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.
Highlights
It is natural to model many financial variables as Wiener processes: examples may be found in asset returns, interest rates, bond yields as well as inflation and commodity prices
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
Summary
It is natural to model many financial variables as Wiener processes: examples may be found in asset returns, interest rates, bond yields as well as inflation and commodity prices. Exotic derivatives are designed to exploit the higher order characteristics through their dependence on the evolution of variables over finite time periods. This is where the behaviour of extrema of Wiener processes becomes relevant and important. For example we may wish to model: the behaviour of the asset returns involved in an overall portfolio; the interplay between bond yields and interest rates in a fixed interest market; the dependence of asset behaviour on economic variables such as inflation, employment and wage growth. The purpose of this paper is to study the distribution of these extrema
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