Elliptical copulas - Normal and T | Vose Software

# Elliptical copulas - Normal and T

Elliptical copulas are simply the copulas of elliptically contoured (or elliptical) distributions. The most commonly used elliptical distributions are the multivariate normal and Student-t distributions. The key advantage of elliptical copula is that one can specify different levels of correlation between the marginals and the key disadvantages are that elliptical copulas do not have closed form expressions and are restricted to have radial symmetry. For elliptical copulas the relationship between the linear correlation coefficient r and Kendall's tau t is given by:

The Normal and Student T copulas are described below.

### Normal copula

The normal copula is an elliptical copula given by:

where F-1 is the inverse of the univariate standard Normal distribution function and r, the linear correlation coefficient, is the copula parameter.

The relationship between Kendall's tau t and the Normal copula parameter r is given by:

This Copula is implemented in ModelRisk as VoseCopulaBiNormal for the bivariate case, and as VoseCopulaMultiNormal for the multivariate case.

### The T copula

The Student-t copula is an elliptical copula defined as:

where n (the number of degrees of freedom) and r (linear correlation coefficient) are the parameters of the copula.

When the number of degrees of freedom n is large (around 30 or so), the copula converges to the Normal copula just as the Student distribution converges to the Normal. But for a limited number of degrees of freedom the behavior of the copulas is different: the t-copula has more points in the tails than the Gaussian one and a star like shape. A Student-t copula with n = 1 is sometimes called a Cauchy copula.

As in the normal case (and also for all other elliptical copulas), the relationship between Kendall's tau t and the T copula parameter r is given by:

This Copula is implemented in ModelRisk as VoseCopulaBiT for the bivariate case, and as VoseCopulaMultiT for the multivariate case.