Credit risk modeling and credit derivatives pricing with copula functions

Abstract

In this dissertation we introduce the copula-based models for the default dependence and efficient methods for portfolio credit derivatives pricing. First of all, we propose an alternative method of finding the $\It{k}$th default time distribution in a portfolio with dependency. Analyzing order statistics of independent and identically distributed random variables, we explicitly derive probability density and cumulative distribution functions of the kth default time based on one factor copula model with three kinds of copulas. Moreover we consider the pricing of portfolio credit derivatives such as the $\It{k}$th to default swaps and $\It{m}$ out of $\It{n}$ default swaps, and derive the relation between prices of single-name CDS and all to default swap within our framework. In order to test efficiency and accuracy we compare the theoretical prediction with Monte Carlo simulation. Secondly, we introduce a new importance sampling method for pricing basket default swaps based on exchangeable Archimedean copulas and nested Gumbel copulas. We establish more realistic dependence structure than the existing copula models for credit risks in the underlying portfolio, and propose an appropriate density for importance sampling by analyzing multivariate Archimedean copulas. To justify efficiency and accuracy of our proposed algorithms, we demonstrate several numerical examples compared with the crude Monte Carlo simulation, and show that our proposed estimators produce remarkably small variance with accurately expected values in pricing basket default swaps. Finally, we propose the kth default time distributions as the semi-analytic and analytic forms based on one factor contagion model with Marshall-Olkin copulas for homogeneous underlying portfolios. In our model, the individual default intensity process are controlled by a systematic shock and an idiosyncratic shock, and also can jump influence by contagion effect. By using proposed distributions we compute prem...