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URN: urn:nbn:de:bsz:25-opus-72539
URL: http://www.freidok.uni-freiburg.de/volltexte/7253/

Grbac, Zorana

Credit risk in Lévy Libor Modeling: rating based approach

Kreditrisiken dargestellt durch Lévy Libor Modelle: Ratingbasierter Ansatz

Kurzfassung in Englisch

Modeling of credit risk has become a very important and rapidly expanding field of mathematical finance in the last fifteen years. In this thesis we study credit risk modeling in the Libor framework and develop the rating based Lévy Libor model and a credit portfolio market model, based on time-inhomogeneous Lévy processes and general semimartingales as driving processes.

In the first part of the thesis, 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. It can be seen as a generalization of the default-free Libor market models found in the literature, and moreover it extends the Lévy Libor model with default risk to multiple credit ratings with migration. Conditional Markov processes are used to model this migration and we provide a detailed analysis of their properties and behavior under forward Libor measures. Furthermore, we calculate the correlations of the rating-dependent Libor rates and study the pricing of credit derivatives. We also consider an application of the defaultable Lévy Libor model to counterparty risk and derive a valuation formula for vulnerable European call options based on Fourier transform methods.

In the second part, we study credit portfolio risk in the Libor market models and introduce a very general market model for CDOs which allows contagion in the portfolio and a direct dependence of the risk-free interest rates and the aggregate loss process. The model is driven by a special semimartingale whose components are time-inhomogeneous Lévy processes and the aggregate loss process of the considered portfolio. We derive conditions for the absence of arbitrage in the model and provide pricing formulae for CDOs in this framework.

Kurzfassung in Deutsch

Kreditrisikomodellierung gehört zu den wichtigsten und sich am schnellsten entwickelnden Gebieten der Finanzmathematik in den letzten fünfzehn Jahren. In dieser Arbeit werden verschiedene Arten von Kreditrisiken durch Libor Modelle dargestellt und zwei Modelle präsentiert: das Lévy Libor Kreditrisikomodell im Ratingbasierten Ansatz und das Libor Kreditportfoliomodell, die von zeit-inhomogenen Lévy-Prozessen und allgemeinen Semimartingalen getrieben werden.