# Archimedean copulas - Clayton, Frank and Gumbel

An important class of copulas - because of the ease with which they can be constructed and the nice properties they possess - are the Archimedean copulas, which are defined by:

where j is the generator of the copula.

There are three Archimedian copulas in common use: the Clayton, Frank and Gumbel.

##### Clayton copula

The Clayton copula is an asymmetric Archimedean copula, exhibiting greater dependence in the negative tail than in the positive. This copula is given by:

And its generator is:

where:

The relationship between Kendall's tau and the Clayton copula parameter is given by:

This Copula is implemented in ModelRisk as VoseCopulaBiClayton.

##### Frank copula

The Frank copula is a symmetric Archimedean copula given by:

And its generator is:

where:

The relationship between Kendall's tau and the Frank copula parameter is given by:

where

is a Debye function of the first kind.

This Copula is implemented in ModelRisk as VoseCopulaBiFrank.

##### Gumbel copula

The Gumbel copula (a.k.a. Gumbel-Hougard copula) is an asymmetric Archimedean copula, exhibiting greater dependence in the positive tail than in the negative. This copula is given by:

And its generator is:

where:

The relationship between Kendall's tau and the Gumbel copula parameter is given by:

This Copula is implemented in ModelRisk as VoseCopulaBiGumbel.