Local weak form meshless techniques based on the radial point interpolation (RPI) method and local boundary integral equation (LBIE) method to evaluate European and American options

TL;DR

This paper introduces two novel meshless methods, LBIE and LRPI, for pricing European and American options, demonstrating their stability, efficiency, and high accuracy without requiring traditional meshes.

Contribution

The paper presents the first application of local weak form meshless methods, LBIE and LRPI, to option pricing, including American options with free boundary problems, using meshless shape functions and iterative solvers.

Findings

Methods are unconditionally stable for implicit schemes.

LBIE and LRPI produce sparse, banded system matrices.

Numerical results show high accuracy and computational speed.

Abstract

For the first time in mathematical finance field, we propose the local weak form meshless methods for option pricing; especially in this paper we select and analysis two schemes of them named local boundary integral equation method (LBIE) based on moving least squares approximation (MLS) and local radial point interpolation (LRPI) based on Wu's compactly supported radial basis functions (WCS-RBFs). LBIE and LRPI schemes are the truly meshless methods, because, a traditional non-overlapping, continuous mesh is not required, either for the construction of the shape functions, or for the integration of the local sub-domains. In this work, the American option which is a free boundary problem, is reduced to a problem with fixed boundary using a Richardson extrapolation technique. Then the $\theta$-weighted scheme is employed for the time derivative. Stability analysis of the methods is analyzed and performed by the matrix method. In fact, based on an analysis carried out in the present paper, the methods are unconditionally stable for implicit Euler (\theta = 0) and Crank-Nicolson (\theta = 0.5) schemes. It should be noted that LBIE and LRPI schemes lead to banded and sparse system matrices. Therefore, we use a powerful iterative algorithm named the Bi-conjugate gradient stabilized method (BCGSTAB) to get rid of this system. Numerical experiments are presented showing that the LBIE and LRPI approaches are extremely accurate and fast.