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An Implied Default Dependency Model of a Credit Portfolio based on the Number of Defaults

by Tomoaki Shouda of Hitotsubashi University

February 28, 2010

Abstract: We propose a market-implied default dependency model that can be exactly calibrated to market quotes of credit index tranches and single name credit default swaps belonging to their reference portfolio. Its calibration does not need simulation even in heterogeneous setting, hence is robust and quite rapidly. The Gaussian copula model is the market standard for default dependency, though it appears to have a problem called {\it correlation smile} that the model parameter changes for each tranche belonging to a credit index. In addition, without the assumption of a large homogeneous portfolio, its calibration generally needs simulation; however, under this assumption we cannot evaluate the risk of tranches with respect to each company. On the other hand, in light of the market crash of 2008, there is a strong demand for default dependency models that have good tractability and high accuracy. The implied factor-copula and top-down approaches are currently applied for research of this field. A model that satisfies the practical demands in a heterogeneous setting has not yet been reported. In this paper, we present a representation of joint default probability of a credit portfolio by a minimal entropy martingale measure. We derive the above model under a feasible assumption related to the market. We calibrate our model to market quotes of CDX NA IG tranches and evaluate their risks.

JEL Classification: C65, D52, G33.

Keywords: default risk, credit index tranche, default dependency, minimal entropy martingale measure.

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