# Adaptive finite differences in option pricing

**Lina von Sydow
**

Division of Scientific Computing

Department of Information Technology

Uppsala University

Uppsala

### Abstract:

Pricing of multi-asset options can be accomplished through the solution of the multi-dimensional Black-Scholes equation. Numerical methods to solve this parabolic PDE often run in to the so called "curse of dimensionality", i.e. that the number of degrees of freedom grows exponentially in the number of dimensions. In this talk we present an adaptive finite difference method that mitigates this problem. By estimating the local truncation error using two coarse spatial grids we place grid-points where they are needed for accuracy reasons. This way we substantially reduce the number of grid-points compared to equidistant grids for the same accuracy in the solution. However, we are still stuck with Cartesian grids and we end by showing why Radial Basis Function generated Finite Differences (RBF-FD) promises a good solution methodology for this type of problems.