Constrained Wiener Processes and Their Financial Applications

Abstract

The extrema of Wiener processes are relevant to the pricing of so-called exotic options, which have many financial applications. The probability densities of such extrema are well known for one dimensional Wiener processes. We employ elementary methods to derive analytical expressions for the densities for multidimensional Wiener processes, with multiple extrema. These take the form of (possibly infinite) series expansions of Gaussian densities. This is undertaken using the characterization of the Wiener process by the heat equation, a well known connection in mathematical physics.