A Top-Down Approach to Multiname Credit

A multi-name credit derivative is a security tied to an underlying portfolio of corporate bonds or other credit-sensitive securities. It enables investors to buy and sell protection against the default losses in the portfolio. The value of a multiname derivative depends on the distribution of portfolio loss at multiple horizons. Intensity-based models of the loss point process that are specified without reference to the portfolio constituents determine this distribution in terms of few economically meaningful parameters, and lead to tractable credit derivatives valuation relations that can be addressed by a variety of efficient methods. This paper proposes random thinning to extend the reach of these models beyond the portfolio level. Random thinning decomposes the portfolio loss process into the sum of its constituent loss processes, and allocates aggregate portfolio risk to sub-portfolios. We show that any loss process can be thinned, and that the associated thinning process is a probabilistic model for the next-to-default. We derive a formula for the constituent default probability in terms of the thinning process and the portfolio intensity, and show how to evaluate it for a large family of portfolio intensity models. This formula facilitates consistent pricing and calibration of constituent and portfolio credit derivatives, which we demonstrate by fitting a familiar stand-alone model of the portfolio loss to market rates of CDX index, tranche and constituent single-name credit swaps. ∗Department of Management Science & Engineering, Stanford University, Stanford, CA 94305-4026, Phone (650) 723 9265, Fax (650) 723 1614, email: [email protected], web: www.stanford.edu/∼giesecke. †MSCI Barra, 2100 Milvia Street, Berkeley, CA 94704-1113, Phone (510) 649 4601, Fax (510) 848 0954, email: [email protected]. ‡We thank Greg Anderson, Aaron Brown, Eymen Errais, Steve Evans, Igor Halperin, Steven Hutt, JyhHuei Lee, Thorsten Schmidt and Pascal Tomecek for illuminating discussions, and seminar participants at the 2007 Applied Probability Society Conference, the 2007 CreditMinds Conference, the 2007 Decision and Risk Analysis Conference, Fields Institute for Research in Mathematical Sciences, the 2007 German Mathematical Society Annual Conference, the 2006 ICBI Global Derivatives and Trading Conference, the 2006 and 2007 INFORMS Annual Meetings, J.P. Morgan, the 2007 Lévy models in Credit Conference, Morgan Stanley, Natixis, the 2007 Newton Institute Conference on Developments in Quantitative Finance, the 2006 Numerical Methods for Finance Conference, the 2006 Quantitative Methods in Finance Conference, the 2006 Quant Congress and the 2006 Stanford-Tsukuba Workshop on Financial Engineering for comments. We are deeply grateful to Xiaowei Ding for his contributions to the development of the material in Section 5 and to Stefan Weber for his contributions to the material in Appendix A.