Elements of Numerical Methods for PDEs - Computational Methods for Quantitative Finance - page 11

Chapter 2
Elements of Numerical Methods for PDEs

In this chapter, we present some elements of numerical methods for partial differ-
ential equations (PDEs). The PDEs are classified into elliptic, parabolic and hy-
perbolic equations, and we indicate the corresponding type of problems that they
model. PDEs arising in option pricing problems in finance are mostly parabolic.
Occasionally, however, elliptic PDEs arise in connection with so-called “infinite
horizon problems”, and hyperbolic PDEs may appear in certain pure jump models
with dominating drift.
Therefore, we consider in particular the heat equation and show how to solve
it numerically using finite differences or finite elements. Finite difference meth-
ods (FDM) consist of finding an approximate solution on a grid by replacing the
derivatives in the differential equation by difference quotients. Finite element meth-
ods (FEM) are based instead on variational formulations of the differential equa-
tions and determine approximate solutions that are usually piecewise polynomials
on some partition of the (log) price domain. We start with recapitulating some func-
tion spaces as well as the classification of PDEs.

2.1 Function Spaces

The variational formulation and the analysis of the finite element method require
tools from functional analysis, in particular Hilbert spaces (see Appendix A). Let G
be a non-empty open subset of

d . If a function u : G → R

R is sufficiently smooth,

we denote the partial derivatives of u by

∂ | n | u(x)
∂x n 1
1

D n u(x) ∂ n 1 ∂ n d := =

x 1 ··· x d u(x),

x = (x 1 ,...,x d ) ∈ G, (2.1)

∂x n d
d

···

d
0

where n = (n 1 ,...,n d ) ∈ N is a multi-index. The order of the partial derivative is

|= i = 1 n i . For any integer n given by | n ∈ N 0 , we define

C n (G) ={ u : D n u exists and is continuous on G for

| n |≤ n } ,