On Meshfree Collocation to Compute the Probability of Default under a Regime-Switching Synchronous-Jump Tempered Stable L\'{e}vy Model

TL;DR

This paper develops a meshfree collocation method using radial basis functions to efficiently solve complex partial integro-differential equations in a generalized regime-switching credit risk model with tempered stable Lévy processes, enhancing computational accuracy.

Contribution

It introduces a de-singularization framework for tempered stable processes and applies a meshfree collocation method to solve the associated equations in credit risk modeling.

Findings

Method accurately computes default probabilities in complex models.

De-singularization improves numerical stability and convergence.

Numerical experiments confirm theoretical accuracy.

Abstract

In the paper [Hainaut, D. and Colwell, D.B., {\rm A structural model for credit risk with switching processes and synchronous jumps}, The European Journal of Finance 22(11) (2016): 1040-1062], the authors exploit a synchronous-jump regime-switching model to compute the default probability of a publicly traded company. Here, we first generalize the proposed L\'{e}vy model to more general setting of tempered stable processes recently introduced into the finance literature. Based on the singularity of the resulting partial integro-differential operator, we propose a general framework based on strictly positive-definite functions to de-singularize the operator. We then analyze an efficient meshfree collocation method based on radial basis functions to approximate the solution of the corresponding system of partial integro-differential equations arising from the structural credit risk model. We show that under some regularity assumptions, our proposed method naturally de-sinularizes the problem in the tempered stable case. Numerical results of applying the method on some standard examples from the literature confirms the accuracy of our theoretical results and numerical algorithm.