Date of Award

Fall 2009

Document Type

Dissertation

Degree Name

Doctor of Philosophy in Mathematical Sciences - (Ph.D.)

Department

Mathematical Sciences

First Advisor

David James Horntrop

Second Advisor

William J. Morokoff

Third Advisor

Eliza Zoi-Heleni Michalopoulou

Fourth Advisor

Michael A. Ehrlich

Fifth Advisor

Roy Goodman

Abstract

Credit risk is the risk of losing contractually obligated cash flows promised by a counterparty such as a corporation, financial institution, or government due to default on its debt obligations. The need for accurate pricing and hedging of complex credit derivatives and for active management of large credit portfolios calls for an accurate assessment of the risk inherent in the underlying credit portfolios. An important challenge for modeling a credit portfolio is to capture the correlations within the credit portfolio. For very large and homogeneous portfolios, analytic and semi-analytic approaches can be used to derive limiting distributions. However, for portfolios of inhomogeneous default probabilities, default correlations, recovery values, or position sizes, Monte Carlo methods are necessary to capture their underlying dynamic evolutions. Since the feasibility of the Monte Carlo methods is limited by their relatively slow convergence rate, methods to improve the efficiency of simulations for credit portfolios are highly desired.

In this dissertation, a comparison of the commonly employed single step models for credit portfolios, referred to as the copula-based default time approach, with our novel applications of multi-step models was made at first. Comparison of simulation results indicates that the dependency structure may be better incorporated by the multi-step models, since the default time models can introduce substantially skewed correlations within credit portfolios, a shortcoming which has become more evident in the recent subprime crisis. Next, to improve the efficiency of simulations, quasi- random sequences were introduced into both the single step and multi-step models by devising several new algorithms involving the Brownian bridge construction and principal component analysis. The simulation results from tests under various scenarios suggest that quasi-Monte Carlo methods can substantially improve simulation effectiveness not only for the problems of computing integrals but also for those of order statistics, indicating significant advantage when calculating a number of risk quantities such as Value at Risk (VaR). Finally, the performance of the simulations based on the above credit portfolio models and the quasi-Monte Carlo methods was examined in the context of modeling and valuation of credit portfolio derivatives. The results suggest that these methods can considerably improve the simulation of complex financial instruments involving portfolio credit risk.

Included in

Mathematics Commons

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