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

TL;DR

This work tackles the computational challenge of first passage times (FPT) for multivariate jump-diffusion processes in finance, where firm defaults occur when asset-value processes cross thresholds. It advances a multivariate uniform sampling (MUNIF) MC method by extending the one-dimensional UNIF approach to correlated, multidimensional jump-diffusion with multiple compound Poisson shocks, leveraging Brownian-bridge crossing probabilities and correlated uniforms via the sum-of-uniforms (SOU) technique. The authors derive an empirical FPT density estimator with kernel smoothing and calibrate the model to historical default data, achieving substantial reductions in CPU time while preserving accuracy, demonstrated through joint-default correlations of two correlated firms. The method offers a scalable, efficient tool for credit risk analysis and can be extended to other jump-diffusion problems in finance, including option pricing.

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.

Figures (4)

• Figure 1: Schematic diagram of (a) the conventional Monte-Carlo and (b) the uniform sampling (UNIF) method.
• Figure 2: Density function (top) and default rate (bottom) of A-rated firm. The simulations were performed with Monte-Carlo runs $N=100,000$, for the conventional Monte-Carlo method, the discretization size of time horizon was $\Delta=0.005$.
• Figure 3: Density function (top) and default rate (bottom) of Ba-rated firm. The simulations were performed with Monte-Carlo runs $N=100,000$, for the conventional Monte-Carlo method, the discretization size of time horizon was $\Delta=0.005$.
• Figure 4: Default correlation (%) of A- and Ba-rated firms. The simulations were performed with Monte-Carlo runs $N=100,000$, for the conventional Monte-Carlo method, the discretization size of time horizon was $\Delta=0.005$.