A trustable shape parameter in the kernel-based collocation method with application to pricing financial options. (1st May 2021)

• Record Type: Journal Article
• Authors: Shirzadi, Mohammad; Dehghan, Mehdi; Bastani, Ali Foroush
• Is Part Of: Engineering analysis with boundary elements. Volume 126(2021)
• Page Start: 108
• Page End: 117
• Publication Date: 2021-05-01
• Subjects: Radial basis function -- Kernel-based collocation method -- Optimal shape parameter -- High-dimensional PDEs -- Partial integro-differential equations -- Option pricing; 45K05 -- 60G51 -- 60H30 -- 65M06 -- 65M70; Boundary element methods -- Periodicals; Engineering mathematics -- Periodicals; Équations intégrales de frontière, Méthodes des -- Périodiques; Mathématiques de l'ingénieur -- Périodiques; Boundary element methods; Engineering mathematics; Periodicals; 620.00151
• DOI: 10.1016/j.enganabound.2021.02.005
• Languages: English
• ISSNs: 0955-7997

Abstract

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.