Elliptical copula that keeps the Gaussian copula's correlation matrix but adds a degrees of freedom parameter producing symmetric, nonzero tail dependence.
The Gaussian copula has a fatal flaw in extreme risk: whatever correlation you pick, strictly below one, its tail dependence is zero, so it asserts that two highly correlated risks stop coinciding once you go far enough into the tail. The Student t copula fixes that without giving up the convenience of a correlation matrix: it adds a degrees of freedom parameter, and its tail dependence is strictly positive, symmetric in both tails, and stronger the lower the degrees of freedom. As they tend to infinity you recover the Gaussian copula. It extends to fifty dimensions without difficulty, which is its decisive advantage over Archimedean copulas when aggregating a whole balance sheet. Its limit is that symmetry: it forces the same dependence intensity in the upper and lower tail, which is false for a book where losses coincide but favorable development does not.
The role of the Gaussian copula in understating joint losses on CDO tranches backed by US mortgages is the most expensive illustration of the problem, exposed from 2007 and 2008 onward. On an aggregate of eight risk modules with an average correlation of 0.25, switching from a Gaussian copula to a Student t copula with four degrees of freedom raises aggregate SCR by roughly 9% without changing a single marginal distribution.
copule elliptique de Student, Student copula, degrés de liberté de la copule