Standard Methods for Standard Options

We now enter the part of the book that is devoted to the numerical solution of equations of the Black–Scholes type. Here we discuss “standard” options in the sense as introduced in Section 1.1 and 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 A4). Ultimately our intention is to solve more general equations and inequalities. In particular, American options will be calculated numerically. The goal is not only to calculate single values V (S0, 0) —for this purpose binomial methods can be applied— but also to approximate the curve V (S, 0), or even the surface defined by V (S, t) on the half strip S > 0, 0 = t = T. Thereby we collect information on early exercise, and on delta hedging by observing the derivative ?V ?S.

American options obey inequalities of the type of the Black–Scholes equation (1.2). To allow for early exercise, the Assumptions 1.2 must be weakened. As a further generalization, the payment of dividends must be taken into account because otherwise early exercise does not make sense for American calls.