Solving Stochastic Differential Equations with Jump-Diffusion Efficiently: Applications to FPT Problems in Credit Risk

TL;DR

This paper develops efficient Monte Carlo methods for solving first passage time problems in jump-diffusion processes, with applications to credit risk default analysis, overcoming computational challenges of traditional approaches.

Contribution

It introduces novel Monte Carlo algorithms tailored for multivariate jump-diffusion processes, improving computational efficiency in FPT problems for credit risk applications.

Findings

Effective Monte Carlo procedures for multivariate jump-diffusion FPT problems.

Application to default rate and correlation analysis in credit risk.

Demonstrated computational efficiency over conventional methods.

Abstract

The first passage time (FPT) problem is ubiquitous in many applications. In finance, we often have to deal with stochastic processes with jump-diffusion, so that the FTP problem is reducible to a stochastic differential equation with jump-diffusion. While the application of the conventional Monte-Carlo procedure is possible for the solution of the resulting model, it becomes computationally inefficient which severely restricts its applicability in many practically interesting cases. In this contribution, we focus on the development of efficient Monte-Carlo-based computational procedures for solving the FPT problem under the multivariate (and correlated) jump-diffusion processes. We also discuss the implementation of the developed Monte-Carlo-based technique for multivariate jump-diffusion processes driving by several compound Poisson shocks. Finally, we demonstrate the application of the developed methodologies for analyzing the default rates and default correlations of differently rated firms via historical data.