A Workout in Computational Finance

3 Finite Differences and the Black-Scholes PDE

In the preceding chapter, we have outlined how to calculate an explicit solution for the price of a European call/put option as the limit of the binomial tree setup. A different method to obtain the same solution is the transformation of the Black-Scholes stochastic differential equation (SDE) into the corresponding partial differential equation (PDE) (Wilmott, 1998).

3.1 A CONTINUOUS TIME MODEL FOR EQUITY PRICES

In this section we summarize mathematical foundations required for the derivation of the Black-Scholes PDE (Hull, 2002). Readers familiar with stochastic differential equations, Wiener processes and the Itô calculus can skip this section.

Returns

Let S_n be the price of an asset at the end of trading day n. Then, we can calculate the log-return,¹

The log-return over a time period of k days is simply calculated by the sum over the respective daily log-returns,

Assuming that log-returns of disjunct time intervals of equal length are independent and identically distributed, the central limit theorem states that the log-returns (3.2) are approximately normally distributed.²

Brownian Motion

The geometric Brownian motion (Shreve, 2008) is a process that is continuous in time and produces normally distributed log-returns at each ...