Introduction

In this chapter we describe some techniques for obtaining analytical solutions to diffusion equations in fixed domains, where the spatial boundaries are known in advance. Free boundary problems, in which the spatial boundaries vary with time in an unknown manner, are discussed in Chapter 7. We highlight in particular one method: we discuss similarity solutions in some detail. This method can yield important information about particular problems with special initial and boundary values, and it is especially useful for determining local behaviour in space or in time. It is also useful in the context of free boundary problems, and in Chapter 7 we see an application to the local behaviour of the free boundary for an American call option near expiry. Beyond this, though, we can also use similarity techniques to derive the fundamental solution of the diffusion equation, and from this we can deduce the general solution for the initial-value problem on an infinite interval. This in turn leads immediately to the Black–Scholes formulæ for the values of European call and put options. Finally, we extend the method to some options with more general payoffs, and we discuss the risk-neutral valuation method.

Similarity Solutions

It may sometimes happen that the solution u(x, τ) of a partial differential equation, together with its initial and boundary conditions, depends only on one special combination of the two independent variables.