The method of images in the pricing of barrier derivatives in three dimensions

The thesis describes the joint distributions of minima, maxima and endpoint values for a three dimensional Wiener process. In particular, the results provide the point cumulative distributions for the maxima and/or minima of the components of the process. The densities are obtained explicitly for special type of correlations by the method of images; the analysis requires a detailed study of partitions of the sphere by means of spherical triangles. The joint densities obtained can be used to obtain explicit expressions for price of options in financial mathematics. We provide closed-form expressions for the price of several barrier type derivatives with a three dimensional geometric Wiener process as underlying. These solutions are found for special correlation matrices and are given by linear combinations of three dimensional Gaussian cumulative distributions. In order to extend the results to a wider set of correlation matrices the method of random correlations is outlined.