Option Pricing and the Dirichlet Problem

Laplace's equation is ubiquitous in physics: it arises in the study of many areas such as electromagnetism, gravity and fluid dynamics since it can be used to describe the behaviour of electric, gravitational and fluid potentials. If we take a body with prescribed temperature on the boundary, then the long-term equilibrium temperature in the interior also satisfies Laplace's equation with Dirichlet boundary con- ditions, that is the value of the solution on the edge is the prescribed temperature. Despite, the equation's importance in physics, it has not been im- portant so far in finance. Here, we will relate it to options' pricing. It is well-known that the Dirichlet problem for the Laplacian on a rea- sonably smooth compact domain in R n can be solved using Brownian motion. Indeed the result was found by Kakutani in 1944, (3, 4). In this note, I want to discuss how this result can be reinterpreted financially. Our objective is to increase our intuition about the problem rather than to attempt to prove new results. We will therefore work in a restricted case rather than in the most general case possible to keep technicalities minimal. For more general cases, and a more rigorous treatment not involving finance, see (5). We will also examine the time-reversed heat equation where the connections are with the Feynman-Kac theorem, (1), (2) and (6). Let be an open bounded subset of