The implementation of credit risk models has largely relied either on the use of historical default dependence, as proxied by the correlation of equity returns, or on equicorrelation, as extracted from CDOs. The drawbacks of equicorrelation are well known from the correlation smile: credit derivative pricing would therefore profit from risk&neutral depen& dence measures without the equicorrelation assumption. Using the copula methodology, we show how to infer them from CDS data, taking counter& party risk into consideration. We also provide a market application and explore its impact on the fees of some higher dimensional credit deriva& tives. Both in the FtD and CDO case, the adoption of a copula with tail dependency instead of the Gaussian one, which has no tail dependency, has the same qualitative effect than the use of (the correct) risk neutral measure instead of equity dependency: therefore, tail dependency com& pensates for the lack of risk neutral correlation, whenever historical equity correlation is adopted.
The assessment of the joint default probability of groups of obligors, as well as related notions, such as the probability that the n&th one of them defaults, is a crucial problem in credit derivatives pricing and hedging. In order to solve it, academics and practitioners have extensively relied on copula methods, which allow to split any joint default probability into the marginal ones and a function, the copula itself, which represents only the dependence between defaults. The splitting up makes both default modelling and calibration much easier, since it permits separate fitting at the univariate and joint level.
Copula techniques require on the one side the choice of a specific dependence or copula function, on the other side the selection of a level of the parameter/s which characterize the copula.
As for the copula choice, structural based models naturally lead to a so&called Gaussian or Student copula, while in intensity&based models the same copulas are very often introduced for analytical convenience, starting from the paper of Li (2000), now an industry&standard.
