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Monte-Carlo Simulations of the First Passage Time for Multivariate Jump-Diffusion Processes in Financial Applications

Di Zhang, Roderick V. N. Melnik

TL;DR

The paper tackles the problem of computing first passage times (FPT) for multivariate jump-diffusion processes in finance, with a focus on correlated default risks. It develops a fast Monte Carlo framework that combines one-dimensional jump-diffusion techniques with correlated multidimensional variates generated via sum-of-uniforms and uniform sampling, exploiting Brownian-bridge dynamics between jumps. A kernel density estimator is used to recover the FPT density, and model calibration aligns simulated default rates with historical data. The approach is demonstrated on multi-firm credit-risk scenarios, producing density and default-rate estimates, as well as joint-default correlations, and is positioned as a computationally efficient tool for credit risk and option pricing in settings with jumps and inter-firm dependencies.

Abstract

Many problems in finance require the information on the first passage time (FPT) of a stochastic process. Mathematically, such problems are often reduced to the evaluation of the probability density of the time for such a process to cross a certain level, a boundary, or to enter a certain region. While in other areas of applications the FPT problem can often be solved analytically, in finance we usually have to resort to the application of numerical procedures, in particular when we deal with jump-diffusion stochastic processes (JDP). In this paper, we propose a Monte-Carlo-based methodology for the solution of the first passage time problem in the context of multivariate (and correlated) jump-diffusion processes. The developed technique provide an efficient tool for a number of applications, including credit risk and option pricing. We demonstrate its applicability to the analysis of the default rates and default correlations of several different, but correlated firms via a set of empirical data.

Monte-Carlo Simulations of the First Passage Time for Multivariate Jump-Diffusion Processes in Financial Applications

TL;DR

The paper tackles the problem of computing first passage times (FPT) for multivariate jump-diffusion processes in finance, with a focus on correlated default risks. It develops a fast Monte Carlo framework that combines one-dimensional jump-diffusion techniques with correlated multidimensional variates generated via sum-of-uniforms and uniform sampling, exploiting Brownian-bridge dynamics between jumps. A kernel density estimator is used to recover the FPT density, and model calibration aligns simulated default rates with historical data. The approach is demonstrated on multi-firm credit-risk scenarios, producing density and default-rate estimates, as well as joint-default correlations, and is positioned as a computationally efficient tool for credit risk and option pricing in settings with jumps and inter-firm dependencies.

Abstract

Many problems in finance require the information on the first passage time (FPT) of a stochastic process. Mathematically, such problems are often reduced to the evaluation of the probability density of the time for such a process to cross a certain level, a boundary, or to enter a certain region. While in other areas of applications the FPT problem can often be solved analytically, in finance we usually have to resort to the application of numerical procedures, in particular when we deal with jump-diffusion stochastic processes (JDP). In this paper, we propose a Monte-Carlo-based methodology for the solution of the first passage time problem in the context of multivariate (and correlated) jump-diffusion processes. The developed technique provide an efficient tool for a number of applications, including credit risk and option pricing. We demonstrate its applicability to the analysis of the default rates and default correlations of several different, but correlated firms via a set of empirical data.

Paper Structure

This paper contains 14 sections, 33 equations, 3 figures, 6 tables.

Table of Contents

  1. Introduction
  2. Model description
  3. Default and default correlation
  4. Multivariate jump-diffusion processes
  5. First passage time distribution of Brownian bridge
  6. The kernel estimator
  7. The methodology of solution
  8. Sum-of-uniforms method
  9. Uniform sampling method
  10. Model calibration
  11. Applications and discussion
  12. Density function and default rate
  13. Correlated default
  14. Conclusion

Figures (3)

  • Figure 1: Schematic diagram of (a) conventional Monte Carlo and (b) uniform sampling (UNIF) method.
  • Figure 2: Estimated density function for differently rated firms. All the simulations were performed with Monte Carlo runs $N=500,000$.
  • Figure 3: Historical (squares), theoretical (circles) and simulated (line) cumulative default rates for differently rated firms. All the simulations were performed with Monte Carlo runs $N=500,000$.