Price calibration and hedging of correlation dependent credit derivatives using a structural model with α-stable distributions

The emergence of Credit Default Swap (CDS) indices and corresponding credit risk transfer markets with high liquidity and narrow bid-ask spreads has created standard benchmarks for market credit risk and correlation against which portfolio credit risk models can be calibrated. Integrated risk management for correlation dependent credit derivatives, such as single-tranches of synthetic Collateralized Debt Obligations (CDOs), requires an approach that adequately reflects the joint default behaviour in the underlying credit portfolios. Another important feature for such applications is a flexible model architecture that incorporates the dynamic evolution of underlying credit spreads. In this article, we present a model that can be calibrated to quotes of CDS index-tranches in a statistically sound way and simultaneously has a dynamic architecture to provide for the joint evolution of distance-to-default measures. This is accomplished by replacing the normal distribution by Smoothly Truncated α-Stable (STS) distributions in the Black/Cox version of the Merton approach for portfolio credit risk. This is possible due to the favourable features of this distribution family, namely, consistent application in the Black/Scholes no-arbitrage framework and the preservation of linear correlation concepts. The calibration to spreads of CDS index tranches is accomplished by a genetic algorithm. Our distribution assumption reflects the observed leptokurtic and asymmetric properties of empirical asset returns since the STS distribution family is basically constructed from α-stable distributions. These exhibit desirable statistical properties such as domains of attraction and the application of the generalized central limit theorem. Moreover, STS distributions fulfill technical restrictions like finite (exponential) moments of arbitrary order. In comparison to the performance of the basic normal distribution model which lacks tail dependence effects, our empirical analysis suggests that our extension with a heavy-tailed and highly peaked distribution provides a better fit to tranche quotes for the iTraxx IG index. Since the underlying implicit modelling of the dynamic evolution of credit spreads leads to such results, this suggests that the proposed model is appropriate to price and hedge complex transactions that are based on correlation dependence. A further application might be integrated risk management activities in debt portfolios where concentration risk is dissolved by means of portfolio credit risk transfer instruments such as synthetic CDOs.