# Solving Option Pricing Problems in High Dimensions

## Christoph Reisinger

**April 29, 2003, 1:30 p.m. T 811**

Urgent questions from computational finance frequently result in high- dimensional tasks, commonly in the shape of integration or PDE problems. The classical example is that of a basket option, where the dimensionality of the equation corresponds to the number of assets in the basket, e. g. 30 for the DAX or 500 for the S & P. Conventional PDE approaches fail due to the "curse of dimensionality", which raises the demand for new approximation and solution techniques in a high-dimensional setting. Clearly a general proceeding cannot be deemed feasible and a priory information about the solution has to be taken into account. The first question is wether the solution is really `fully high-dimensional' or can be represented - or at least approximated - by combinations of lower-dimensional functions. Another less specific yet very strong piece of information is smoothness, which can be used for approximation by sparse hierarchical subspaces or extrapolation techniques. We combine both approaches. On the example of a big Black-Scholes basket we show how the correlation structure of the assets can be utilised to approximate the full system in lower dimensions via principal component analysis. The remaining problems of dimension less than ten are solved by a new combination technique of high order, which allows for estimation of sensitivities of the solution that are important for trading purposes. Since the resulting discrete equations, which can be solved fully in parallel, are inherently anisotropic, we employ a multigrid method with semicoarsening and robust (hyper-)plane smoothing. In the case of American options, where the solution is subject to complementarity conditions, grid transfer operators have to be adjusted in a suitable way. Accompanying numerical examples will demonstrate the efficiency and approximation quality of this technique.