ); j = ; ; M; have he following form (; T j ) = j ; L(; Tj ); L(; T j+ ); ; L(; T M ) ; 8 2 [; T j ]; j [; T j ] R Mj+! R d are known funcions. In his way, we obain a sysem of SDEs dl(; T j ) = jx k=m() k+ k (; L k ()) j (; L j ()) + k+ L(; T k ) d + j (; L j ()) dw L ; we wrie L j () = (L(; T j ); L(; T j+ ); ; L(; T M )) Under mild regulariy assumpions, his sysem can be solved recursively, saring from L(; T M ) The lognormal model of forward LIBOR raes corresponds o he choice of (; T j ) = (; T j )L(; T j ); (; T j ) [; T j ]! R d is a deerminisic funcion for every j

7 Marek Rukowski 7 4 Models of Forward Swap Raes We assume, as before, ha he enor srucure T = < T < < T M = T is given. Recall ha j = T j T j for j = ; ; M; and hus T j = P j i= i for every j = ; ; M 4. Fixed-Mauriy Swaps We will rs analyse he model of forward swap raes developed by Jamshidian (996, 997). For any xed j; we consider a xed-for-oaing forward (payer) swap which sars a ime T j and has M j accrual periods, whose consecuive lenghs are j+ ; ; M The xed ineres rae paid a each of rese daes T l for l = j + ; ; M equals ; and he corresponding oaing rae, L(T l ); is found using he formula B(T l ; T l+ ) = + (T l+ T l )L(T l ) = + l+ L(T l ); i.e., i coincides wih he LIBOR rae L(T l ; T l ) I is no dicul o check, using no-arbirage argumens, ha he value of such a swap equals (by convenion, he noional principal equals ) FS () = B(; T j ) MX l=j+ c l B(; T l ); 8 2 [; T j ]; (8) c l = l for l = j + ; ; M ; and c M = + M Consequenly, he associaed forward swap rae, Mj (; T j ); ha is, ha value of a xed rae for which such a swap is worhless a ime ; is given by he formula Mj (; T j ) = B(; T j ) B(; T M ) j+ B(; T j+ ) + + M B(; T M ) (9) for every 2 [; T j ]; j = ; ; M In his secion, we consider he family of forward swap raes ~(; T j ) = Mj (; T j ) for j = ; ; M Le us sress ha he underlying swap agreemens dier in lengh, however, hey all have a common expiraion dae, T = T M Suppose for he momen ha we are given a family of bond prices B(; T m ); m = ; ; M; on a lered probabiliy space (; F; P) equipped wih a Brownian moion W As in Secion 3, we nd i convenien o posulae ha P = P T is he forward measure for he dae T ; and he process W = W T is he corresponding Brownian moion. For any m = ; ; M ; we inroduce he xed-mauriy coupon process G(m) ~ by seing (recall ha T l = T Ml ; in paricular, T = T M ) ~G (m) = MX l=mm+ l B(; T l ) = m X k= Mk B(; T k ); 8 2 [; T Mm+ ] () A forward swap measure is ha probabiliy measure equivalen o P; which corresponds o he choice of he xed-mauriy coupon process as a numeraire asse. We have he following deniion. Deniion 4. For a xed j = ; ; M; a probabiliy measure ~ PTj on (; F Tj ); equivalen o P; is said o be he xed-mauriy forward swap measure for he dae T j if, for every k = ; ; M; he relaive bond price Z Mj+ (; T k ) def = B(; T k ) ~G (M j + ) = B(; T k ) j B(; T j ) + + M B(; T M ) ; 8 2 [; T k ^ T j ]; follows a local maringale under ~ PTj Pu anoher way, for any xed m = ; ; M ; he relaive bond prices Z m (; T k ) = B(; T k ) ~G (m) = B(; T k ) Mm+ B(; T m ) + + MB(; T ) ; 8 2 [; T k ^ T m];

8 8 Models of Forward LIBOR and Swap Raes are bound o follow local maringales under he forward swap measure ~ PT m I follows immediaely from (9) ha he forward swap rae for he dae T m equals ~(; T m) = B(; T m) B(; T ) Mm+ B(; T m ) + + M B(; T ) ; 8 2 [; T m]; or equivalenly, ~(; T m) = Z m (; T m) Z m (; T ); 8 2 [; T m] Therefore ~(; T m) also follows a local maringale under he forward swap measure ~ PT m Moreover, since obviously ~ G () = M B(; T ); i is eviden ha Z (; T k ) = M F B(; T k ; T ); and hus he probabiliy measure ~ PT can be chosen o coincide wih he forward maringale measure P T Our aim is o consruc a model of forward swap raes hrough backward inducion. As one migh expec, he underlying bond price processes will no be explicily specied. We assume ha we are given a family of bounded adaped processes ~(; T j ); j = ; ; M ; which represen he volailiies of processes ~(; T j ) In addiion, an iniial erm srucure of ineres raes, given by a family B(; T j ); j = ; ; M; of bond prices, is known. We wish o consruc a family of forward swap raes in such a way ha d~(; T j ) = ~(; T j )~(; T j ) d ~ W T j+ () for any j = ; ; M ; each process W ~ T j+ follows a sandard Brownian moion under he corresponding forward swap measure PTj+ ~ The model should also be consisen wih he iniial erm srucure of ineres raes, meaning ha ~(; T j ) = B(; T j ) B(; T ) j+ B(; T j+ ) + + M B(; T M ) (2) We proceed by backward inducion. The rs sep is o inroduce he forward swap rae for he dae T by posulaing ha ~(; T ) solves he SDE wih he iniial condiion d~(; T ) = ~(; T )~(; T ) d ~ W T ; 8 2 [; T ]; (3) ~(; T ) = B(; T ) B(; T ) M B(; T ) To specify he process ~(; T 2 ); we need rs o inroduce a forward swap measure PT ~ and an associaed Brownian moion W ~ T To his end, noice ha each process Z (; Tk ) = B(; T k )= MB(; T ) follows a sricly posiive local maringale under PT ~ = P T More specically, we have dz (; T k ) = Z (; T k ) (; T k ) dw T (4) for some adaped process (; Tk ) According o he deniion of a xed-mauriy forward swap measure, we posulae ha for every k; he process Z 2 (; T k ) = B(; Tk ) M B(; T ) + M B(; T ) = Z (; Tk ) + M Z (; T ) follows a local maringale under ~ P T Applying Lemma 3. o processes G = Z (; T k ) and H = M Z (; T ); i is easy o see ha for his propery o hold, i suces o assume ha he process W T ; which is given by he formula ~W T = ~ W T Z M Z (u; T ) + M Z (u; T ) (u; T ) du; 8 2 [; T ];

9 Marek Rukowski 9 follows a Brownian moion under ~ PT (probabiliy measure ~ PT is ye unspecied, bu will be soon found hrough Girsanov's heorem). Noe ha Z (; T ) = B(; T ) M B(; T ) = ~(; T ) + Z (; T ) = ~(; T ) + M (5) Diereniaing boh sides of he las equaliy, we ge (cf. (3) and (4)) Consequenly, ~ W T ~W T Z (; T ) (; T ) = ~(; T )~(; T ) is explicily given by he formula = ~ W T Z M ~(u; T ) + M M + M~(u; T ) ~(u; T ) du; 8 2 [; T ] We are in a posiion o dene, using Girsanov's heorem, he associaed forward swap measure ~ PT Subsequenly, we inroduce he process ~(; T 2 ); by posulaing ha i solves he SDE wih he iniial condiion d~(; T 2 ) = ~(; T 2 )~(; T 2 ) d ~ W T ~(; T 2 ) = B(; T 2 ) B(; T ) M B(; T ) + MB(; T ) For he reader's convenience, le us consider one more inducive sep, in which we are looking for ~(; T 3 ) We now consider processes Z 3 (; T k ) = B(; T k ) M2 B(; T 2 ) + MB(; T ) + M B(; T ) = Z 2 (; T k ) + M2 Z 2 (; T 2 ) ; so ha Z ~W T 2 = W ~ T M2 Z 2 (u; T 2 ) + M2 Z 2 (u; T ) 2(u; T 2 ) du 8 2 [; T 2 ] 2 I is useful o noe ha in urn Z 2 (; T 2 ) = B(; T 2 ) M B(; T ) + M B(; T ) = ~(; T 2 ) + Z 2 (; T ); (6) Z 2 (; T ) = Z (; T ) + M Z (; T ) + M ~(; T ) and he process Z (; T ) is already known from he previous sep (clearly, Z (; T ) = =d M ). Diereniaing he las equaliy, we may hus nd he volailiy of he process Z 2 (; T ); and consequenly, o dene ~ P T 2 Le us now urn o he general case. We proceed by inducion wih respec o m Suppose ha we have already found forward swap raes ~(; T ); ; ~(; T m); he forward swap measure ~ PT m ; and he associaed Brownian moion ~ W T m Our aim is o deermine he forward swap measure ~P T m ; he associaed Brownian moion ~ W T m; and he forward swap rae ~(; T m+) To his end, we posulae ha processes Z m+ (; T k ) = B(; T k ) ~G (m + ) = B(; T k ) Mm B(; T m) + + M B(; T ) = Z m (; T k ) + Mm Z m (; T m) follow local maringales under PT ~ m In view of Lemma 3., applied o processes G = Z m (; Tk ) and H = Z m (; Tm); i is clear ha we may se ~W T m = ~ W T Z Mm Z m (u; T m) + Mm Z m (u; T m) m(u; T m) du; 8 2 [; T m] (7)

10 Models of Forward LIBOR and Swap Raes Therefore i is sucien o analyse he process Z m (; T m) = To conclude, i is enough o noice ha B(; T m) Mm+ B(; T m ) + + M B(; T ) = ~(; T m) + Z m (; T ) Z m (; T ) = Z m (; T ) + Mm+ Z m (; T ) + Mm+ ~(; T m ) Indeed, from he preceding sep, we know ha he process Z m (; T ) is a (raional) funcion of forward swap raes ~(; T ); ; ~(; T m) Consequenly, he process under he inegral sign on he righ-hand side of (7) can be expressed using he erms ~(; T ); ; ~(; T m) and heir volailiies (since he explici formula is raher lenghy, i is no repored here). Having found he process ~ W T m and probabiliy measure ~ PT m ; we inroduce he forward swap rae ~(; T m+) hrough (){(2), and so forh. If all volailiies are deerminisic, he model is ermed he lognormal model of xed-mauriy forward swap raes. 4.2 Amorising Swaps So far, i was implicily assumed ha we deal wih a xed-for-oaing payer swap, wih consan noional principal. Le us now consider he so-called amorising swap, in which he noional principal P j varies over ime (we assume, however, ha i is deerminisic). We assume he cash ows a ime T j of he underlying swaps are P j j L(T j ; T j ) and P j j Therefore he value a ime 2 [; T j ] of a swap which sars a ime T j equals FS () = MX i=j+ The associaed forward swap rae, a (; T j ); equals or equivalenly, a (; T j ) = P j+ P i B(; Ti ) ( + i )B(; T j ) ; 8 2 [; T j ] B(; Tj ) B(; T j+ ) + + P M B(; TM ) B(; T M ) P j+ j+ B(; T j+ ) + + P M M B(; T M ) P a (; T j ) = P m j+b(; T j ) + (P i=j+ i+ P i )B(; T i ) P M B(; T M ) P M ~ ; i=j+ i B(; T i ) ; ~ i = P i i Assume ha processes G(m) ~ are given by formula (), wih 's subsiued wih ~'s. We may apply he recursive procedure based on he same se of general ideas as before. Le us focus, for simpliciy, on he dierences which occur in he rs wo seps. Insead of (5), we now have Z (; T ) = B(; T ) ~ M B(; T ) = P M a (; T ) + M (8) This allows us o nd ~ PT ; and hus also o inroduce he swap rae a (; T 2 ) Similarly, equaliy (6) now akes he following form Z 2 (; T 2 ) = B(; T 2 ) ~ M B(; T ) + ~ M B(; T ) = P M a (; T 2 )+(P M P M )Z 2 (; T )+P M Z 2 (; T ) ; in urn Z 2 (; T i ) = Z (; T i ) + ~ ; i = ; M Z (; T )

11 Marek Rukowski The auxiliary process Z 2 (; T 2 ) may hus be expressed in erms of processes which have been found in he rs sep (Z (; T ) = = ~ M and Z (; T ) is given by (8)). More generally, we have m X Z m (; Tm) = P Mm+ a (; Tm) + (P Mi P Mi+ )Z m (; Ti ) + P M Z m (; T ) i= ; Z m (; T i ) = Z m (; T i ) ; i = ; ; m + Mm+ ~ Z m (; T m ) Since processes Z m (; T i ); i = ; ; m are already known from he preceding sep, we conclude ha he backward inducion is sill feasible, subjec o suiable minor modicaions. 4.3 Fixed-mauriy Swapions For a xed, bu oherwise arbirary, dae T j ; j = ; ; M; we consider a swapion wih expiraion dae T j ; wrien on a payer swap seled in arrears, wih xed rae ; which sars a dae T j and has M j accrual periods (i is referred o as he j h swapion in wha follows). Noice ha he j h swapion can be seen as a conrac which pays o is owner he amoun l P ( Mj (T j ; T j ) ) + a each selemen dae T l ; l = j +; ; M (we wrie P o denoe a consan noional principal). Equivalenly, he j h swapion pays an amoun ~Y = MX l=j+ l P B(T j ; T l ) ~(T j ; T j ) + a mauriy dae T j I is useful o observe ha ~ Y admis he following represenaion ~Y = ~ GTj (M j)p ~(T j ; T j ) + Recall ha he model of xed-mauriy forward swap raes species he dynamics of he process ~(; T j ) hrough he following SDE d~(; T j ) = ~(; T j )~(; T j ) d ~ W T j+ ; W ~ T j+ follows a sandard d-dimensional Brownian moion under he corresponding forward swap measure PTj+ ~ The deniion of he forward swap measure PTj+ ~ implies ha any process of he form B(; T k )= G ~ (M j) is a local maringale under PTj+ ~ Furhermore, from he general consideraions concerning he choice of a numeraire (see, e.g. Geman e al. (995) or Musiela and Rukowski (997b)) i is easy o see ha, he arbirage price (X) of an aainable claim X = g(b(t j ; T j+ ); ; B(T j ; T M )); which seles a ime T j ; equals (X) = ~ G (M j) E ~ P T j+ ~G T j (M j)x j F ; 8 2 [; Tj ] Applying he las formula o he swapion's payo ~ Y ; we obain fps j = ( ~ Y ) = ~ G (M j)p E ~ P T j+ (~(Tj ; T j ) ) + j F ; 8 2 [; Tj ]; PS f j sands for he price a ime of he j h swapion. We assume from now on ha ~(; T j ) [; T j ]! R d is a bounded deerminisic funcion; ha is, we place ourselves wihin he framework of he lognormal model of xed-mauriy forward swap raes. We have he following resul, due o Jamshidian (996).

12 2 Models of Forward LIBOR and Swap Raes Proposiion 4. For any j = ; ; M ; he arbirage price a ime 2 [; T j ] of he j h swapion equals fps j = MX l=j+ l P B(; T l ) ~(; T j )N h ~ (; T j ) N h2 ~ (; T ) ; N denoes he sandard Gaussian probabiliy disribuion funcion, and ~h ;2 (; T j ) = ln(~(; T j)=) 2 ~v2 (; T j ) ~v(; T j ) R wih ~v 2 T (; T j ) = j j~(u; T j )j 2 du To obain a closed-from soluion for he price of a swapion wrien on a forward swap rae associaed wih a given family of amorising swaps, i is convenien o assume he framework inroduced in Secion 4.2. In his seing, i is enough o replace in he valuaion formula above he consan principal P by he variable principal P l ; he swap rae (; T j ) by a (; T j ); and he volailiy funcion (; T j ) by he (deerminisic) volailiy a (; T j ) of he process a (; T j ) 4.4 Fixed-lengh Swaps In his secion, we no longer assume ha he underlying swap agreemens have dieren lenghs bu he same mauriy dae. On he conrary, he lengh, K; of a swap will now be xed, however, is mauriy dae will vary. For insance, for a forward swap which sars a ime T j ; j = ; ; M K; he rs selemen dae is T j+ ; and is mauriy dae equals T K+j The value a ime 2 [; T j ] of such a swap equals (cf. (8)) FS j () = B(; T j ) j+k X l=j+ c l B(; T l ); c l = l for l = j + ; ; j + K ; and c j+k = + j+k The forward swap rae of lengh K; for he dae T j ; is ha value of he xed rae which makes he underlying forward swap saring a T j worhless a ime Using he las formula, we nd easily ha K (; T j ) = B(; T j ) B(; T j+k ) j+ B(; T j+ ) + + j+k B(; T j+k ) For convenience, we will wrie ^(; T j ) insead of K (; T j ) Also, we denoe M = M K + Remarks. I is worhwhile o noe ha he forward LIBOR rae L(; T j ) coincides wih he oneperiod forward swap rae over he ime period [T j ; T j+ ] Formally, he model of forward LIBOR raes presened in Secion 3 may be seen as a special case of he model of xed-lengh forward swap raes. However, since in his case K = ; he consrucion is simpler, and requires less assumpions. For any m = ; ; M ; we inroduce he xed-lengh coupon process ^G(j) by seing ^G (m) def = X Mm+ l=m m+ l B(; T l ) = X m+k2 k=m ; Mk B(; T k ); 8 2 [; T j ] As one migh expec, a modied forward swap measure which we are now going o inroduce corresponds o he choice of a xed-lengh coupon process as a numeraire asse. Deniion 4.2 For a xed j = ; ; M ; a probabiliy measure ^PTj on (; F Tj ); equivalen o he underlying probabiliy measure P; is called he xed-lengh forward swap measure for he dae T j if, for every k = ; ; M; he relaive bond price U M j+(; T k ) def = follows a local maringale under ^PTj B(; T k ) ^G (M j + ) = B(; T k ) j B(; T j ) + + j+k B(; T j+k ) ; 8 2 [; T k ^ T j ];

13 Marek Rukowski 3 I is clear ha each forward swap rae ^(; T j ); j M ; follows a local maringale under he forward swap measure for he dae T j+ ; since ^(; T j ) = U M j+(; T j ) U M j+(; T j+k ) If we assume, in addiion, ha ^(; T j ) is a sricly posiive process, i is naural o posulae ha i saises T d^(; T j ) = ~(; T j )^(; T j ) d ^W j+ ; ^W T j+ follows a sandard Brownian moion under ^PTj+ The iniial condiion is ^(; T j ) = B(; T j ) B(; T j+k ) j+ B(; T j+ ) + + j+k B(; T j+k ) As expeced, he consrucion of xed-lengh forward swap raes model also relies on he backward inducion. In conras o boh previously sudied models, he erminal forward swap measure ^PTM does no coincide wih he \usual" forward measure for he dae T M ; however. Neverheless, we sill nd i convenien o proceed backwards in ime { ha is, o sar by posulaing ha he underlying probabiliy measure P represens he forward swap measure for he dae T M ; and hen o dene recursively he forward swap measures corresponding o he preceding daes, T M ; T M 2; ; T To proceed furher, we need o assume ha we are given a family ^(; T j ); j = ; ; M K; of volailiies of forward swap raes ^(; T j ); as well as he iniial erm srucure of bond prices B(; T j ); j = ; ; M As one migh easily guess, he rs sep is o posulae ha ^(; T M ); which equals saises ^(; T M ) = B(; T M ) B(; T M ) M B(; T M ) + + M B(; T M ) = B(; T M ) B(; T M ) ; ^G () d^(; T M ) = ^(; T M )^(; T M ) dw ; wih obvious iniial condiion. This means, in paricular, ha we have chosen ^PTM = P and ^W T M = W Our nex goal is o inroduce he forward swap measure for he dae T M Case of equal accrual periods. Assume rs, for simpliciy, ha j = is consan. In his case he backward inducion goes hrough as in previous cases. Recall ha ^(; T M 2) = On he oher hand, we have B(; T M 2) B(; T M ) B(; T M ) + + B(; T M ) = B(; T M 2) B(; T M ) ^G (2) U 2 (; T k ) = B(; T k) ^G (2) = B(; T k ) B(; T M ) + + B(; T M ) ; so ha he relaive bond price U 2 (; T k ) admis he following represenaion U 2 (; T k ) = U (; T k ) + ^(; T M ) In view of Lemma 3., i is obvious ha he process U 2 (; T k ) follows a local maringale under ^PTM whenever he process Z ^W T M T = ^W M ^(u; T M ) + ^(u; T M ) ^(u; T M ) du follows a sandard Brownian moion under his measure. To nd ^PTM, i is enough o apply Girsanov's heorem. A general inducion sep relies on he following relaionship U M m+(; T k ) = B(; T k ) ^G (M m + ) = U M m(; T k ) + ^(; T m ) ;

14 4 Models of Forward LIBOR and Swap Raes which allows us o dene ^PTM m and ^W T M m provided ha he measure ^PTm+ and processes ^(; T m ); W Tm+ are already known. We se ^W Tm = ^W T m+ Z ^(u; T m ) + ^(u; T m ) ^(u; T m) du; 8 2 [; T m ]; and we deermine ^PTm hrough Girsanov's heorem. Given ^W T m and ^PTm ; we posulae ha he forward swap rae ^(; T m ) is governed by he SDE d^(; T m ) = ~(; T m )^(; T m ) d ^W T m This shows ha he knowledge of all swap rae volailiies (and, of course, he iniial erm srucure) is sucien if he purpose is o consruc an arbirage-free family of forward swap raes. Somewha surprisingly, his is no longer he case if he lengh of he accrual period varies over ime. General case. We rs deal wih he dae T M 2 Noe ha ^(; T M 2) = Furhermore, for any k we have B(; T M 2) B(; T M ) M B(; T M ) + + M B(; T M ) = B(; T M 2) B(; T M ) ^G (2) U 2 (; T k ) = B(; T k) ^G (2) The las equaliy may be given he following equivalen form U 2 (; T k ) = = B(; T k ) M B(; T M ) + + M B(; T M ) U (; T k ) + U (; T M ) + M ^(; T M ) = U (; T k ) + V () ; we wrie = M M ; and he deniion of he process V is clear from he conex. To proceed furher, we need o assume ha he dynamics of he process U (; T M ) = B(; T M )= ^G() under ^P TM is known; more precisely, i is enough o specify is volailiy, since i is clear ha U (; T M ) is governed by he SDE of he form du (; T M ) = U (; T M ) (; T M ) d ^W T M for some process (; T M ) Suppose ha (; T M ); and hus also U (; T M ); are known. denoe v () = U (; T M ) (; T M ) + M ^(; T M )^(; T M ) Then ^W T M Z T = ^W M v (u) + V (u) du; 8 2 [; T M ]; Le us and hus he probabiliy measure ^PTM can be found explicily. Nex, he swap rae ^(; T M 2) is dened by seing d^(; T M 2) = ^(; T M 2)^(; T M 2) d ^W T M Le us now consider a general inducion sep. We assume ha we have already found ^PTm+ and ^W T m+ ; and hus also ^(; T m ) In order o dene ^(; T m ); we need rs o specify ^P Tm and ^W Tm We have he following relaionship U M m+(; T k ) = B(; T k ) ^G (M m + ) = U M m(; T k ) + V M m() ;

15 Marek Rukowski 5 V M m() = M mu M m(; T m+k ) + m^(; T m ) and M m = m+k m We have, as expeced ^W Tm = ^W T m+ Z v M m(u) + V M m(u) du; 8 2 [; T m]; v M m() = U M m(; T m+k ) M m(; T m+k ) + m^(; T m )^(; T m ) and M m(; T Mm+ ) represens he volailiy of he process U M m(; T m+k ) = ^G (M m + )B(; T m+k ) Concluding, in order o consruc a model of xed-lengh forward swap raes hrough backward inducion, we nd i convenien o assume, in addiion, ha he volailiies of relaive bond prices are exogenously given. U m (; T Mm+ ) = ^G (m)b(; T Mm+ ); m = ; ; M 2; 4.5 Fixed-lengh Swapions For any dae T j ; j = ; ; M ; le us consider a payer swapion wih expiraion dae T j ; wrien on a payer swap, wih xed rae ; which sars a dae T j and has K accrual periods. The j h swapion may now be seen as a conrac ha pays ^Y = a ime T j I is useful o noice ha j+k X l=j+ l P B(T j ; T l ) ^(T j ; T j ) + ^Y = ^GTj (M j)p ^(T j ; T j ) + In he framework of he xed-lengh forward swap raes model, he forward swap rae ^(; T j ) saises d^(; T j ) = ^(; T j )^(; T j ) d ^W T j+ ; ^W T j+ follows a sandard d-dimensional Brownian moion under ^P Tj+ In view of he deniion of he forward swap measure PTj+ ~ ; any process of he form B(; T k )= ^G (M j) is a local maringale under ^PTj+ Therefore he price of an aainable claim X; which seles a ime T j and depends on he bond prices, equals (X) = ^G (M j) E ^P T j+ ^G T j (M j)x j F ; 8 2 [; Tj ] In paricular, for he j h xed-lengh swapion we ge cps j = ( ^Y ) = ^G (M j)p E ^P T j+ (^(Tj ; T j ) ) + j F ; 8 2 [; Tj ] Le us place ourselves wihin he framework of he lognormal model of xed-lengh forward swap raes, in which he volailiies ^(; T j ) of xed-lengh forward swap raes are assumed o follow bounded deerminisic funcions. Then we have he following resul, which is a counerpar of Proposiion 4.. I is useful o noe ha if K = ; he j h swapion may be idenied wih he j h caple. Therefore Proposiion 4.2 covers also he valuaion of caples (and hus also caps) in he lognormal model of forward LIBOR raes presened in Secion 3.

16 6 Models of Forward LIBOR and Swap Raes Proposiion 4.2 For any j = ; ; M ; he arbirage price a ime 2 [; T j] of he j h swapion equals cps j = l=j+k X l=j+ l P B(; T l ) ^(; T j )N ^h (; T j ) N ^h2 (; T ) ; N denoes he sandard Gaussian probabiliy disribuion funcion, and wih ^v 2 (; T j ) = R T j j^(u; T j )j 2 du ^h ;2 (; T j ) = ln(^(; T j)=) 2 ^v2 (; T j ) ^v(; T j ) I is worhwhile o observe he swapion's price a ime depends only on he iniial erm srucure and he volailiy of he underlying swap. In paricular, he volailiies M m(; T m+k ) of relaive bond prices, which were used as auxiliary inpus in model's consrucion, do no ener he valuaion formula. For his reason, o ge he valuaion formula above, we may equally well ake hese volailiies equal idenically o zero. On he oher hand, i is reasonable o expec ha he choice of hese volailiies would inuence he price processes of underlying bonds, and hus also he swapion's price a ime > The choice of he underlying bond price srucure is, of course, even more imporan in valuaion { also a ime { of derivaive securiies wih American feaures, such as Bermudan swapions. 4.6 Bond Prices All models of forward raes examined in his noe are arbirage-free, in he following sense i is possible o nd a family B(; T j ); j = ; ; n of bond prices, which is compaible wih forward raes specied by a model, and which possesses he arbirage-free feaures. The las propery means ha all bond prices discouned by he price of he bond wih he longes mauriy follow local maringales under some probabiliy measure equivalen o he original probabiliy measure. The acual procedure of nding such a family depends on a paricular model and requires furher sudies (of pracical, raher han heoreical, naure). In he case of he lognormal model of forward LIBOR raes, an ineresing mehod of a direc consrucion of underlying bond prices { no only for he daes T j ; bu indeed for any mauriy { was proposed recenly by Sawa (997). Essenially, his mehod combines he HJM mehodology wih he (linear or non-linear) inerpolaion. I should be sressed, however, ha his approach akes also ino accoun he properies of a nie family of forward LIBOR raes prediced by he lognormal model. Since various models of forward marke raes are inconsisen wih each oher, in general (his propery is obvious for lognormal models), anoher imporan issue which arises in his conex is he model choice. We refer o Jamshidian (997) for an ineresing discussion of his poin. Le us only menion ha, essenially, he advocaes he use of dieren models for dieren classes of ineres-rae derivaives. When combined wih each oher, hese models may give rise o a srucure which is no arbirage-free { his feaure should no be seen as heir major deciency, however. Indeed, his is a ypical siuaion in mos real-life applicaions of he arbirage pricing heory. On he posiive side, since he focus is on he modelling of ineres raes which are direcly observed in he marke, he model calibraion should hopefully prove easier han in he case of radiional models of insananeous raes. Sill, if for whaever reasons we insis on he absence of arbirage in he combined model as a compulsory requiremen, a judiciously chosen arbirage-free combinaion of various models seems o be a plausible soluion. A rs glance, i would be reasonable o focus on a combined model, which would encompass he forward swap raes model for longer mauriies, he forward LIBOR raes model for shorer mauriies, and he classical by now Heah-Jarrow-Moron (992) bond pricing mehodology for mauriies from ouside a given enor srucure. Deails of such a consrucion would depend on he feaures of a paricular marke and he purpose of he modelling. ;

17 Marek Rukowski 7 5 Fuures Raes and Prices Le us inroduce he discree-ime savings accoun B by he formula B Tj = jy + k L(T k ) = k= jy k= B (T k ; T k ) Noice ha B T = + L() = =B(; T ) is deerminisic (consequenly, he \spo" maringale measure will coincide wih he forward Libor measure for he dae T ). I is clear ha B Tj = G Tj for any j = ; ; M (he process G is given by (3)). Remarks. I is worhwhile o poin ou ha B is he unique discree-ime savings accoun implied by he model of forward LIBOR raes, a coninuous-ime savings accoun is no uniquely deermined. The coninuous-ime process G; which was inroduced by Jamshidian (997) (see Secion 3. of his noe), is merely an example of a coninuous-ime savings accoun implied by he family of forward LIBOR raes. In general, here is no reason o expec ha a coninuous-ime savings accoun B ~ compaible wih he family L(; T j ) of forward LIBOR raes would necessarily saisfy BTj ~ = B Tj for j = ; ; M In paricular, i seems plausible ha a realisic model of bond prices species B T as a random variable raher han as a deerminisic consan. I should hus be sressed ha in wha follows we work as if he process G was indeed chosen as he coninuous-ime savings accoun implied by he family of forward LIBOR raes (which are lognormally disribued under he forward measures). Such a choice implies ha prices of all bond wih mauriies less han T are known and hey follow necessarily deerminisic funcions. The boom line is ha he resuls of his secion concerning Eurodollar fuures and Eurodollar fuures opions are valid only under our specic assumpions. 5. Spo Maringale Measure For our furher purposes, i is essenial o invesigae he concep of a spo maringale measure wihin he framework of he model of forward LIBOR rae. We posulae ha he spo maringale measure P is dened hrough he Radon-Nikodym derivaive c is a consan. Equivalenly dp T = ; P -a.s., dp c B T dp = c B T ; P T -a.s., (9) dp T c = =E P T (B T ) = B(; T ) Of course, we need now o jusify his convenion. Remarks. In he sandard HJM mehodology, he modelling is done under he spo measure, P say. The forward measure P T for any dae T is hen inroduced by seing dp T dp = ; B(; T )B T P -a.s. In he presen conex, he order is reversed; ha is, forward measures for discree se of daes are known by consrucion, and \spo" measure is dened wih reference o he forward measures. Le us emphasize once again ha we did no specify a family of bond prices, excep for bonds wih mauriies in he rs inerval and bond prices B(T j ; T j+ ) This makes he concep of a spo probabiliy measure more ambiguous han in he radiional HJM framework, in which he savings accoun process is uniquely specied hrough he shor-erm rae r = f(; ) As one migh expec, he probabiliy measure P given by (9) is closely relaed o spo LIBOR measure P L ; inroduced in Secion 3.. To be more specic, we have he following resul.

18 8 Models of Forward LIBOR and Swap Raes Lemma 5. Le P be a probabiliy measure such ha dp = B(; T )G T ; P T -a.s. (2) dp T Then he process B(; T k )=G follows a maringale under P for k = ; ; M Proof. For simpliciy, we assume ha all processes B(; T k )=B(; T ) follow a maringale under P T (his allows us o work direcly wih condiional expecaions raher han o apply I^o's rule). We need o show ha B(Tk ; T k ) E P F = B(; T k) ; 8 2 [; T k ] G Tk G The lef-hand-side in he las formula equals E P B(Tk ; T k ) G Tk F = E P T (G T G T k j F ) E P T (G T j F ) From he properies of he forward LIBOR measure, we obain Therefore, i is enough o show ha E P T (G T j F ) = G B (; T ); 8 2 [; T ] J def = E P T (G T G T k j F ) = B(; T k )B (; T ) If k = M ; he las equaliy is obvious. If k < M ; hen for every T k < T ; we have J = E P T B(T k ; T k+ ) B(T ; T F ) = E P T B(T k ; T k+ ) B(T B(T2 ; T ) E ; T ) P T B(T ; T F T F ) 2 = E P T B(T k ; T k+ ) B(T3 ; T 2 ) B(T2 ; T F ) Repeaing, if necessary, his procedure we nd easily ha E P T (G T G T k j F ) = E P T B(T k ; T ) F = B(; T k) B(; T ) ; as expeced. 2 In view of Lemma 5., probabiliy measure P given by (2) (or equivalenly, by (9)) saises he deniion of he spo LIBOR measure P L If i was known, in addiion, ha he spo LIBOR measure is uniquely deermined, his would mean, of course, ha P = P L Since he uniqueness of P L is unclear, we prefer o work in wha follows wih he measure P (which is, by deniion, unique). I should be made clear ha a spo maringale measure is no well-dened in he model of forward LIBOR raes, since he model does no uniquely specify he implied (coninuous-ime) savings accoun. I seems ha his can be done in various way (for insance, by specifying all bond prices). Noice ha P L corresponds essenially o he choice of he process G as a savings accoun. By choosing G as a savings accoun, one implicily denes also all bond prices hrough he sandard formula B(; T ) = G E P L(G T j F ); 8 T T For convenience, we shall frequenly refer o P as he spo probabiliy measure. Le us now consider he daes T j j = ; ; M We denoe by P j he resricion of P o he -eld F Tj { ha is, P j = P j ; 8 j = ; ; M FTj

19 Marek Rukowski 9 Our aim is o show ha he probabiliy measure P j can be inroduced wihou making reference o P ; we shall use insead he forward LIBOR measure P Tj for a given dae as he reference probabiliy measure. Remarks. Le us place ourselves for he momen wihin he sandard HJM seup. For any mauriies U T; we have Consequenly Furhermore, Concluding, we have dp U dp T = B(; T )B T B(; U)B U ; P T -a.s. dp U dp T jfu = E P T B(; T )BT B(; U)B U FU = B(; T ) B(; U) E P T BT B U FU E P T BT B U FU c is he normalizing consan c = = E P (B U j F U) E P (B T j F U) = E P (B U B T j F U) = B(U; T ) dp U dp T = dp U c = dp T jfu B(U; T ) ; (2) he P T B (U; T ) i = E PU (B(U; T )) = B(; T ) B(; U) I seems reasonable o expec ha relaionship (2) is universal, in paricular, ha i also holds in he framework of he forward LIBOR raes model. This is indeed rue, as he following resul shows. 4 Lemma 5.2 For any j = ; ; M we have dp Tj dp Tj+ = B(; T j+) B(; T j ) B(T j ; T j+ ) = + j+l(t j ; T j ) + j+ L(; T j ) (22) and or equivalenly dp Tj dp T dp Tj dp T = B(; T ) M Y B(; T j ) E P T k=j = B(; T ) B(; T j ) E P T M Y k=j B(T k ; T k+ ) F Tj ; (23) + k+ L(T k ; T k ) FTj (24) Proof. Recall ha we have (see Secion 3) dp Tj dp Tj+ = exp Z Tj (u; T j ) dw Tj+ u 2 Z Tj j(u; T j )j 2 du ; P Tj+ -a.s., (u; T j ) = j+l(u; T j ) + j+ L(u; T j ) (u; T j); 8 u 2 [; T j ] 4 Noice ha formula (22) agrees wih wha we already know, namely, ha i is possible o deermine he forward LIBOR measure P Tj if he forward LIBOR measure P Tj and he process L(; T j ) are specied.

20 2 Models of Forward LIBOR and Swap Raes Since T j is xed, we nd i convenien o wrie L() = L(; T j ); () = (; T j ); ec. in wha follows. Le us dene he process J by seing def J = exp Z for 2 [; T j ] I is clear ha J = and On he oher hand, he process saises I = and di = j+ L(u) Tj+ (u) dwu + j+ L(u) 2 dj = J Z j+ L() Tj+ () dw + j+ L() def I = + j+l() + j+ L() j+ L(u) + j+ L(u) (u) 2 du j+dl() + j+ L() = j+()l() Tj+ j+ L() Tj+ dw = I () dw + j+ L() + j+ L() This shows ha I = J for every 2 [; T j ] By seing = T j ; we obain equaliy (22). To prove he second formula, we proceed by inducion. For j = M he formula follows immediaely from (22). Le us consider j = M 2 For any A 2 F T 2 we have B(; T ) B(; T ) 2 ZA B(T ; T ) 2 B(T ; T ) dp T = B(; T ) B(; T ) 2 ZA B(T ; T ) dp T 2 More generally, for any j and arbirary even A 2 F Tj ; we have B(; T ) B(; T j ) Z Y M A k=j = B(; T ) B(; T j ) B(T k ; T k+ ) dp T Z = B(; T j+) B(; T j ) A Z B(T j ; T j+ ) E P T A M Y k=j+ B(T k ; T k+ ) B(T j ; T j+ ) dp T j+ = P Tj (A); = P T 2 (A) F Tj+ dp T he second equaliy follows from he inducion sep, and he las one is an immediae consequence of (22). This complees he proof of he lemma 2 We are in a posiion o prove he following auxiliary resul. Lemma 5.3 For any j = ; ; M; we have c j = =E P T j (B T j ) = B(; T j ) Proof. We have (cf. (2)) dp j = c j B Tj dp Tj ; P Tj -a.s., (25) dp = E P dp T (B(; T )G TM j F Tj ) = B(; T )G Tj E P T T jft j Using Lemma 5.2, we nd easily ha M Y k=j B(T k ; T k+ ) F Tj dp dp Tj = B(; T j )G Tj dp T jft dp j T This proves (25) upon simplicaion. 2 In view of he las resul, when dealing wih fuures conracs wih mauriy T j ; we may inroduce he spo maringale measure on (; F Tj ) using equaliy (25) raher han (2). I appears ha such a choice considerably simplies he calculaions.

21 Marek Rukowski Fuures LIBOR Rae and Eurodollar Fuures Price By he marke convenion, he selemen price of he Eurodollar fuures conrac equals (cf. Amin and Ng (997)) E(T ; T ) = 2 L(T ; T ) a he conrac's expiraion dae T Consequenly, a any dae 2 [; T ] he Eurodollar fuures price (or index) equals E(; T ) = E P 2 L(T ; T ) j F = 2 E P L(T ; T ) j F Le us formally dene he fuures LIBOR rae L f (; T ) for he dae T by seing L f (; T ) = E P (L(T ; T ) j F ) = E P 2 (L(T ; T ) j F ); 8 2 [; T ]; so ha E(; T ) = 2 L f (; T ) Recall ha B T2 = + L(; ) + 2 L(T ; T ) and dp 2 = B(; T 2 )B T2 dp T2 Therefore, using he absrac Bayes rule, we obain L f (; T ) = E P T2 ( + 2 L(T ; T ))L(T ; T ) j F = I E P T2 + 2 L(T ; T ) j F J I is eviden ha Furhermore, J = E P T2 + 2 L(T ; T ) F = + 2 L(; T ) I = E P T2 ( + 2 L(T ; T ))L(T ; T ) F = E PT2 + 2 L(; T )e 2 v2 L(; T )e 2 v2 F ; Z T Z T = (u; T ) dwu T2 ; v 2 = Var ( ) = j(u; T )j 2 du (26) Noice ha he random variable is independen of he -eld F and i has he Gaussian law wih zero mean and he variance v 2 Consequenly, we have I = L(; T ) + 2 L 2 (; T )E P T2 e 2 v 2 = L(; T ) + 2 L 2 (; T )e v2 (27) Combining he formulae above, we conclude ha L f (; T ) = L(; T ) + 2 L(; T ) + 2 L(; T )e R T j(u;t )j 2 du for every 2 [; T ] We are in a posiion o formulae he following resul. Proposiion 5. The fuures LIBOR rae for he dae T equals L f (; T ) = L(; T ) + 2 g ()L(; T ) + 2 L(; T ) and he Eurodollar fuures price, for he fuures conrac ha seles a ime T ; equals (28) E(; T ) = 2g 2 ()L 2 (; T ) ; (29) + 2 L(; T ) Z T g () = exp j(u; T )j 2 du ; 8 2 [; T ] (3)

22 22 Models of Forward LIBOR and Swap Raes The nex resul deals wih he dynamics of he fuures LIBOR rae and Eurodollar fuures price under he spo measure P I is useful o observe ha, under our presen assumpions, equaliy P = P T is saised. For his reason, we do no need o inroduce he \spo" Brownian moion W ; since we may and do assume ha W = W T Proposiion 5.2 The fuures LIBOR rae saises W = W T ; and he process Z is given by he formula Z = The Eurodollar fuures price saises dl f (; T ) = L f (; T )Z (; T ) dw ; (3) + 2 L(; T ) + 2g ()L(; T ) + 2 g ()L(; T ) ; 8 2 [; T ] (32) de(; T ) = E(; T ) Z (; T ) dw (33) Proof. Since L f (; T ) is given by an explici formula (29), i is enough o apply he I^o formula. Le us denoe L f = L f (; T ); L() = L(; T ); () = (; T ) and g() = g () I is clear ha L f = X Y ; X = L() + 2 L() ; Y = + 2 L()g() I is useful o observe ha Using I^o's rule, we nd ha and This in urn implies ha dl f = 2 g()l()x () dw T 2 dl() = L()() dw T2 ; dg() = g()j()j 2 d dx = 2 X X () dw T2 2 X 2 j()j 2 d dy = 2 g()l() () dw T2 j()j 2 d + 2 X 2 X g()l()j()j 2 d Afer rearranging, we obain Observe ha j()j 2 d + 2 X Y X () dw T2 dl f = ( 2 X )Y + 2 g()l() X dw T2 2 X () d dw T2 Furhermore, by simple algebra we obain 2 X () d = dw T2 2L() T () d = dw + 2 L() 2 X Y + 2 g()l() X = L f 2 2 X Y 2 X 2 j()j2 d Upon subsiuion, his yields he dynamics (3). To derive (33), i is enough o observe ha and de(; T ) = 2 dl f (; T ) = 2 L f (; T )Z (; T ) dw T 2 L f (; T ) = E(; T ) This complees he proof of he proposiion. 2