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SurferD
Topic Author
Posts: 35
Joined: February 22nd, 2004, 2:00 pm

### Clayton Copula

Hi all,does anyone have any experience with the Clayton copula - theoretical or practical?Thanks+regards,SurferD
Last edited by SurferD on September 21st, 2004, 10:00 pm, edited 1 time in total.

Mircea
Posts: 19
Joined: July 9th, 2004, 6:04 pm

### Clayton Copula

Check Roger B. Nelsen book “An Introduction to Copulas” for a mathematical presentation of Archimedean family of copulas (Clayton is a member of this family). In finance it can be used for building joint distributions from marginal distributions. A typical example is the single mixture-factor model for basket default swaps where one can use a Clayton copula for the joint distribution instead of a Gaussian copula (the most common approach). The Clayton copula prescribes stronger tail dependencies, a feature that one may view it as important for describing multiple defaults. For example the coefficient lambda_l = lim_{u->0} P( u2<u | u1 < u ) (that measure the lower tail dependency) is 2^{-1/theta} for Clayton and 0 for Gaussian (theta being the Clayton parameter). However lambda_u = lim_{u->1} P( u2<u | u1 < u ) (that measures the upper tail dependency) is zero in both cases.

SurferD
Topic Author
Posts: 35
Joined: February 22nd, 2004, 2:00 pm

### Clayton Copula

Thanks for your answer - do you have a recipe how to simulate Clayton-distributed Random Variables fo higher dimensions (e.g. 100-dim)?Regards,SurferD

scholar
Posts: 832
Joined: October 17th, 2001, 8:03 pm

### Clayton Copula

A good algorithm is Marshall-Olkin, it is described e.g. in Schonbucher's book.

luebke
Posts: 51
Joined: August 17th, 2004, 5:40 pm

### Clayton Copula

QuoteOriginally posted by: scholarA good algorithm is Marshall-Olkin, it is described e.g. in Schonbucher's book.I do not have Schonbucher's book at hand, and I am not sure what the Marshall-Olkin method is. I assume it is the standard method to generate random numbers from copulas, using the nice property of the generator of the Clayton copula.My question now is: are there any articles / methods on generating random numbers from Gumbel or Frank copula in more then two dimensions? E.g. 8 or 10? There does not seem to be a nice formula for the n-th derivative of the (inverse) generator, as it is with Clayton.All hints are welcome.

ClosetChartist
Posts: 190
Joined: July 17th, 2003, 4:41 pm

### Clayton Copula

See Schonbucher's paper "Taken to the Limit: Simple and Not So Simple Loan Loss Distributions" for a cookbook discussion of the Marshall-Okin method.

rpfeynman
Posts: 9
Joined: July 14th, 2002, 3:00 am

### Clayton Copula

I wrote the S-Plus script below for 2 dimensions - cause I wanted to plot 2 dimensions. Should be straightforward to do for n-dimensions (except for the plotting).genClaytonCopulaSamples = function(n, theta){#Effects: Draw n pairs from a Clayton copula.#--------------------------------------------------------- #Generate n pairs of independent uniforms. x = matrix(0, n, 2) x[, 1] = runif(n) x[, 2] = runif(n) #Generate n mixing variables. y = rgamma(n, 1 / theta, 1) #Generate n pairs from a Clayton copula. for(i in 1:2) { x[, i] = (1 / (1 - log(x[, i]) / y))^ (1 / theta) } return (x)}main = function(){ par(mfcol = c(1, 2)) x = genClaytonCopulaSamples(1000, 2) plot(x[, 1], x[, 2], main = "Clayton (theta = 2)", xlab = "u1", ylab = "u2") x = genClaytonCopulaSamples(1000, 50) plot(x[, 1], x[, 2], main = "Clayton (theta = 50)", xlab = "u1", ylab = "u2")}main()

luebke
Posts: 51
Joined: August 17th, 2004, 5:40 pm

### Clayton Copula

QuoteOriginally posted by: ClosetChartistSee Schonbucher's paper "Taken to the Limit: Simple and Not So Simple Loan Loss Distributions" for a cookbook discussion of the Marshall-Okin method.Thank you very much for the link. Meanwhile I have done some literature review, concerning the construction of multivariate random numbers from Archimedean copulas. I think that the above method is ok for the Clayton copula but is not that easy to implement for Gumbel and Frank. For Clayton, one can even use the standard algorithm using the n-th derivative of the generator function, which in case of Clayton is easy to compute and rather easy to invert. For Gumbel and Frank, this does not work this way. In my opinion, the most promising implementation can be found in Lindskog's master thesis. This method works fine for ALL Archimedean copulas, I think.Anyone out there who shares this point of view or sees things differently?

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