In this dissertation, we consider the numerical challenge of pricing credit derivatives that depend on rare event probabilities. In particular, much of our attention will be devoted to Collateralized Debt Obligations (CDOs). CDOs are highly dimensional credit derivatives for which many computational challenges exist. One of the main difficulties in pricing CDOs is the high-dimensional nature of the problem. This fact usually precludes analytical solutions and so numerical techniques, such as Monte Carlo, are employed. Due to the slow converge of Monte Carlo techniques, variance reduction becomes critical and importance sampling is the main variance reduction technique used. However, the change of measure that is required to perform importance sampling can be difficult to find and to simulate.

Therefore, we are interested in the intersection of two events; a complicated model that doesn't lend itself to importance sampling, and the computation of rare probabilities under such a model. We develop the methodology to use Feynman-Kac path measures and their interacting particle system interpretation to the field of credit derivatives pricing under the first passage model. Our main result is the derivation of the asymptotic variance of the interacting particle system for a particular choice of weight function that naturally arises in a credit derivatives pricing context. We show that the variance of using interacting particle systems for this class of models is substantially less than traditional MC. In addition, we derive many different probability densities on the path to computing the variance. We also develop and show how to apply interacting particle systems to obtain Value-at-Risk estimates and compute expected shortfall. We also will present several numerical results that confirm our theoretical results.