A trustable shape parameter in the kernel-based collocation method with application to pricing financial options

Abstract

In this paper, we focus on the kernel-based solution of high-dimensional elliptic PDEs and propose an efficient algorithm to compute a trustable value of the shape parameter for a given positive definite kernel. The proposed strategy is based on the norm-minimal generalized interpolation problem coming from the optimality of radial basis function interpolation. Also, the effectiveness of the suggested algorithm is assessed by solving parabolic partial integro-differential equations arising in option pricing theory when the dynamic of the underlying asset is driven by a Lévy process. The performance of the proposed algorithm is evaluated by some high-dimensional PDEs on regular and irregular computational domains. Indeed, the numerical results obtained with European put options under Merton’s model show the trustability of the suggested optimal shape parameter.