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

Abstract

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.

Acknowledgements

Svetlozar Rachev gratefully acknowledges research support by grants from the Division of Mathematical, Life and Physical Sciences, College of Letters and Science, University of California, Santa Barbara, the Deutschen Forschungsgemeinschaft, and the Deutscher Akademischer Austausch Dienst.

Notes

¹ See for example Burtschell et al. (2005), p. 17.

² See Burtschell et al. (2005), p. 17.

³ See Wang et al. (2009), p. 12.

⁴ For these reasons, Wang et al. (2007) introduce two new one-factor heavy-tailed copula models: (1) the one-factor double-t distribution with fractional degrees of freedom copula model and (2) the one-factor double mixture distribution of t and Gaussian distribution copula model. In each model, there is a parameter to continuously control the tail-fatness of the copula function. Moreover, the maximum tail-fatnesses of our two models are much larger than that for Hull and White's one-factor double-t copula model.

⁵ See Duffie (2004).

⁶ Menn and Rachev (2005).

⁷ Similar to the concept of implied volatility in option pricing, the Gaussian copula has become the market standard to communicate prices of synthetic CDO tranches.

⁸ See Hull et al. (2005), p. 11.

⁹ See Hull et al. (2005), p. 7.

¹⁰ See, for example, Zolotarev (1966).

¹¹ See Rachev et al. (2005), figure 1.

¹² See, for example, Höchstötter et al. (2005).

¹³ This is comprehensively quantified in Menn and Rachev (2004).

¹⁴ See Menn and Rachev (2005), p. 11.

¹⁵ There exist similar models in practice and it is often assumed that both M and Z_i have distributions with the same tail index.

¹⁶ Fortunately, performance can be strongly improved to restrict the grid to a smaller abscissa range. This is possible since the truncation produces negligible small values for the normal distributions in the tails due to their non-heavy-tailed character. All procedures mentioned so far – including the numerical convolution – consume 12 s for one specific parameter tuple (α, σ) in the symmetric case in C++ on a 1.5 GHz processor and 512 MB of RAM.

¹⁷ See Hull et al. (2005), p. 5.

¹⁸ There is an exception concerning the up-front fee of the equity tranche which results in a different default time risk profile.

¹⁹ See Amato and Gyntelberg (2005).

²⁰ See Amato and Gyntelberg (2005), p. 74.

²¹ See Amato and Gyntelberg (2005), p. 77, footnote 10.

²² This is due to standardized deal conventions.

²³ See for example Press et al. (1992), p. 412.

²⁴ An additional parameter will slow down the optimization speed of the GA but variation of the evolutionary search might speed the procedure up. Also, further effort could be undertaken with respect to the implementation of the model. Operations to build up the evaluation function of standardized STS distributions and the barrier calibration for a homogeneous portfolio require about 35 s. The Monte-Carlo simulation with 10,000 scenarios consumes about 105 s. This part leaves room for various performance improvements.

²⁵ Hull et al. (2005) successfully extend their model in these directions.