Valuation of Basket Credit Derivatives in the Credit Migrations Environment

by Tomasz R. Bielecki of the Illinois Institute of Technology,
Stéphane Crépey of the Université d'Évry Val d'Essonne,
Monique Jeanblanc of the Université d'Évry Val d'Essonne, and
Marek Rutkowski of the University of New South Wales & Warsaw University of Technology

January 2006

Introduction: The goal of this work is to present some methods and results related to the valuation and hedging of basket credit derivatives, as well as of portfolios of credits/loans, in the context of several possible credit ratings of underlying credit instruments. Thus, we are concerned with modeling dependent credit migrations and, in particular, with modeling dependent defaults. On the mathematical level, we are concerned with modeling dependence between random times and with evaluation of functionals of (dependent) random times; more generally, we are concerned with modeling dependence between random processes and with evaluation of functionals of (dependent) random processes. Modeling of dependent defaults and credit migrations was considered by several authors, who proposed several alternative approaches to this important issue. Since the detailed analysis of these methods is beyond the scope of this text, let us only mention that they can be roughly classified as follows:

• modeling correlated defaults in a static framework using copulae (Hull and White (2001), Gregory and Laurent (2004)[REF]),
• modeling dependent defaults in a "dynamic" framework using copulae (Schönbucher and Schubert (2001), Laurent and Gregory (2003), Giesecke (2004),
• dynamic modelling of credit migrations and dependent defaults via proxies (Douady and Jeanblanc (2002), Chen and Filipovic (2003), Albanese et al. (2003), Albanese and Chen (2004a, 2004b)),
• factor approach (Jarrow and Yu (2001), Yu (2003), Frey and Backhaus (2004), Bielecki and Rutkowski (2003)),
• modeling dependent defaults using mixture models (Frey and McNeil (2003), Schmock and Seiler (2002)[REF]),
• modeling of the joint dynamics of credit ratings by a voter process (Giesecke and Weber (2002)),
• modeling dependent defaults by a dynamic approximation (Davis and Esparragoza (2004)).

The classification above is rather arbitrary and by no means exhaustive. In the next section, we shall briefly comment on some of the above-mentioned approaches. In this work, we propose a fairly general Markovian model that, in principle, nests several models previously studied in the literature. In particular, this model covers jump-diffusion dynamics, as well as some classes of Lévy processes. On the other hand, our model allows for incorporating several credit names, and thus it is suitable when dealing with valuation of basket credit products (such as, basket credit default swaps or collateralized debt obligations) in the multiple credit ratings environment. Another practically important feature of the model put forward in this paper is that it refers to market observables only. In contrast to most other papers in this field, we carefully analyze the issue of preservation of the Markovian structure of the model under equivalent changes of probability measures.

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