The Mathematics of Financial Modelingand Investment Management

9-DifferntEquations Page 263 Wednesday, February 4, 2004 12:51 PM

Differential Equations and Difference Equations 263

L

C^2 πk 

k = ----∫φξ()sin ------ξdξ

L 0 L 

The previous method applies to the first boundary value problem

but cannot be applied to the Cauchy problem, which admits only an ini-

tial condition. It can be demonstrated that the solution of the Cauchy

problem can be expressed in terms of a convolution with a Green’s func-

tion. In particular, it can be demonstrated that the solution of the

Cauchy problem can be written in closed form as follows:

1

∞

()  (x – ξ)

(^2) 
( , -----------exp– -------------------dξ
t ^4 t 

ft x)= -----------∫

φξ

2 π–∞

for t > 0 and f(0,x) = φ(x). It can be demonstrated that the Black-Scholes

equation (see Chapter 15), which is an equation of the form

∂f 1 2 ∂^2 f ∂f

-----+ ---σ^2 x ---------+ rx------– rf = 0

∂t (^2) ∂x^2 ∂x
can be reduced through transformation of variables to the standard dif-
fusion equation to be solved with the Green’s function approach.
Numerical Solution of PDEs
There are different methods for the numerical solution of PDEs. We
illustrate the finite difference methods which are based on approximat-
ing derivatives with finite differences. Other discretization schemes,
such as finite elements and spectral methods are possible but, being
more complex, they go beyond the scope of this book.
Finite difference methods result in a set of recursive equations when
applied to initial conditions. When finite difference methods are applied
to boundary problems, they require the solution of systems of simulta-
neous linear equations. PDEs might exhibit boundary conditions, initial
conditions or a mix of the two.
The Cauchy problem of the diffusion equation is an example of initial
conditions. The simplest discretization scheme for the diffusion equation
replaces derivatives with their difference quotients. As for ordinary differ-
ential equations, the discretization scheme can be written as follows: