Pricing Derivatives under Multiple Stochastic Factors by Localized Radial Basis Function Methods

Abstract

We propose two localized Radial Basis Function (RBF) methods, the Radial Basis Function Partition of Unity method (RBF-PUM) and the Radial Basis Function generated Finite Differences method (RBF-FD), for solving financial derivative pricing problems arising from market models with multiple stochastic factors. We demonstrate the useful features of the proposed methods, such as high accuracy, sparsity of the differentiation matrices, mesh-free nature and multi-dimensional extendability, and show how to apply these methods for solving time-dependent higher-dimensional PDEs in finance. We test these methods on several problems that incorporate stochastic asset, volatility, and interest rate dynamics by conducting numerical experiments. The results illustrate the capability of both methods to solve the problems to a sufficient accuracy within reasonable time. Both methods exhibit similar orders of convergence, which can be further improved by a more elaborate choice of the method parameters. Finally, we discuss the parallelization potentials of the proposed methods and report the speedup on the example of RBF-FD.

Pricing foreign exchange options under stochastic volatility and interest rates using an RBF–FD method

“…As suggested in [35], an alternative way to impose (4.7)-(4.9) in order to save computational time is as follows. One can consider the original PDE for the collocation nodes located at the boundaries, and discretize it by using the Gaussian RBF-FD approach.…”