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A Workout in Computational Finance by Michael Aichinger, Andreas Binder

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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 Sn 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 ...

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