Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

First Passage Time for Multivariate Jump-diffusion Stochastic Models With Applications in Finance

Di Zhang, Roderick V. N. Melnik

TL;DR

This paper tackles the first passage time (FPT) problem for multivariate jump-diffusion processes in finance by developing a Monte Carlo-based framework that leverages an affine jump-diffusion model and a fast uniform-sampling method. It derives a computable multi-dimensional FPT density via interval decomposition and Brownian-bridge crossing probabilities, and uses a kernel-density estimator with an optimally chosen bandwidth to estimate the FPT distribution. The authors extend the UNIF method to multivariate settings and demonstrate substantial efficiency gains over conventional MC in a two-dimensional test, with accurate density replication and favorable CPU times. The approach enables practical analysis of default correlations and barrier-option pricing, and offers a scalable tool for broader financial applications that involve multivariate FPTs.

Abstract

The ``first passage-time'' (FPT) problem is an important problem with a wide range of applications in mathematics, physics, biology and finance. Mathematically, such a problem can be reduced to estimating the probability of a (stochastic) process first to reach a critical level or threshold. 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 develop a Monte-Carlo-based methodology for the solution of the FPT problem in the context of a multivariate jump-diffusion stochastic process. The developed methodology is tested by using different parameters, the simulation results indicate that the developed methodology is much more efficient than the conventional Monte Carlo method. It is an efficient tool for further practical applications, such as the analysis of default correlation and predicting barrier options in finance.

First Passage Time for Multivariate Jump-diffusion Stochastic Models With Applications in Finance

TL;DR

This paper tackles the first passage time (FPT) problem for multivariate jump-diffusion processes in finance by developing a Monte Carlo-based framework that leverages an affine jump-diffusion model and a fast uniform-sampling method. It derives a computable multi-dimensional FPT density via interval decomposition and Brownian-bridge crossing probabilities, and uses a kernel-density estimator with an optimally chosen bandwidth to estimate the FPT distribution. The authors extend the UNIF method to multivariate settings and demonstrate substantial efficiency gains over conventional MC in a two-dimensional test, with accurate density replication and favorable CPU times. The approach enables practical analysis of default correlations and barrier-option pricing, and offers a scalable tool for broader financial applications that involve multivariate FPTs.

Abstract

The ``first passage-time'' (FPT) problem is an important problem with a wide range of applications in mathematics, physics, biology and finance. Mathematically, such a problem can be reduced to estimating the probability of a (stochastic) process first to reach a critical level or threshold. 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 develop a Monte-Carlo-based methodology for the solution of the FPT problem in the context of a multivariate jump-diffusion stochastic process. The developed methodology is tested by using different parameters, the simulation results indicate that the developed methodology is much more efficient than the conventional Monte Carlo method. It is an efficient tool for further practical applications, such as the analysis of default correlation and predicting barrier options in finance.

Paper Structure

This paper contains 9 sections, 23 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Mathematical Model
  3. Affine jump-diffusion
  4. First passage time distribution
  5. The kernel estimator
  6. Algorithms
  7. Uniform sampling method
  8. Simulation results
  9. Conclusion

Figures (3)

  • Figure 1: Example 1 ($\lambda=1$): One-dimensional (top) and two-dimensional density function (bottom) estimate using $100,000$ iterations for UNIF and conventional Monte Carlo approaches. The discretization size of time horizon is $\Delta=0.0002$ for conventional Monte Carlo method.
  • Figure 2: Example 2 ($\lambda=3$): One-dimensional (top) and two-dimensional density function (bottom) estimate using $100,000$ iterations for UNIF and conventional Monte Carlo approaches. The discretization size of time horizon is $\Delta=0.0002$ for conventional Monte Carlo method.
  • Figure 3: Example 3 ($\lambda=8$): One-dimensional (top) and two-dimensional density function (bottom) estimate using $100,000$ iterations for UNIF and conventional Monte Carlo approaches. The discretization size of time horizon is $\Delta=0.0002$ for conventional Monte Carlo method.

Theorems & Definitions (1)

  • Definition 1