Correlation Smile, Volatility Skew, and Systematic Risk Sensitivity of Tranches

Abstract

In collateralized debt obligations and other securitized credit derivatives, the expected tranche payoffs depend heavily on the default correlations among the securities in the pool. Like volatility, correlation is not directly observable, but it is possible to infer it from the market price of a tranche. But unfortunately, also like volatility, these implied correlations don’t behave well. They should be equal across all of the tranches created from a given pool, but they never are. Instead, implied correlations exhibit the pattern known as correlation skew. In this article, Hamerle, Igl, and Plank consider two ways in which the real world departs from the assumptions of the Gaussian copula model. Different correlations extracted from different tranches is one, but the other departure is that investors require expected risk premia for bearing a security’s downside exposure. This is not the same as a (symmetrical) distaste for “volatility,” say, and it is reflected in an asymmetrical implied volatility skew exhibited by options on a bond issuer”s equity. The standard Gaussian copula model allows for the first effect, but not the second. In this article, the authors look at both. Their most successful model estimates downside risk premia from the risk-neutral probability densities extracted from the issuers’ equity options and then imposes a fixed and moderate degree of correlation. This combination captures market pricing very well for all of the tranches above the equity tranche.