Master of Science (MSc)
Faculty of Science
The first passage time (FPT) problems are ubiquitous in many applications, from physics to finance. Mathematically, such problems are often reduced to the evaluation of the probability density of the time for a process to cross a certain level, a boundary, or to enter a certain region. While in other areas of applications the FPT problems 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). The application of the conventional Monte-Carlo procedure is possible for the solution of the resulting model, but it becomes computationally inefficient which severely restricts its applicability in many practically interesting cases. In this dissertation, we are interested in the development of efficient Monte-Carlo-based computational procedures for the estimation of the probability density of the time for a random process to cross a specified threshold level. Our main application is the credit risk analysis where we focus on a case of several “coupled” companies for which we attempt to evaluate their dependent defaults. In particular, we consider a situation where individual companies are linked together via certain economic conditions, so the default events of companies are correlated. This is usually the case, for example, when the companies are in the same industry or in supply chain management problems. In this dissertation, we have successfully developed such efficient computational procedures that can be carried out for multivariate (and correlated) jump-diffusion processes. We have also provided details of the implementation of the developed Monte-Carlo-based technique for a subclass of multidimensional Levy processes with several compound Poisson shocks. Finally, we have demonstrated the applicability of the developed methodologies to the analysis of the default rates and default correlations of several different, but correlated firms via a set of empirical data.
Zhang, Di, "First passage time problem for multivariate jump-diffusion processes: Models, computation, and applications in finance" (2007). Theses and Dissertations (Comprehensive). 51.