Introduction to Bayesian Estimation and Copula Models of Dependence

CHAPTER 7 Statistics of Copulas

7.1 The Formula that Killed Wall Street

This is how copulas became infamous. Very few mathematical formulas ever achieved such a huge popular recognition, even with clearly negative connotations, as the title of this section may suggest. Formulas rarely kill. Gaussian copulas (or, rather, applications of Gaussian copulas to pricing credit derivatives) have earned this distinction. As Felix Salmon [38] tells the story, it all started in the year 2000 when David X. Li, a former actuary turned financial mathematician, suggested a smart way to estimate default correlations using historical credit swap rates [29].

The approach suggested by Li was indeed very attractive. Being able to use correlation structure to define dependence, as was discussed in Chapter 5, really simplifies the problem of building a model for joint distribution. As we have mentioned, this is how it works for multivariate normal distribution: all dependence that may exist between the components of a Gaussian random vector can be expressed in terms of the correlation matrix. In the bivariate case the correlation matrix degenerates into just one number: the correlation coefficient.

By that time (the turn of the new millennium) it was already well known that Gaussian approximation was not working very well for many financial variables. Empirical observations of the long-term behavior of daily stock price returns, market indexes, and currency exchange rates had exposed some strange ...