| The field of credit risk in finance concerns the analysis of default risk for firms. There are many facets to such analysis. The modeling of default probabilities, the recovery process and default correlation are the prominent examples. Nevertheless, modeling default correlation remains one of the more difficult problems which are still unanswered.;This dissertation focuses on the default correlation modeling aspect of credit risk. First, a survey of existing models, as well as their shortcomings and strengths, is presented. Second, a simple graphical model for correlation modeling is proposed. In the model, firms and their interactions are modeled through an undirected graph, where nodes represent firms and edges represent their interactions. For a static single period version, a probability distribution based on tonic models building on the underlying graph is proposed. Third, algebraic geometry techniques are employed to show that this model is well posed for default dependence: it can represent any given marginal distribution for single firms and any pairwise correlation matrix. These techniques also provide a calibration algorithm based on maximum likelihood estimation. Explicit formulas for the loss distribution are derived. Fourth, a multi-period generalization based on a discrete-time discrete-state Markov chain is proposed to introduce a temporal component and for pricing. Fifth, suitability of the model in financial applications is investigated through analysis of numerical properties of both the static and multi-period versions. Finally, advantages of the model over the standard normal copula model in terms of the tails of the loss distribution and the implied correlation smile is exhibited. Overall, a new framework to correlation modeling that rectifies certain shortcomings of existing models is presented. |
