Archimedean copulas - Clayton, Frank and Gumbel

See also: Modeling correlation introduction, Copulas, Copulas in ModelRisk

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.

Read on: Elliptical copulas - Normal and T

 

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