Chapter 8: Option Pricing Using Finite Difference Method

In this chapter, we consider the finite difference method of numerically solving the option pricing problem. We start by considering the European call option. Pricing of American options is harder due to the possibility of early exercise. We later consider the American put option to show how the problem can be tackled.

The Black-Scholes equation and boundary condition for a European call option C(S, t) are the following:

with

Here σ is the volatility parameter and r is the risk-free rate of interest. In (8.1), the first two boundary conditions are with respect to S and specify the values of C(S, t) for S = 0 and for S → ∞. The last boundary condition is with respect to t. Note that the boundary condition with respect to t is a final condition. Due to this, the Black-Scholes equation is said to be in the backward form and we are required to obtain the value of C for time t < T.