Risk Latte - Living with the Gaussian Copula-Devil in the Derivatives

Living with the Gaussian Copula-Devil in the Derivatives

Rahul Bhattacharya
Jun 05, 2006

The Gaussian copula was first used by the RiskMetricsTM group in 1997 to outline their credit risk estimation methodology (reference can be found in Gupton, et al., 1997). Greg Gupton, formerly with the RiskMetricsTM Group (and now with the Moodys KMV Group) is a big name in the field of credit risk, and rightly so. He was (to the best of our knowledge) one of the main architects of the technical document on credit risk that was produced by RiskMetricsTM in 1997.

However, RiskMetricsTM technical document and Gupton's team did not mention Gaussian Copula by name. That was done by David Li in his near revolutionary paper on the pricing of credit derivatives in the year 2000 (see the related article on Broken Hearts and CDO in this website). David Li first identified and used the Gaussian ( Normal ) Copula approach to generate default times and price correlation products in the credit markets. Ever since then, the Normal Copula approach has become the industry standard amongst bankers to price credit derivatives products such as CDOs, first-to-default-baskets, etc.

Many researchers and credit derivatives structurers have since experimented with other kinds of copulas (t-copulas, Archimedean copulas, etc.) but the Gaussian copula has ruled so far. Though the implementation of the copula approach in any programming language is fairly simple enough once you get past the matrix math (we at Risk Latte Company use Excel/VBA as well as C++ to write codes to implement the Gaussian copula in a Monte Carlo simulation model of default times) understanding the deeper functional issues as well as philosophical issues in the use of Copulas to model the default times of different companies is not straightforward at all. At least we find it quite vexing. OK, here is a VBA code in Excel to calculate the Cholesky and the Gaussian Copula and here are the simulated default times,....that part is easy. It is called translating math into computer code. But what is a copula? We can explain correlation in simple English, can we explain copulas in simple English? We can't but we are learning.

We also believe that most traders and structurers around the world don't understand the copulas well enough. It is a devil that beats us all.

An excellent book on Copulas and Credit Derivatives is Credit Derivatives by Geoff Chaplin. We have four copies in our very small company.

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