Finite Differences and Standard Options

Abstract

We now enter the part of the book that is devoted to the numerical solution of equations of the Black-Scholes type. Accordingly, let us assume the scenario characterized by the Assumptions 1.2. In case of European options the function V (S, t) solves the Black-Scholes equation (1.2). It is not really our aim to solve this partial differential equation because it possesses an analytic solution (→ Appendix A3). Ultimately it will be our intention to solve more general equations and inequalities. In particular, American options will be calculated numerically. The primary goal of this chapter is not to calculate single values V (S ₀, 0) —for this purpose binomial methods are recommended— but to approximate the surfaces that are defined by V (S, t) on the half strip S > 0, 0 ≤ t ≤ T.