Credit Risk in Lévy Libor Modeling: Rating Based Approach

Preface Modeling of credit risk has become a very important and rapidly expanding field of mathematical finance in the last fifteen years. Apart from a purely academic interest, the credit derivatives industry clearly needs advanced mathematical models to objectively assess and hedge this kind of risk, which was only underlined by the recent financial crisis. Although there exist several credit risk extensions of the default-free HJM framework in the literature, modeling of credit risk within the Libor market models seems to be far less studied. The emphasis in this study is therefore put on the application of time-inhomogeneous Lévy processes and semimartingales to credit risk modeling in the Libor framework. We consider modeling of the defaultable forward Libor rates related to defaultable bonds with credit ratings and develop the rating based Lévy Libor model with a time-inhomogeneous Lévy process as a driving process. Furthermore , we study credit portfolio risk in the Libor market models and introduce a market model for CDOs driven by time-inhomogeneous Lévy processes. The choice of time-inhomogeneous Lévy processes as driving processes is quite natural having in mind that Lévy processes, and more recently time-inhomogeneous Lévy processes, have already been proven to be an excellent tool for modeling price processes and term structures of interest rates in mathematical finance. The thesis is divided into five chapters. Chapter 1 provides the necessary mathematical background on semimartingales and Lévy processes. In particular, we study exponential semimartingales since they are used for modeling of interest rates throughout the thesis. Moreover, we present an overview of two different risk-free interest rate models: the Lévy forward rate model and the Lévy Libor market model. In Chapter 2 we study conditional Markov chains and their applications to credit risk modeling. We present several results concerning conditional Markov chains under forward Libor measures. In Chapter 3 we develop a Libor model for defaultable bonds with credit ratings. We call this model the rating based Lévy Libor model since it is driven by a time-inhomogeneous Lévy process. The model introduces the concept of defaultable forward Libor rates with credit migration, and therefore can be seen as a generalization of the default-free Libor market models found in the literature. Moreover, it extends the Lévy Libor model with default risk to the multiple rating case with credit migration. The correlations of the rating-dependent Libor rates are also calculated and pricing of credit derivatives in this framework …