The Mathematics of Financial Modelingand Investment Management

20-Term Structure Page 628 Wednesday, February 4, 2004 1:33 PM

628 The Mathematics of Financial Modeling and Investment Management

We will now describe a mathematical technique for computing this

expectation using the Feynman-Kac formula.

To understand the reasoning behind the Feynman-Kac formula,

recall that there are two basic ways to represent stochastic processes.

The first, which was presented in Chapter 8, is a direct representation of

uncertainty pathwise through Itô processes. Itô processes can be

thought of as modifications of Brownian motions. One begins by defin-

ing Brownian motions and then defines a broad class of stochastic pro-

cesses, the Itô processes, as Itô integrals obtained from the Brownian

motion. Discretizing an Itô process, one obtains equations that describe

individual paths.

An equivalent way to represent stochastic Itô processes is through

transition probabilities. Given a process Xtthat starts at X 0 , the transi-

tion probabilities are the conditional probability densities p(Xt/X 0 ).

Given that the process is a Markov process, these densities also describe

the transition between the value of the process at time sto time t:

p(XtXs) that we write p(x,t,y,s). The Markov nature of the process

means that, given any function h(y), the expectation Es[h(XtXs)] is the

same as if the process started anew at the value Xs.

It can be demonstrated that the transition density p(x,t,y,s) obeys

the following partial differential equation (PDE) which is called the for-

ward Kolmogorov equation or the Fokker-Planck equation:

2

∂ (^1 ∂

(^2) [σ (xt, )pxtys( ,, , )] ∂μ[ (xt, )pxtys( ,, ,)]
-----pxtys)= --------------------------------------------------------------– -----------------------------------------------------
∂x
,, ,
∂t (^2) ∂x^2
with boundary conditions p(x,t,y,s) = δs(y) where δs(y) is Dirac’s delta
function.^9 The numerical solution of this equation, after discretization,
gives the required probability density.
For example, consider the Brownian motion whose stochastic differ-
ential equation is
dXt = dBt, μ= 0, σ= 1
The associated Fokker-Planck equation is the diffusion equation in one
dimension:
(^9) Strictly speaking Dirac’s delta function is not a function but a distribution. In a
loose sense, it is a function that assumes value zero in all points except one where it
becomes infinite. It is defined only through its integral which is finite.