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Dynamic Copula Processes: A new way of modelling CDO tranches

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Niveau: Supérieur, Doctorat, Bac+8
Dynamic Copula Processes: A new way of modelling CDO tranches Daniel Totouom1 & Margaret Armstrong2 First draft November 2005 Abstract We have developed a new family of Archimedean copula processes for modeling the dynamic dependence between default times in a large portfolio of names and for pricing synthetic CDO tranches. After presenting their general properties, we show that there is a class of processes where default is not predictable. Then we study a new Clayton copula process in detail. Using CDS data as at July 2005, we show that the base correlations given by this model at the standard detachment points are very similar to those quoted in the market for a maturity of 5 years. JEL Classification: G 13 Key words : default risk, CDOs, correlation smile, Archimedean copulas, multivariate stochastic processes Introduction Over the past five years the one factor Gaussian copula initially developed by Li (2000) has become a market standard for pricing CDOs but as the market for standard CDO tranches became more liquid, it became clear that a flat correlation model did not price these tranches correctly. See Burtschell et al (2005b) for an example. The recent literature on modelling the correlation skew seeks to overcome these shortcomings. At the same time base correlation (McGinty & Ahulwalia, 2004 a & b) has become increasingly popular among market practioners. While this approach is satisfactory at any point in time, it does not provide any way of linking prices/spreads at different points in time.

  • density model

  • transforms corresponding

  • process effectively

  • specified independent

  • copulas no corresponding

  • archimedean copula

  • archimedean copulas

  • gamma process

  • default times


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  Dynamic Copula Processes: A new way of modelling CDO tranches    21Daniel Totouom & Margaret Armstrong  First draft November 2005    Abstract  We have developed a new family of Archimedean copula processes for modeling the dynamic dependence between default times in a large portfolio of names and for pricing synthetic CDO tranches. After presenting their general properties, we show that there is a class of processes where default is not predictable. Then we study a new Clayton copula process in detail. Using CDS data as at July 2005, we show that the base correlations given by this model at the standard detachment points are very similar to those quoted in the market for a maturity of 5 years.  JEL Classification: G 13 Key words : default risk, CDOs, correlation smile, Archimedean copulas, multivariate stochastic processes   Introduction  Over the past five years the one factor Gaussian copula initially developed by Li (2000) has become a market standard for pricing CDOs but as the market for standard CDO tranches became more liquid, it became clear that a flat correlation model did not price these tranches correctly. See Burtschell et al (2005b) for an example. The recent literature on modelling the correlation skew seeks to overcome these shortcomings.  At the same time base correlation (McGinty & Ahulwalia, 2004 a & b) has become increasingly popular among market practioners. While this approach is satisfactory at any point in time, it does not provide any way of linking prices/spreads at different points in time. The two-fold aims of this work have been to develop a family of multivariate Archimedean copula processes, and a dynamic model for pricing synthetic CDO tranches. This will be required for pricing forward starting CDOs.                                                       1 BNPParibas: daniel.totouom-tangho@bnpparibas.com, daniel.totouom@mines-paris.org  2 CERNA, Ecole des Mines de Paris: margaret.armstrong@ensmp.fr  The authors would like to thank Alain Galli, Jon Gregory, Jean-Paul Laurent & Marek Musiela for their helpful comments. The usual disclaimer applies.  1
Recent work by Andersen, Sidenius & Basu (2003), Gregory & Laurent (2003), Hull & White (2004) and Burtschell Gregory & Laurent (2005a) has confirmed that factor copulas are a powerful tool for pricing CDOs within a semi-analytical framework. One of their strong points is that the dependence structure between the default times can be specified independent of the marginal credit curves. In a recent paper, Burtschell et al (2005b) studied different extensions of the well-known Gaussian and factor copulas and split them into two broad classes. In the first group, correlation is treated as being stochastic (rather constant) and is independent of the factor whereas in the second approach the correlation depends upon the factor. The authors refer to the latter as the local correlation model to distinguish from the stochastic correlation in the previous case.  In this paper we develop a family of dynamic Archimedean copula processes to model the default times. For simplicity we assume that the portfolio is large but not necessarily homogeneous. We require a stochastic process in which the copula defining the defaults amongst the n names is a valid n-copula at any point in time. While there is a vast literature on bivariate copulas, much less has been published on multivariate copulas. The canonical textbook on copulas, Nelsen (1999) gives some results for exchangeable Archimedean copulas, that is, those where the variables can be permuted. It also contains several counter-examples which demonstrate how wrong “intuition” can be and how difficult it is to find multivariate Archimedean copulas. The other major reference book, Joe (1997), provides some results for the trivariate & quadrivariate cases. Lindskog (2000) provides some interesting extensions and shows the level of constraints on the parameter values in the non-exchangeable case. But these are all static copulas.  One school of thought has developed dynamic copula models from a time series point of view. Patton (2001 & 2003) developed an approach based on an ARMA-type process and applied it to foreign exchange data. Fermanian & Wegkamp (2004) extended the approach proposed by Patton (2001) and based on pseudo-copulas. Duan (1995) and more recently Van den Goorbergh et al (2005) have used GARCH processes but they only priced bivariate options. A major shortcoming of these papers have is that they only consider bivariate cases. Chen & Fan (2005) consider a class of semiparametric copula based multivariate dynamic models (SCOMDY) which they apply to three and higher dimensional daily exchange rates. None of these seems to suitable for pricing CDOs on large portfolios.  Berd, Engle & Voronov (2005) have developed a hydrid model in which the dynamics of the underlying latent variables is governed by a GARCH or a TARCH process. This has the advantage of producing aggregate return distributions that are asymmetric and clearly non-Gaussian. The authors have used historical data on the SP500 going back to 1962 as a proxy for market returns (pre and post 1990). While they mention using some market data (e.g. the level of hazard rates and the expected default probabilities) to calibrate parameters, they do not seem to use the available CDS data, which is a pity.  Fewer papers have tackled the question from a continuous-time point of view. The earliest paper on copula processes seems to be Darsow et al (1992) who studied Archimedean copulas and Markov processes from a theoretical point of view. Several authors have modelled credit risk dynamics using default intensities rather than default times. Rogge & Schonbucher (2003) developed a method for modelling dynamic portfolio credit risk based on Archimedean copulas. They provide some very useful results that link Archimedean copulas with Laplace transforms, and also a fast and efficient method for simulating realisations, one that is not mentioned by Nelsen (1999)3.  Following Madan’s work on stock prices (Madan & Seneta, 1990, and Madan & Milne, 1991) and that of Cariboni & Schouten (2004) on the value of the firm, Joshi & Stacey (2004) used a gamma process when modelling default intensities and pricing CDOs. They found that a double gamma process was required to match the base correlations observed in the market correctly. One disadvantage of working with intensities is that it requires calibrating the default functions for each of the names.                                                      3 Nelsen (1999) gives general methods for simulating bivariate Archimedean copulas and some ad-hoc methods for specific copulas. These are presented in Exercises N° 4.13, 4.14 & 4.15 p108  2