ON PARTIAL DIFFERENTIAL EQUATIONS RELATED TO ... - CiteSeerX

ON PARTIAL DIFFERENTIAL EQUATIONS RELATED TO TERM STRUCTURE MODELS B. Goldys

ph.: (612)3853082, e-mail: [email protected] and

M. Musiela

ph.:(612)3852959, e-mail: [email protected] School of Mathematics, The University of New South Wales Sydney 2052, Australia FAX: (612)3851071

Abstract. Recently a reparametrized version of HJM model has been proposed which leads

naturally to the in nite dimensional Markov process of forward curves. In this paper we discuss some consequences of the Markovian structure of forward rate dynamics. In particular, we obtain price of the swaption as a solution to the in nite dimensional "Black-Scholes" partial dierential equation.

1

0. Introduction The classical Black-Scholes formula for a European option price has been derived by solving a partial dierential equation identi ed by means of heuristic arguments (cf. [BS]). Later on a probabilistic interpretation of the above arguments allowed to make the derivation rigorous [HP]. Let us recall brie y the main ideas of this approach. Assume that the price X (t) of a stock is a positive continuous semimartingale such that the logarithm of the stock price has a deterministic quadratic variation hlog X it = 2 t: Then some mild technical conditions imply existence of a unique probability measure under which for every t  0

X (t) = X0 +

Zt 0

rX (s) ds +

Zt 0

X (s) dW (s):

Moreover, for a given maturity T and a strike price K we can calculate the price of a European put option by taking the conditional expectation of the discounted option payo, i.e.,

?

VT (t; x) = e?r(T ?t)E (K ? X (T ))+jX (t) = x




for t  T . Since X is a strong Feller process with the in nitesimal generator

@ + 1 2 x2 @ 2 L = rx @x 2 @x2 we can apply the Feynman-Kac formula and identify the function VT with a unique solution of the Backward Kolmogorov Equation

@u (t; x) + 1 2 x2 @ 2 u (t; x) + rx @u (t; x) ? ru(t; x) = 0 @t 2 @x2 @x

(0:1)

with the terminal condition u(T; x) = (K ? x)+ . The aim of this paper is to investigate if this strategy can be applied to interest rate options in general term structure models. The common feature of these models is that taking the Heath, Jarrow and Morton model [HJM] as a starting point they naturally lead to in nite dimensional Markov processes which describe the arbitrage free dynamics of forward rates. By a forward rate r(t; x) we mean the continuously compounded forward rate prevailing at time t over the time interval [t+x; t+x+dx]. In all these models the time evolution of forward curves r(t;
) is completely determined by the initial curve and the volatility structure. The question how to determine the volatility structure is a delicate one and dierent approaches can be chosen to address this problem; for possible answers see [M], [BM], [GMS] or [BGM]. In this paper, however, we assume that the volatility structure f(t; x) : t  0; x  0)g is a known vector-valued stochastic process. In that 2

case the forward rate process fr(t; x) : t  0; x  0g must satisfy the following stochastic partial dierential equation

@ dr(t; x) = @x





r(t; x) + 12 j(t; x)j2 dt + (t; x)dW (t)



(0:2)

for all t; x  0, where W is a d-dimensional Brownian Motion. It has been shown in [M] that (0.2) is sucient for the non-arbitrage condition. Let BT (t) denote the price of a zero coupon bond with maturity T . Then

BT (t) = exp ?

Z T ?t 0

!

r(t; u) du :

(0:3)

Consider a European swaption, an option with maturity T on a swap with the cash ows Ci ; i = 1; : : : ; n at times T < T1 < : : : < Tn . In Section 1 we will show that, under some technical conditions the process fr(t;
) : t  0g of forward curves, given by equation (0.2), is a strong Markov and Feller process in the state space H = L2 (0; 1). We will also identify the form of its generator L. Because the time t price of the swaption is given by the formula

0 R T VT (t; ) = E @e? t r s;

( 0)

ds

K?

n X i=1

1 ! CiBTi (T ) r(t; x) = (x); x  0A ; +

we can expect that in analogy with the nite dimensional case (0.1) the Feynman-Kac formula should lead to a parabolic dierential equation for VT (
;
) of the form

@u (t; ) + Lu(t; ) ? (0)u(t; ) = 0: @t

3

1. The Markov process of forward rates

We start with a more rigorous de nition of a solution to equation (0.2). To this end we shall exploit an approach presented in [DaZ]. We are going to work with the state space H = L2 (0; 1) equipped with the inner product

hf; gi =

Z1 0

f (x)g(x) dx:

In this space we consider the operator A = @x@ with the domain H 1 (0; 1) endowed with the graph norm kk21 = kk2 + kAk2 . It is well known that A is a generator of the strongly continuous semigroup of left shifts on H , i.e., for every t; x  0 S (t)f (x) = f (x + t): Let us notice that the choice of L2 (0; 1) as the state space of the process of forward curves is not crucial. This space is relatively easy to work with, however, it is not convenient for certain choices of the volatility . Let ( ; F ; (Ft ); P) be a probability space with the ltration satisfying usual conditions and let fW (t) : t  0g be an Rd -valued Brownian Motion de ned on it. The norm of a vector u 2 Rd will be denoted by juj. We are going to consider the following stochastic dierential equation on H: dr(t) = [Ar(t) + 21 Aj(t; r(t))j2 ] + A(t; r(t))dW (t) (1:1)

with the initial condition r(0) =  2 H . In order to simplify formulae we introduce the following notation  (t; ) = A(t; ) and G (t; ) = 21 Aj(t; )j2 :

For every t  0 and  2 H the mapping  (t; ) : Rd ! H is linear. Equivalently, there exist 1 (t; ); : : : ; d (t; ) 2 H such that for every u = (ui )di=1 2 Rd

 (t; )u =

d X i=1

ui i (t; ):

We require that the functions  and G satisfy the following conditions. (A1) The mapping G : R+  H ! H is measurable and kG (t; ) ? G (t; )k  C k ? k: (A2) The mapping  : R+  H ! L(Rd; H ) is measurable and k (t; ) ?  (t; )k  C k ? k: An H -valued predictable process r is said to be a solution to (1.1) if for every t  0 the following integral equation is satis ed. Zt Zt 1 2 r(t) = S (t) + 2 AS (t ? s)j(s; r(s))j ds + S (t ? s) (s; r(s)) dW (s): (1:2) 0 0 The proposition below is an immediate consequence of the general theory developed in [DaZ]. 4

Proposition 1.1. If (A1) and (A2) hold then for every H -valued random variable  independent of the Wiener process W there exists a unique continuous solution r(t;  ) to (1.1) which is a strong Markov process on H . Moreover, if E k kp < 1 for a certain p  2 then for any T > 0 sup E kr(t;  )kp  CT;p (1 + E k kp ) : tT

If  and G are Lipschitz transformations on H 1 (0; 1) and  2 H 1 (0; 1) then r(t) 2 H 1 (0; 1) for all t  0. Remark 1.2. In some cases a more general version of equation (1.1) with more singular coecients  and G has to be considered. For the discussion of such models see [GMS] or [BGM]. In general the solution to (1.1) is not a semimartingale but for every 2 dom (A ) = H01 (0; 1).

hr(t); i = h; i + +

Zt 0

hG (s; r(s)); i ds +

Zt 0

Zt 0

hr(s); A i ds

h  (s; r(s)) ; dW (s)i

(1:3)

and hence hr(t); i is a semimartingale and so is the multidimensional process (hr(t); 1 i; : : : ; hr(t); n i) for any n and arbitrary collection of 1 ; : : : ; n 2 dom (A ). It follows that the process r is an L2 ([0; T ]  ;  P )-limit of semimartingales for every T > 0. This property will be used later on in the discussion of the Kolmogorov equation. We shall also make use of the fact that if there exists a continuous version of r then it can be de ned for every x  0 as a unique solution to the equation

r(t; x) = (t + x) +

Zt @ Zt  (s; r(s; x + t ? s)) dW (s): j  ( s; r ( s; x + t ? s )) j ds + @x 2

0

0

The following property of the process

R(t; ) =

Zt 0

r(s; ) ds

will be used in Section 2. Lemma 1.3. For every T > 0 there exists cT > 0 such that sup E k(R(t; ) ? R(t; )k1  cT k ? k: tT

Proof The standard proof of this lemma is ommited.

5

2. Kolmogorov equation

In this section we assume for simplicity that the process r is time homogenous,i.e., (t; ) =(). R R In view of Lemma 1.3 we use a simpler notation 0t (r(s; )) ds instead of  0t r(s; ) ds . We will need additional assumption.

(A3) There exists m  0 such that for every t > 0 and a > 0



 Zt

sup E jr(t; )j2m exp ?2

kka

0

(r(s; )) ds



< 1:

(2:1)

Remark 2.1. In view of Proposition 1.1 Assumption (A3) is always satis ed if r is nonnegative.

In section 2 we shall show that (A3) holds in the Gaussian case as well. For m  0 we de ne the space Cm (H ) of functions F : H ! R such that the function

F () 1 + kkm is bounded and uniformly continuous on H . The space Cm (H ) is endowed with the norm

 jF ()j  kF km = sup 1 + kkm : 2H

Let Cmn (H ) denote the subspace of Cm (H ) containing all functions F which are n times Frechet dierentiable on H and

Dk F ()

! sup < 1; k = 1; : : : ; n; 2H 1 + kkm where Dk F () denotes the k-th Frechet derivative of F . If F 2 Cm1 (H ) then the derivative DF () of F at  2 H can be identi ed with an element of H and therefore DF : H ! H is a continuous mapping. If F 2 Cm2 (H ) then the second derivative D2 F () is a symmetric linear operator in H and we can de ne a continuous mapping D2 F : H ! L(H ) from H into the algebra of bounded operators on H . Finally by C n (H ) we denote the vector space of all real-valued continuous functions on H with n continuous Frechet derivatives. Let 1 ; : : : ; n 2 dom (A ) and let Pn denote the orthogonal projection on the linear span Hn of the vectors 1 ; : : : ; n . Moreover, let

D0 = fF 2 C (H ) : F = f  Pn ; f 2 Cm2 (Rn ); n = 1; : : :g: Note that every F 2 Cm (H ) is a pointwise limit of elements of D0 . In view of Proposition 1.1. we can introduce the transition semigroup

Pt F () = EF (r(t; )) 6

of the Markov process r. Clearly Pt F (
) is well de ned for any F : H ! R which is bounded and measurable. Let us go back now to the problem of pricing interest rate dependent options. To begin with note that in the present terminology the formula (0.3) for the price of zero coupon can be rewritten as follows. Let BT (t; ) = e?hS(t)IT ;i; with IT denoting the indicator function of the interval [0; T ]. Then BT (t) = BT (t; r(t)): In general, any measurable mapping F : H ! R such that E jF (r(T ))j < 1 de nes a contingent claim F (r(T )) with maturity T . Note that if the process r is not nonnegative then typical contingent claims correspond to functions F which are unbounded on H . Due to the Markov property of the process r the time t ( T ) price of the claim is

! VT (t) = E exp ? r(u; 0) du F (r(T )) Ft t ! ! ZT = E exp ? r(u; 0) du F (r(T )) r(t) : t !

ZT

In view of (A3) the above equation can be rewritten in the form

VT (t; ) = E exp ?

ZT t

! (r(u)) du F (r(T )) r(t) =  ; !

(2:2)

where  denotes the Dirac measure at zero. The transformation F ! VT is closely related to the following "Feynman-Kac semigroup"

  Zt

Pt F () = E exp

?

0



(r(u; )) du F (r(t; ))



by a simple equation VT (t; ) = PT ?t F (). Clearly P0 F = F and by the Markov property Pt+s = Pt Ps . It becomes obvious than in analogy with the nite dimensional case the problem of pricing interest rate dependent options is equivalent to the problem of calculating the semigroup Pt for a suciently rich class of initial conditions F .

Proposition 2.2. If (A1)-(A3) hold then Pt (Cm (H ))  C (H ) for every t  0. Proof Let F 2 Cm (H ). Then F () = (1 + kkm ) G() with G 2 C (H ) and 0

P  F ()  kGk E (1 + kr(t; )km) exp ? Z t (r(u; ) )du : t 0

0

7

Hence in view of (A3) Pt F () is well de ned. Moreover (A3) yields that the family of random variables     Zt m kr(t; )k exp ? (r(u; ) )du : kk  a 0

is uniformly integrable. Hence the proposition follows from the continuity of F and Lemma 1.2. We shall identify now the in nitesimal generator L of the Markov process r. Because the process r is not a semimartingale we can not apply the Ito formula to the function F (r(t; )) even if F 2 Cb2 (H ). However, it turns out that the property (1.3) is sucient for our needs. If F 2 D0 then in view of (1.3) the process F (r(t; )) is a semimartingale and

F (r(t; )) = F () + where

Zt 0

LF (r(s; )) ds +



Zt 0

DF (r(s; )) (r(s; ))dW (s);

(2:3)



LF () = 12 D2 F () ();  () + hA + G (); DF ()i

for  2 dom (A). If F 2 D0 then the function A DF () is well-de ned for all  2 H and therefore LF () is a well-de ned continuous function on H . Above considerations show that the generator of the Markov process r coincides on D0 with the operator L. Therefore we can expect that VT as de ned in (2.2) is a Feynman-Kac formula for the solution of the following equation

 @u

@t (t; ) + Lu(t; ) ? ()u(t; ) = 0;

(2:4)

u(T; ) = F ():

In other words the operator L = L ?  when considered on an appropriate domain is a generator of the semigroup Pt . However, equation (2.4) is not valid in general because VT (t;
) need not be dierentiable. Proposition 2.3. Assume that  and G are twice dierentiable on H and of linear growth, i.e.,

k ()k + kG ()k  C (1 + kk): Then for every F 2 Cm2 (H ) the function VT is a unique solution of the Backward Kolmogorov equation (2.4) in the following sense. i) The function VT : R+  H ! R is bounded and continuous with respect to each variable. ii) For every t  0 we have VT (t;
) 2 C 2 (H ). iii) VT 2 C 1 ([0; T ]; H 1 ). iv) Equation (2.4) holds for every  2 dom (A) and t  0. Moreover, VT is given by (2.2). Proof Let n denote a sequence of C 2 functions on R such that hn ; i ! () for every continuous  and let Ln = L ? n . If we denote by Ptn the semigroup

  Zt

Ptn F () = E exp



? hn; r(u; )i du F (r(t; )) 0

8



then by a simple modi cation of the proof of Theorem 9.17 in [DaZ] we can show putting un (t; ) = Ptn F () that  @un (t; ) = Lun (t; ) ? hn ; iun (t; ); @t (2:5) n u (0; ) = F () and moreover un is a unique solution of (2.5). We shall show rst that for every  2 H n  nlim !1 Pt F () = Pt F ():

Indeed,

 kF kmE



(2:6)

jPtnF () ? Pt F ()j

 Zt     Zt ? hn; r(u; )i du ? exp ? (r(u; )) du

(1 + jr(t; )jm ) exp

0

0

and therefore (A3) and the de nition of n yield (2.6). Using (2.6) and Theorem 9.16 in [DaZ] we obtain easily that the right-hand side of (2.5) converges (along the subsequence nk ) to the expression LPt F () ? ()Pt F () for every  2 H 1 uniformly in t  T . Hence nk



@u (t; ) = @Pt () lim k!1 @t @t and therefore Pt F satis es (2.4). Unfortunately, this theorem has too strong assumptions to be applicable to some important contingent claims like swaptions. Stronger results can be obtained in Gaussian case.

Proposition 2.4. The mapping u is a solution of (2.4) if and only if u(t; ) = BT (t; )RT (t; ), RT (T; ) = F () and

@RT (t; ) + 1 D2 R (t; ) ();  () + hDR (t; ); A + G ()i T T  @t 2

? hDRT (t; );  ()i h (); S (t)IT i = 0;

(2:7)

where the solution is de ned in the sense of Theorem 2. Proof Let u satisfy (2.4) and de ne the function RT by the formula u(t; ) = BT (t; )RT (t; ). Then RT is smooth and

@u (t; ) = (T ? t)B (t; )R (t; ) + B (t; ) @RT (t; ); T T T @t @t

(2:8)

Du(t; ) = ?BT (t; )RT (t; )S (t)IT + BT (t; )DRT (t; );

(2:9)

9

D2u(t; ) = BT (t; )RT (t; ) (S (t)IT ) (S (t)IT ) ? 2BT (t; )DRT (t; ) S (t)IT +BT (t; )D2 RT (t; ): Hence by (2.9)

(2:10)

hDu(t; ); A + G ()i



= ?BT (t; )RT (t; ) (T ? t) ? (0) + 21 hS (t)IT ;  ()i2



(2:11)

and by (2.10)

D u(t; ) ();  () = B (t; )R (t; ) hS (t)I ;  ()i ?2B (t; ) hDR (t; );  ()i hS (t)I ;  ()i T T T T T T

 +B (t; ) D R (t; ) ();  () : (2:12) 2

2

T

2

T

Finally, taking into account (2.8), (2.11) and (2.12) we nd that

@u (t; ) + 1 D2 u(t; ) ();  () + hDu(t; ); A + G ()i ? ()u(t; )  @t 2 = BT (t; )

 @R

T (t; ) + 1 D2 R (t; ) ();  () + hDR (t; ); A + G ()i T T  @t 2

? hDRT (t; );  ();  ()i hS (t)IT ;  ()i) and (2.7) follows. Using similar arguments we show that if RT satis es (2.7) then u(t; ) = BT (t; )RT (t; ) is a solution to (2.4). Remark 2.5. Proposition 2.4 describes the forward measure transformation performed at the level of the Kolmogorov equation. Note that equation (2.7) is the Kolmogorov equation for the process Y (say) de ned as a solution to the stochastic dierential equation

dY = (AY + G (Y ) ? h (Y ); S (t)IT i  (Y )) dt +  (Y )dW or in a more explicit form

 @Y

dY (t; x) = @x (t; x) +  (Y (t))(x)

? (Y (t))(x)

Z T ?t 0

Zx 0



 (Y (t))(u) du dt

 (Y (t))(u) dudt +  (Y (t))(x)dW (t):

10

3. Gaussian case

In this section we assume that the function  (and hence G ) is constant. This case has been discussed in [M] and [BM]. Note that now

r(t) = S (t) +

Zt

S (s)G ds +

0

Zt 0

S (t ? s) dW (s):

Hence for every t  0 the random variable r(t) is Gaussian with the mean

Er(t) = S (t) + and the covariance operator

Qt =

Zt 0

Zt 0

S (s)G ds

S (s)  S  (s) ds

Note that conditions (A1), (A2) are trivially satis ed. Moreover, because r(t; ) is Gaussian so is R(t; )(0). Hence using the Holder inequality and Proposition 1.1 we check by direct calculations that for t  T ?
E jr(t; )j2m exp (?2R(t; )(0))  CT kk2m exp ( T kk) for some constants CT ; T > 0. Therefore (A3) holds as well. We shall need the nite dimensional parabolic PDE n @h (t; x ; : : : ; x ) + 1 X @ 2 h (t; x ; : : : ; x ) = 0  b ( t ) b ( t ) x x n 1 n @t 1 2 i;j =1 i j i j @xi @xj

(3:1)

with the terminal condition h (T; x1 ; : : : ; xn ) = h0 (x1 ; : : : ; xn ) and

bi (t)bj (t) =

Z Ti ?t T ?t

  (x) dx

Z Tj ?t T ?t

 (x) dx:

It is well known that equation (3.1) has a unique solution for every bounded and measurable terminal condition h0 . Let FT;Ti (t; ) = exp (? hS (t)IT;Ti ; i) ; where IT;Ti is an indicator function of the interval [T; Ti ].

Theorem 3.1. If the function U (t; x ; : : : ; xn ) is a solution to (3.1) with the terminal condition U0 (x1 ; : : : ; xn ) then the function

1

u(t; ) = BT (t; )U (t; FT;T1 (t; ); : : : ; FT;Tn (t; )) 11

is a solution to the Cauchy problem (2.4) with the terminal condition

u(T; ) = U0 (BT1 (T; ); : : : ; BTn (T; )) Proof It is enough to consider the case n = d = 1. The general argument is exactly the same. In view of Proposition 2.4 we need to show that the function

R(t; ) = U (t; FT;T1 (t; ); : : : ; FT;Tn (t; ))

(3:2)

is a solution to equation (2.7). Note rst that

dFT;T1 (t; ) = ( (T ? t) ?  (T ? t)) F (t; ); 1 T;T1 dt DFT;T1 (t; ) = ?FT;T1 (t; )lt with lt = I[T ?t;T1 ?t] and

D2FT;T1 (t; ) = FT;T1 (t; )lt lt :

Hence denoting l = I[0;T ?t] we nd that for  2 dom (A)

@R (t; ) = @U (t; F (t; )) T;T1 @t @t

and Hence and

+FT;T1 (t; )((T1 ? t) ? (T ? t)) @U @x (t; FT;T1 (t; ))

(3:3)

DR(t; ) = ?FT;T1 (t; ) @U @x (t; FT;T1 (t; )) lt:

hDR(t; );  i = ?FT;T1 (t; ) @U @x (t; FT;T1 (t; )) hlt;  i hDR(t; ); A + G i = ?FT;T1 (t; ) @U @x (t; FT;T1 (t; )) hlt; A + G i = ?FT;T1 (t; ) @U @x (t; FT;T1 (t; )) ( (T1 ? t) ?  (T ? t))

Z



Z T1 ?t 1 d x 2 @U ?FT;T1 (t; ) @x (t; FT;T1 (t; )) dx  ( u ) du T ?t 2 dx 0 = ?FT;T1 (t; ) @U @x (t; FT;T1 (t; )) ( (T1 ? t) ?  (T ? t)) 12

(3:4)

0 Z ! Z T ?t !1 T ? t @ ? 12 FT;T (t; ) @U  (u) du ?  (u) du A : @x (t; FT;T (t; )) 1

1

Thereby

1

2

2

0

0

hDR(t; ); A + G i = ?FT;T1 (t; ) @U @x (t; FT;T1 (t; )) ( (T1 ? t) ?  (T ? t))

@U 2 ? 12 FT;T1 (t; ) @U @x (t; FT;T1 (t; )) h; li ? FT;T1 (t; ) @x (t; FT;T1 (t; )) h; li h; lt i : Next

(3:5)

D2 R(t; ) = FT;T1 (t; ) @U @x (t; FT;T1 (t; )) lt lt @ U (t; F (t; )) l l : 2 +FT;T T t t 1 (t; ) @x2 2

Hence

D R(t; );   = 2

2 FT;T1 (t; ) @U @x (t; FT;T1 (t; )) hlt ;  i

@ U (t; F (t; )) hl ;  i2 : 2 +FT;T T;T1 t 1 (t; ) @x2 2

(3:6)

Now, taking into account (3.3), (3.4), (3.5) and (3.6) we nd that

@R (t; ) + 1 D2 R (t; ) ();  () + hDR (t; ); A + G ()i T T  @t 2

? hDRT (t; );  ()i h (); S (t)IT i 1 F 2 (t; ) @ 2 U (t; F (t; )) hl ;  i2 ; ( t; F ( t;  )) + = @U T;T1 T;T1 t @t 2 T;T1 @x2 where R(t; ) is de ned by (3.2). Therefore by (3.1) the function R satis es equation (2.7) and the theorem follows.

13

4. Console bond and elliptic equation

In this section we assume the conditions of Section 2. A bond which pays a coupon of $1 per unit of time forever, and never repays the principal is called a "console". The price of the console and its yield are the reciprocal of one another. The price at time t, say C (t), is given by

Z1

C (t) = If we de ne the function F by

F () =

Z1 0

0

Bt+x (t) dx:

 Zx

exp ?

0



(u)du dx

(4:1)

then

C (t) = F (r(t)): It has been shown in [M] that EC (t) < 1 provided C (0) < 1 and

Z1 0

 (u) du < 1:

(4:2)

The mapping F is not well de ned for all bounded continuous functions on H . Let dom (F ) = f 2 Cb (H ) : F () < 1g :

Proposition 4.1. Let (A1)-(A3) hold. Then for every  2 dom (F ) the function F de ned by (4.1) is a solution to the equation

(L ? )F = ?1;

where 1() = 1 for all . In other words

F =?

Z1 0

for  2 dom (F ). Proof Note that for 2 dom (DF ())

hDF (); i =

Z 1 Z x 0

Therefore

hDF (); A + G ()i =

?

0

Pt 1 dt



 Zx

(u) du exp ?

Z 1 Z x 0

(4:3)

?

0

14

A ?

0

Zx  0



(u) du dx:

 Zx 

G exp ?

0

 dx

=?

Z1 0

 Zx 

((x) ? (0)) exp ?

Z1 1  dx ? 2 0

0

and

hD F ()G ; G i = 2

Z 1 Z x  0

0

 Zx 

Z x 

2

0

G exp ?

0

 dx

 Zx 

2

G exp ?

0

 dx:

(4:4) (4:5)

In view of (4.2) all terms in the above equations are well de ned. Finally, taking together (4.4) and (4.5) we obtain (L ? )F () = provided

Z 1 0

 Z x 

?(x) exp ?

0

Z1 0



dx =

Z 1 d   Z x  0

dx exp ?

0



dx = ?1

(u) du = 1

and (4.3) follows.

References

[BS] Black F. and Scholes M.: The pricing of options and corporate liabilities, J. Political Economy 81 (1973), 637-659 [BM] Brace A. and Musiela M.: A multifactor Gauss-Markov implementation of Heath, Jarrow and Morton, Mathematical Finance 2 (1994), 259-283 [BGM] Brace A., Gatarek D. and Musiela M.: The market model of interest rate dynamics, Report No.S95-2, School of Mathematics, The University of New South Wales, 1995 [DaZ] Da Prato, G. and Zabczyk, J.: Stochastic equations in in nite dimensions, Cambridge University Press 1992. [GMS] Goldys B., Musiela M. and Sondermann D.: Lognormality of rates and term strucutre models (1994), to appear in Mathematical Finance [HP] Harrison J.M. and Pliska S.R.: Martingales and stochastic integrals in the theory of continuous trading, Stochastic Process. Appl. 11 (1981), 215-260 [HJM] Heath, D. Jarrow, R. and Morton, A.: Bond pricing and the term structure of interest rates: a new methodology. Econometrica 61(1) (1992), 77- 105. [M] Musiela M.: Stochastic PDEs and term structure models , (1994), submitted

15