Heath Jarrow Morton (HJM) Methodology

Transcription

1 Chaper 8 Heah Jarrow Moron (HJM Mehodology original aricle by Heah, Jarrow and Moron [1, MR[2(Chaper 13.1, Z[28(Chaper 4.4, ec. As we have seen in he previous secion, shor rae models are no always flexible enough o calibraing hem o he observed iniial yield curve. Heah, Jarrow and Moron (HJM, 1992 [1 proposed a new framework for modelling he enire forward curve direcly. The seup is as in Chaper 6. We fix a sochasic basis (Ω, F, (F, P, where P is considered as objecive probabiliy measure, and a d-dimensional Brownian moion W. 8.1 Forward Curve Movemens We assume ha we are given an R-valued and R d -valued sochasic process α(,t and σ(,t = (σ 1 (,T,...,σ d (,T, respecively, wih wo indices,,t, such ha HJM.1 for every ω, he maps (,T α(,t,ω and (,T σ(,t,ω are coninuous for T, HJM.2 for every T, he processes α(,t and σ(,t, T, are adaped, HJM.3 E α(s, ds d < for all T, 97

2 98 CHAPTER 8. HJM METHODOLOGY HJM.4 E σ(s, 2 ds d < for all T. For a given coninuous iniial forward curve T f(,t i is hen assumed ha, for every T, he forward rae process f(,t follows he Iô dynamics f(,t = f(,t + α(s,tds + σ(s,tdw(s, T. (8.1 This is a very general seup. The only subsanive economic resricions are he coninuous sample pahs assumpion for he forward rae process, and he finie number, d, of random drivers W 1,...,W d. Noice ha he inegrals in (8.1 are well defined by HJM.1 HJM.4. Moreover, i follows from Corollary ha he shor rae process r( = f(, = f(, + α(s,ds + σ(s,dw(s has a predicable version (again denoed by r( and, by Schwarz inequaliy and he Iô isomery, E r( d + T T 1 2 f(, d + ( f(, d + ( E α(s, ds d ( 2 E σ(s,dw(s d E σ(s, 2 ds d 1 2 E α(s, ds d 1 2 <. Hence he savings accoun B( = e R r(sds is well defined. More can be said abou he zero-coupon bond prices. Lemma ( For every mauriy T, he zero-coupon bond price process P(,T = exp T f(,udu, T, is an Iô process of he form P(,T = P(,T + P(s,T (r(s + b(s,tds + P(s,Ta(s,TdW(s (8.2

3 8.1. FORWARD CURVE MOVEMENTS 99 where a(s,t := b(s,t := s s σ(s,udu, α(s,udu a(s,t 2. Proof. Using he classical Fubini Theorem and Theorem for sochasic inegrals wice, we calculae logp(,t = f(,udu = f(,udu = = + = + f(,udu f(,udu s α(s,uds du α(s,ududs α(s,ududs s σ(s,udw(sdu σ(s,ududw(s σ(s,ududw(s f(,udu + α(s,ududs + σ(s,ududw(s s s f(,udu + (b(s,t 12 a(s,t 2 ds + a(s,tdw(s ( u u f(,udu + α(s,uds + σ(s,udw(s du }{{} = log P(,T + + a(s,tdw(s, =r(u (r(s + b(s,t 12 a(s,t 2 ds and we have used he fac ha σ(s,u1 {s u}dw(s = u σ(s,udw(s. Iô s formula now implies (8.2 ( exercise. We wrie Z(,T = P(,T B( for he discouned bond price processes.

4 1 CHAPTER 8. HJM METHODOLOGY Corollary We have, for T, Z(,T = P(,T + Proof. Iô s formula ( exercise. Z(s,Tb(s,Tds Absence of Arbirage Z(s,Ta(s,TdW(s. In his secion we invesigae he resricions on he dynamics (8.1 under he assumpion of no arbirage. In wha follows we le γ L be such ha E(γ W is a uniformly inegrable maringale wih E (γ W >. Girsanov s Change of Measure Theorem hen implies ha dq dp = E (γ W defines an equivalen probabiliy measure Q P, and W( := W( γ(s ds is a Q-Brownian moion. We call Q an ELMM for he bond marke if he discouned bond price processes, Z(,T = P(,T, T, are Q-local B( maringales, for all T. Theorem (HJM Drif Condiion. Q is an ELMM if and only if b(,t = a(,t γ( T d dp a.s. (8.3 In his case, he Q-dynamics of he forward raes f(, T, T, are of he form ( f(,t = f(,t + σ(s,t σ(s,udu ds + σ(s,td W(s. } s {{ } HJM drif (8.4 Proof. In view of Corollary we find ha dz(,t = Z(,T (b(,t + a(,t γ( d + Z(,Ta(,Td W(.

5 8.2. ABSENCE OF ARBITRAGE 11 Hence Z(,T, T, is a Q-local maringale if and only if b(,t = a(, T γ( d dp-a.s. Since a(, T and b(, T are coninuous in T, we deduce ha Q is an ELMM if and only if (8.3 holds. Differeniaing boh sides of (8.3 in T yields α(,t + σ(,t Insering his in (8.1 gives (8.4. σ(,udu = σ(,t γ( T d dp a.s. Remark I follows from (8.2 and (8.3 ha dp(,t = P(,T (r( + a(,t ( γ(d + P(,Ta(,TdW(. Whence he inerpreaion of γ as he marke price of risk for he bond marke. The sriking feaure of he HJM framework is ha he disribuion of f(,t and P(,T under Q only depends on he volailiy process σ(,t (and no on he P-drif α(,t. Hence opion pricing only depends on σ. This siuaion is similar o he Black Scholes sock price model. We can give sufficien condiions for Z(,T o be a rue Q-maringale. Corollary Suppose ha (8.3 holds. Then Q is an EMM if eiher 1. he Novikov condiion holds; OR E Q [exp ( 1 T σ(,t 2 d < T ( he forward raes are posiive: f(,t T. Proof. We have Z(,T = P(,TE (σ(,t W. Hence he Novikov condiion (8.5 is sufficien for Z(,T o be a Q-maringale (see [24, Proposiion (1.26, Chaper IV. If f(,t, hen P(,T 1 and B( 1. Hence Z(,T 1. Bu a uniformly bounded local maringale is a rue maringale.

6 12 CHAPTER 8. HJM METHODOLOGY 8.3 Shor Rae Dynamics Wha is he inerplay beween he shor rae models of he las chaper and he presen HJM framework? Le us consider he simples HJM model: a consan σ(,t σ >. Suppose ha Q is an ELMM. Then (8.4 implies ( f(,t = f(,t + σ 2 T + σ 2 W(. Hence for he shor raes r( = f(, = f(, + σ2 2 This is jus he Ho Lee model of Secion In general, we have he following 2 + σ W(. Proposiion Suppose ha f(,t, α(,t and σ(,t are differeniable in T wih uf(,u du < and such ha HJM.1 HJM.4 are saisfied when α(,t and σ(,t are replaced by T α(,t and T σ(,t, respecively. Then he shor rae process is an Iô process of he form where r( = r( + ζ(u := α(u,u + u f(,u + Proof. Recall firs ha r( = f(, = f(, + ζ(udu + u u α(s,uds + α(s,ds + σ(u, u dw(u (8.6 u u σ(s,udw(s. σ(s,dw(s. Applying he Fubini Theorem o he sochasic inegral gives σ(s,dw(s = = = σ(s,sdw(s + σ(s,sdw(s + σ(s,sdw(s + s u (σ(s, σ(s,sdw(s u σ(s,ududw(s u σ(s,udw(sdu.

7 8.4. FUBINI S THEOREM 13 Moreover, from he classical Fubini Theorem we deduce, similarly, α(s,ds = α(s,sds + u and finally f(, = r( + Combining hese formulas, we obain (8.6. u f(,udu. u α(s,uds du, 8.4 Fubini s Theorem In his secion we prove Fubini s Theorem for sochasic inegrals. For he classical version of Fubini s Theorem, we refer o he sandard exbooks in inegraion heory. In wha follows we le A denoe a closed convex subse in [,T 2 and A c = [,T 2 \ A is complemen, e.g. A = {(,s [,T 2 s}. Theorem (Fubini s Theorem for sochasic inegrals. Consider he R d -valued sochasic process φ = φ(,s wih wo indices,,s T, saisfying he following properies: 1. for every ω, he map (,s φ(,s,ω is coninuous on he inerior of A and A c 2. for every s [,T, he process φ(,s, T, is adaped, 3. E φ(,s 2 d ds <. Then he sochasic process ψ(s := φ(,sdw( has a B[,T F T- measurable version (denoed again by ψ, and λ( := φ(,sds is piecewise coninuous. Moreover, ψ(sds = λ(dw(, ha is, ( ( φ(,sdw( ds = φ(,sds dw(. (8.7

8 14 CHAPTER 8. HJM METHODOLOGY Proof. We may assume ha φ N, oherwise we replace φ by φ1 { φ N} and le N. Noice ha τ(s,ω := inf{ φ(,s,ω > N} T is B[,T F T -measurable and a sopping ime for every fixed s ( exercise. Tha λ is piecewise coninuous and adaped follows from he classical Fubini Theorem ( exercise. Now le = < 1 < < n = T be a pariion of he inerval [,T, and define n 1 φ n (,s := φ( i,s1 (i, i+1 (. I is hen clear ha ψ n (s := i= n 1 φ n (,sdw( = φ( i,s (W( i+1 W( i (8.8 is B[,T F T -measurable. Moreover, ( T n 1 ψ n (sds = φ( i,s (W( i+1 W( i ds i= n 1 ( = = i= ( i= φ( i,sds (W( i+1 W( i φ n (,sds dw(. (8.9 From he Iô isomery and dominaed convergence we have lim E[ (ψ n (s ψ(s 2 [ = lim E (φ n (,s φ(,s 2 d = s. n n (8.1 Le A := {(s,ω lim n ψ n (s,ω exiss}. Then A is B[,T F T -measurable and so is he process ψ(s,ω := { lim n ψ n (s,ω, if (s,ω A, oherwise. (8.11 Bu in view of (8.1 we have ψ(s = ψ(s a.s. Hence ψ has a B[,T F T -measurable version, which we denoe again by ψ, so ha he inegral ψ(sds is well defined and F T-measurable.

9 8.4. FUBINI S THEOREM 15 From he Iô isomery and dominaed convergence again we hen have [ ( E ψ n (sds = T On he oher hand, [ ( ( E 2 ψ(s ds T E [ (ψ n (s ψ(s 2 ds [ E (φ n (,s φ(,s 2 d ds for n. (8.12 φ n (,sds dw( [ ( = E φ n (,sds 2 λ(dw( [ TE (φ n (,s φ(,s 2 ds d φ(,sds 2 d Combining (8.12 and (8.13 wih (8.9 proves he heorem. for n. (8.13 Corollary Le φ be as in Theorem Then he process ψ(s := has a predicable version. s φ(,sdw(, s [,T, Proof. This follows by similar argumens as in he proof of Theorem Replace (8.8 by he predicable process ψ n (s := s n 1 φ n (,sdw( = φ( i s,s (W( i+1 s W( i s, and modify (8.1 and (8.11 accordingly ( exercise. i=

10 16 CHAPTER 8. HJM METHODOLOGY

11 Chaper 9 Forward Measures We consider he HJM seup (Chaper 8 and assume here exiss an EMM Q P of he form dq/dp = E(γ W, as in Secion 8.2, under which all discouned bond price processes P(,T B(, [,T, are sricly posiive maringales. 9.1 T-Bond as Numeraire Fix T >. Since 1 P(,TB(T > and E Q [ 1 P(,TB(T = 1 we can define an equivalen probabiliy measure Q T Q on F T by For T we have dq T dq F = E Q dq T dq = 1 P(,TB(T. [ dq T dq F = P(,T P(,TB(. This probabiliy measure has already been inroduced in Secion 7.6. I is called he T-forward measure. 17

12 18 CHAPTER 9. FORWARD MEASURES Lemma For any S >, P(,S P(,T, [,S T, is a Q T -maringale. Proof. Le s S T. Bayes rule gives E Q T [ P(,S P(,T F s = = P(,S P(,TB( E Q [ P(,T P(s,S B(s P(s,T B(s P(s,T P(,TB(s = P(s,S P(s,T. P(,T F s We hus have an enire collecion of EMMs now! Each Q T corresponds o a differen numeraire, namely he T-bond. Since Q is relaed o he risk-free asse, one usually calls Q he risk neural measure. T-forward measures give simpler pricing formulas. Indeed, le X be a T-claim such ha X B(T L1 (Q, F T. (9.1 Is arbirage price a ime T is hen given by [ π( = E Q e R T r(s ds X F. [ To compue π( we have o know he join disribuion of exp T r(s ds and X, and inegrae wih respec o ha disribuion. Thus we have o compue a double inegral, which in mos cases urns ou o be raher hard work. If B(T/B( and X were independen under Q (which is no realisic! i holds, for insance, if r is deerminisic we would have a much nicer formula, since π( = P(,TE Q [X F, we only have o compue he single inegral E Q [X F ;

13 9.2. AN EXPECTATION HYPOTHESIS 19 he bond price P(,T can be observed a ime and does no have o be compued. The good news is ha he above formula holds no under Q hough, bu under Q T : Proposiion Le X be a T-claim such ha (9.1 holds. Then E Q T [ X < (9.2 and π( = P(,TE Q T [X F. (9.3 Proof. Bayes s rule yields [ X E Q T [ X = E Q < (by (9.1, P(,TB(T whence (9.2. And [ X π( = P(,TB(E Q P(,TB(T F which proves (9.3. P(,T = P(,TB( P(,TB( E Q T [X F = P(,TE Q T [X F, 9.2 An Expecaion Hypohesis Under he forward measure he expecaion hypohesis holds. Tha is, he expression of he forward raes f(,t as condiional expecaion of he fuure shor rae r(t. To see ha, we wrie W for he driving Q-Brownian moion. The forward raes hen follow he dynamics ( f(,t = f(,t + σ(s,t σ(s,udu ds + σ(s,tdw(s. s (9.4

14 11 CHAPTER 9. FORWARD MEASURES The Q-dynamics of he discouned bond price process is P(,T B( = P(,T + P(s,T B(s This equaion has a unique soluion P(,T B( ( σ(s,udu dw(s. (9.5 s = P(,TE (( σ(,udu W. We hus have (( dq T dq F = E σ(,udu W. (9.6 Girsanov s heorem applies and W T ( = W( + ( s σ(s,udu ds, [,T, is a Q T -Brownian moion. Equaion (9.4 now reads f(,t = f(,t + σ(s,tdw T (s. Hence, if hen E Q T [ (f(,t [,T σ(s,t 2 ds < is a Q T -maringale. Summarizing we have hus proved Lemma Under he above assumpions, he expecaion hypohesis holds under he forward measures f(,t = E Q T [r(t F.

15 9.3. OPTION PRICING IN GAUSSIAN HJM MODELS Opion Pricing in Gaussian HJM Models We consider a European call opion on an S-bond wih expiry dae T < S and srike price K. Is price a ime = (for simpliciy only is [ π = E Q e R T r(s ds (P(T,S K +. We proceed as in Secion 7.6 and decompose π = E Q [ B(T 1 P(T,S 1(P(T,S K KE Q [ B(T 1 1(P(T,S K = P(,SQ S [P(T,S K KP(,TQ T [P(T,S K. This opion pricing formula holds in general. We already know ha dp(,t P(,T = r(d + v(,tdw( and hence [ P(,T = P(,T exp v(s,tdw(s + where v(,t := ( We also know ha P(,T is a Q S -maringale and P(,S [,T Q T -maringale. In fac ( exercise P(,T P(,S = P(,T P(,S [ exp where = P(,T P(,S exp σ T,S (sdw(s 1 2 [ σ T,S (sdw S (s 1 2 (r(s 12 v(s,t 2 ds σ(, u du. (9.7 σ T,S (s := v(s,t v(s,s = ( P(,S P(,T [,T is a ( v(s,t 2 v(s,s 2 ds S T σ T,S (s 2 ds σ(s, u du, (9.8

16 112 CHAPTER 9. FORWARD MEASURES and P(,S P(,T = P(,S P(,T [ exp = P(,S P(,T exp σ T,S (sdw(s 1 2 [ σ T,S (sdw T (s 1 2 ( v(s,s 2 v(s,t 2 ds σ T,S (s 2 ds. Now observe ha [ P(T,T Q S [P(T,S K = Q S P(T,S 1 K [ P(T,S Q T [P(T,S K = Q T P(T,T K. This suggess o look a hose models for which σ T,S is deerminisic, and hence P(T,T and P(T,S are log-normally disribued under he respecive P(T,S P(T,T forward measures. We hus assume ha σ(,t = (σ 1 (,T,...,σ d (,T are deerminisic funcions of and T, and hence forward raes f(,t are Gaussian disribued. We obain he following closed form opion price formula. Proposiion Under he above Gaussian assumpion, he opion price is π = P(,SΦ[d 1 KP(,TΦ[d 2, where [ log P(,S ± 1 T σ KP(,T 2 T,S(s 2 ds d 1,2 =, T σ T,S(s 2 ds σ T,S (s is given in (9.8 and Φ is he sandard Gaussian CDF. Proof. I is enough o observe ha log P(T,T P(T,S P(,T log + 1 T σ P(,S 2 T,S(s 2 ds σ T,S(s 2 ds

17 9.3. OPTION PRICING IN GAUSSIAN HJM MODELS 113 and log P(T,S P(T,T P(,S log + 1 T σ P(,T 2 T,S(s 2 ds σ T,S(s 2 ds are sandard Gaussian disribued under Q S and Q T, respecively. Of course, he Vasicek opion price formula from Secion can now be obained as a corollary of Proposiion ( exercise.

18 114 CHAPTER 9. FORWARD MEASURES

19 Chaper 1 Forwards and Fuures B[3(Chaper 2, or Hull (22 [11 We discuss wo common ypes of erm conracs: forwards, which are mainly raded OTC, and fuures, which are acively raded on many exchanges. The underlying is in boh cases a T-claim Y, for some fixed fuure dae T. This can be an exchange rae, an ineres rae, a commodiy such as copper, any raded or non-raded asse, an index, ec. 1.1 Forward Conracs A forward conrac on Y, conraced a, wih ime of delivery T >, and wih he forward price f(;t, Y is defined by he following paymen scheme: a T, he holder of he conrac (long posiion pays f(;t, Y and receives Y from he underwrier (shor posiion; a, he forward price is chosen such ha he presen value of he forward conrac is zero, hus [ E Q e R T r(s ds (Y f(;t, Y F =. This is equivalen o 1 f(;t, Y = P(,T E Q [e R T r(s ds Y F = E Q T [Y F. 115

20 116 CHAPTER 1. FORWARDS AND FUTURES Examples The forward price a of 1. a dollar delivered a T is 1; 2. an S-bond delivered a T S is P(,S P(,T ; 3. any raded asse S delivered a T is S( P(,T. The forward price f(s;t, Y has o be disinguished from he (spo price a ime s of he forward conrac enered a ime s, which is E Q [ e R T s r(u du (Y f(;t, Y F s 1.2 Fuures Conracs = E Q [ e R T s r(u du Y F s P(,Tf(;T, Y. A fuures conrac on Y wih ime of delivery T is defined as follows: a every T, here is a marke quoed fuures price F(;T, Y, which makes he fuures conrac on Y, if enered a, equal o zero; a T, he holder of he conrac (long posiion pays F(T;T, Y and receives Y from he underwrier (shor posiion; during any ime inerval (s, he holder of he conrac receives (or pays, if negaive he amoun F(;T, Y F(s;T, Y (his is called marking o marke. So here is a coninuous cash-flow beween he wo paries of a fuures conrac. They are required o keep a cerain amoun of money as a safey margin. The volumes in which fuures are raded are huge. One of he reasons for his is ha in many markes i is difficul o rade (hedge direcly in he underlying objec. This migh be an index which includes many differen (illiquid insrumens, or a commodiy such as copper, gas or elecriciy, ec. Holding a (shor posiion in a fuures does no force you o physically deliver he underlying objec (if you exi he conrac before delivery dae, and selling shor makes i possible o hedge agains he underlying.

21 1.2. FUTURES CONTRACTS 117 Suppose Y L 1 (Q. Then he fuures price process is given by he Q-maringale F(;T, Y = E Q [Y F. (1.1 Ofen, his is jus how fuures prices are defined. We now give a heurisic argumen for (1.1 based on he above characerizaion of a fuures conrac. Firs, our model economy is driven by Brownian moion and changes in a coninuous way. Hence here is no reason o believe ha fuures prices evolve disconinuously, and we may assume ha F( = F(;T, Y is a coninuous semimaringale (or Iô process. Now suppose we ener he fuures conrac a ime < T. We face a coninuum of cashflows in he inerval (,T. Indeed, le = < < N = be a pariion of [, T. The presen value of he corresponding cashflows F( i F( i 1 a i, i = 1,...,N, is given by E Q [Σ F where Σ := N i=1 1 B( i (F( i F( i 1. Bu he fuures conrac has presen value zero, hence E Q [Σ F =. This has o hold for any pariion ( i. We can rewrie Σ as N i=1 1 B( i 1 (F( i F( i 1 + N i=1 ( 1 B( i 1 (F( i F( i 1. B( i 1 If we le he pariion become finer and finer his expression converges in probabiliy owards 1 T 1 T B(s df(s + d B,F = s 1 B(s df(s, since he quadraic variaion of 1/B (finie variaion and F (coninuous is zero. Under he appropriae inegrabiliy assumpions (uniform inegrabiliy we conclude ha [ 1 E Q B(s df(s F =,

22 118 CHAPTER 1. FORWARDS AND FUTURES and ha M( = [ 1 T B(s df(s = E Q is a Q-maringale. If, moreover hen 1 B(s df(s F, [,T, [ E Q B(s 2 d M,M s = E Q [ F,F T < F( = B(sdM(s, is a Q-maringale, which implies ( Ineres Rae Fuures [,T, Z[28(Secion 5.4 Ineres rae fuures conracs may be divided ino fuures on shor erm insrumens and fuures on coupon bonds. We only consider an example from he firs group. Eurodollars are deposis of US dollars in insiuions ouside of he US. LIBOR is he inerbank rae of ineres for Eurodollar loans. The Eurodollar fuures conrac is ied o he LIBOR. I was inroduced by he Inernaional Money Marke (IMM of he Chicago Mercanile Exchange (CME in 1981, and is designed o proec is owner from flucuaions in he 3-monhs (=1/4 years LIBOR. The mauriy (delivery monhs are March, June, Sepember and December. Fix a mauriy dae T and le L(T denoe he 3-monhs LIBOR for he period [T,T + 1/4, prevailing a T. The marke quoe of he Eurodollar fuures conrac on L(T a ime T is 1 L F (,T [1 per cen where L F (,T is he corresponding fuures rae (compare wih he example in Secion As ends o T, L F (,T ends o L(T. The fuures price, used for he marking o marke, is defined by F(;T,L(T = L F(,T [Mio. dollars.

23 1.4. FORWARD VS. FUTURES IN A GAUSSIAN SETUP 119 Consequenly, a change of 1 basis poin (.1% in he fuures rae L F (,T leads o a cashflow of = 25 [dollars. We also see ha he final price F(T;T,L(T = 1 1 L(T = Y is no 4 P(T,T + 1/4 = 1 1 L(TP(T,T + 1/4 as one migh suppose. In fac, he 4 underlying Y is a synheic value. A mauriy here is no physical delivery. Insead, selemen is made in cash. On he oher hand, since L F(,T = F(;T,L(T = E Q [F(T;T,L(T F = E Q [L(T F, we obain an explici formula for he fuures rae L F (,T = E Q [L(T F. 1.4 Forward vs. Fuures in a Gaussian Seup Le S be he price process of a raded asse. Hence he Q-dynamics of S is of he form ds( = r(d + ρ(dw(, S( for some volailiy process ρ. Fix a delivery dae T. The forward and fuures prices of S for delivery a T are f(;t,s(t = S( P(,T, F(;T,S(T = E Q[S(T F. Under Gaussian assumpion we can esablish he relaionship beween he wo prices. Proposiion Suppose ρ( and v(, T are deerminisic funcions in, where v(,t = σ(,udu

24 12 CHAPTER 1. FORWARDS AND FUTURES is he volailiy of he T-bond (see (9.7. Then ( F(;T,S(T = f(;t,s(t exp (v(s,t ρ(s v(s,tds for T. Hence, if he insananeous correlaion of ds( and dp(,t is negaive d S,P(,T d = S(P(,Tρ( v(,t hen he fuures price dominaes he forward price. Proof. Wrie µ(s := v(s, T ρ(s. I is clear ha f(;t,s(t = S( ( P(,T exp µ(sdw(s 1 2 ( exp µ(s v(s,tds, and hence f(t;t,s(t = f(;t,s(t exp ( exp µ(s v(s,tds ( µ(sdw(s 1 2 By assumpion µ(s is deerminisic. Consequenly, and as desired.. µ(s 2 ds ( E Q [exp µ(sdw(s 1 µ(s 2 ds F = 1 2 F(;T,S(T = E Q [f(t;t,s(t F ( = f(;t,s(t exp µ(s v(s,tds, µ(s 2 ds

25 1.4. FORWARD VS. FUTURES IN A GAUSSIAN SETUP 121 Similarly, one can show ( exercise Lemma In a Gaussian HJM framework (σ(, T deerminisic we have he following relaions (convexiy adjusmens beween insananeous and simple fuures and forward raes ( f(,t = E Q [r(t F σ(s,t σ(s,udu ds, s F(;T,S = E Q [F(T,S F for T < S. P(,T (S TP(,S (e R T ( R S T σ(s,v dv R S s σ(s,u duds 1 Hence, if σ(s,v σ(s,u for all s min(u,v hen fuures raes are always greaer han he corresponding forward raes.

26 122 CHAPTER 1. FORWARDS AND FUTURES