A novel local meshless scheme based on the radial basis function for pricing multi-asset options

A novel local meshless scheme based on the radial basis function (RBF) is introduced in this article for price multi-asset options of even European and American types based on the Black-Scholes model. The proposed approach is obtained by using operator splitting and repeating the schemes of Richardson extrapolation in the time direction and coupling the RBF technology with a finite-difference (FD) method that leads to extremely sparse matrices in the spatial direction. Therefore, it is free of the ill-conditioned difficulties that are typical of the standard RBF approximation. We have used a strong iterative idea named the stabilized Bi-conjugate gradient process (BiCGSTAB) to solve highly sparse systems raised by the new approach. Moreover, based on a review performed in the current study, the presented scheme is unconditionally stable in the case of independent assets when spatial discretization nodes are equispaced. As seen in numerical experiments, it has a low computational cost and generates higher accuracy. Finally, the proposed local RBF scheme is very versatile so that it can be used easily for solving numerous models and obstacles not just in the finance sector, as well as in other fields of engineering and science.

Keywords:

Pseudo-differential operators , Separation-Preserving operators , Adjoints