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In the Core of Correlation

by Jon Gregory of BNP Paribas, and
Jean-Paul Laurent of the University of Lyon & BNP Paribas

April 2004

Abstract: The modelling of dependence between defaults is a key issue for the valuation and risk management of multi-name credit derivatives. The Gaussian copula model seems to have become an industry standard for pricing. It's appeal is partly due to its ease of implementation via Monte Carlo simulation and the fact that the underlying dependence structure has for a long time been linked to equity returns correlation. Furthermore, a big driving force behind the adoption of this approach has been the tractability in reduced dimension, with fast (semi-)analytical calculations of prices and deltas of CDO tranches and basket default swaps. The simplest form of the model is the so-called one factor Gaussian copula.

There do, however, remain some drawbacks in the applicability of the one factor Gaussian model:

  • There is a reported "correlation smile" in the CDO market similar to the well-known Black-Scholes implied volatility smile. Whether this is due to liquidity effects or a more theoretical issue such as the choice of copula (e.g. fat tails in the loss distribution) remains an open question.

  • The one-factor correlation structure is rather limited, for example, we cannot have separated "regions" such as a highly correlated domain amongst a background of low correlation.

  • There are some practical issues with the representation and aggregation of correlation risk, for example calibration done at the transaction level means that a name can exist in different portfolios with different associated correlation parameters. It is more realistic to view risk to changes in underlying correlations rather than the factor(s) themselves.

This paper aims to provide practical results in light of such issues. We show that in a Gaussian copula framework we can keep the appeal of analytical tractability and:

  • Provide a more intuitive correlation structure, leading itself readily to correlation risk analysis.

  • Compute correlation sensitivities either via the above structure or analytically in an extension of the one-factor model.

  • Introduce some dependence between recovery rates and between recovery rates and defaults.

While this paper is dedicated to CDO tranches, the results can be directly applied to k th to default swaps and to portfolio credit risk analysis.

Published in: RISK, Vol. 17, No. 10, (October 2004), pp. 87-91.

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