Evaluation of Cumulative Random Shocks Generated from a Semi-markov Modulated Poisson Process and Its Application to Cdo Pricing

During the past decade, a structured financial product called “Collateralized Debt Obligation (CDO)” has been drawing much attention of researchers and practitioners, and is now traded with growing liquidity. However, the approach for CDO pricing has been rather limited in the literature, largely because it is necessary to evaluate the time dependent distribution of the underlying cumulative loss so as to find the pricing scheme satisfying the non-arbitrage condition of the derivatives market. The purpose of this paper is to fill this gap by describing the CDO model in terms of a semi-Markov modulated Poisson process. Based on the theoretical results of Huang and Sumita [6] as well as the Laguerre transform method, numerical algorithms are developed for evaluating the time dependent distribution of the cumulative loss up to time t, which in turn enables one to evaluate the price of a CDO tranche. Some numerical results are presented, demonstrating the power of the algorithms.