A Bottom-Up Dynamic Model of Portfolio Credit Risk. Part II: Common-Shock Interpretation, Calibration and Hedging Issues

Abstract

In this paper, we prove that the conditional dependence structure of default times in the Markov model of [4] belongs to the class of Marshall-Olkin copulas. This allows us to derive a factor representation in terms of “common-shocks”, the latter beeing able to trigger simultaneous defaults in some pre-specified groups of obligors. This representation depends on the current default state of the credit portfolio so that fast convolution pricing schemes can be exploited for pricing and hedging credit portfolio derivatives. As emphasized in [4], the innovative breakthrough of this dynamic bottomup model is a suitable decoupling property between the dependence structure and the default marginals as in [10] (like in static copula models but here in a full-flesh dynamic “Markov copula” model). Given the fast deterministic pricing schemes of the present paper, the model can then be jointly calibrated to single-name and portfolio data in two steps, as opposed to a global joint optimization procedures involving all the model parameters at the same time which would be untractable numerically. We illustrate this numerically by results of calibration against market data from CDO tranches as well as individual CDS spreads. We also discuss hedging sensitivities computed in the models thus calibrated.