The Mathematics of Financial Modelingand Investment Management

20-Term Structure Page 630 Wednesday, February 4, 2004 1:33 PM

630 The Mathematics of Financial Modeling and Investment Management

with boundary conditions F(x,T) = Ψ(x). Consider now the stochastic

differential equation

dXs= μ(Xs , t)dt+ σ(Xs, t)dBs , s∈ [t,T], Xt = x

There is a fundamental relationship between the two equations given by

the Feynman-Kac formula, which states that

Fxt( , ) = Et [Ψ(XT ) Xt = x]

The meaning of this relationship can be summarized as follows. A

PDE with the related boundary conditions F(x,T) = Ψ(x) is given. The

solution of this PDE is a function of two variables F(x,t), which assumes

the value Ψ(x) for t = T. A stochastic differential equation (SDE) is asso-

ciated to this equation. The two coefficients of the PDE are the drift and

the volatility of the SDE. The solution of the SDE starts at (x,t). For

each starting point (x,t), consider the expectation Et[Ψ(XT)]. This

expectation coincides with F(x,t).

One might wonder how it happened that a conditional expectation—

which is a random variable—has become the perfectly deterministic solu-

tion of a PDE. The answer is that F(x,t) associates the expectation of a

given function Ψ(XT) to each starting point (x,t). This relationship is

indeed deterministic while the starting point depends on the evolution

of the stochastic process which solves the SDE. It is thus easy to see why

the above is a consequence of the backward Kolmogorov equation

which associates to each starting point (x,t) the conditional probability

density of XT.

We can now make the final step and state the Feynman-Kac equa-

tion in a more general form. In fact, it can be demonstrated that, given

the following PDE:

∂Fxt( , ) 1 ∂^2 Fxt( , ) ( ,

,

∂Fxt)

--------------------+ ---σ^2 (xt, )----------------------+ μ(xt)--------------------– fxt( , )Fxt( , ) = 0

∂t (^2) ∂x^2 ∂x
with boundary conditions F(x,T) = Ψ(x) and given the stochastic equa-
tion
dXs= μ(Xs , t)dt+ σ(Xs, t)dBs , s∈ [t,T], Xt = x
the following relationship holds: