Publication: First passage time for multivariate jump-diffusion processes in finance and other areas of applications

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Title First passage time for multivariate jump-diffusion processes in finance and other areas of applications
Authors/Editors* D. Zhang and R. Melnik
Where published* Applied Stochastic Models in Business and Industry
How published* Journal
Year* 2008
Pages to appear
Keywords First passage time problems; Stochastic differential equations; Jump-diffusion processes; Modified Monte Carlo algorithms; Default correlations; Multiscale problems; Brownian bridge simulations
The first passage time (FPT) problem is an important problem with a wide range of applications in science, engineering, economics, and industry. Mathematically, such a problem can be reduced to estimating the probability of a stochastic process first to reach a boundary level. In most important applications in the financial industry, the FPT problem does not have an analytical solution and the development of efficient numerical methods becomes the only practical avenue of its solution. Most of our examples in this contribution are centered around the evaluation of default correlations in credit risk analysis, where we are concerned with the joint defaults of several correlated firms, the task that is reducible to a FPT problem. This task represents a great challenge for jump-diffusion processes (JDP). In this contribution, we develop further our previous fast Monte-Carlo method in the case of multivariate (and correlated) jump-diffusion processes. This generalization allows us, among other things, to evaluate the default events of several correlated assets based on a set of empirical data. The developed technique is an efficient tool for a number of financial, economic, and business applications, such as credit analysis, barrier option pricing, macroeconomic dynamics, and the evaluation of risk, as well as for a number of other areas of applications in science and engineering where the first passage time problem arises.
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