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Topic Title: puzzle?? Why copula?
Created On Thu Mar 03, 05 04:36 PM

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quantworm
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Posts: 36
Joined: Jul 2004

Thu Mar 03, 05 04:36 PM

I am really confused what is advantage by using copula compared to multivariate.

I know sometime it is hard to get a multivariate distribution when you know marginal distribution and correlation matric. but for copula, it is the same situation which it is difficult to find out a closed form copula. I know I must miss something , anyone helps me out?

Thanks in advance.

BTW, I know margianls--->copula----->Multiple distribution and Multiple--->copula as well. For example, for find out the default probability of basket(2 tranche) credit products, we know the marginal default probability and try to locate a joint one, and pricing it. Why not try to find out a propriate joint distribution by trying some joint distribution? If using the copula, we still do not have a closed form joint distribution, that is , the same method without th copula one.

Thanks again!

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genkideska
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Joined: May 2004

Fri Mar 04, 05 08:19 AM

copulas allow you to model and estimate the univariate margins and the dependence structure independently. i.e. you can for example decide to take exp-dist. for the margins and gauss-copula for the dependence structure (see Li 1999). furthermore you can also take different margins for each rv or use for example the empirical margins. if you choose a multivariate distr. you also have to fix the margins, e.g. multivariate normal => margins must be also normal. copulas allow to study the dependence structure independly of the marginal behaviour.

hope it helps you

marcel

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balaji
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Posts: 480
Joined: Dec 2003

Fri Mar 04, 05 05:21 PM

To add, Copula allow us to look beyond the usual correlation. There are many interesting scale free measures of dependence that come out of Copula.

Yes, fitting Copula is as difficult as fitting a joint DF. The advantages are, as previously mentioned, decoupling of dependence structure and more general dependence measures than the usual correlation.

Posts: 36
Joined: Jul 2004

Fri Mar 04, 05 07:51 PM

Thanks for your guys help!

"if you choose a multivariate distr. you also have to fix the margins, e.g. multivariate normal => margins must be also normal. copulas allow to study the dependence structure independly of the marginal behaviour."

I know, from statistical theory,if you have a multivatiate normal, then the marginals got to be normal, so do you mean, for example, I have a bivaiate Gaussian Copula, then the marginal does not have to be normal, could be some other distributions? and then could build the dependence structure independly? Or, you just pick up the dependence stucture totally separate with the marginals. Maybe my understandingn is totally wrong, sorry about that.

I have another question that Li(1999) paper mentioned " There are many different techniques in statistics with allow us to specify a joint distribution with given marginal and a correlation structure. Among them copula function is a simple and convient approach".

Who can give me a couple of examples on the techniques he talked about ?

Thanks a lot

Posts: 36
Joined: Jul 2004

Sun Mar 06, 05 12:06 AM

I am still looking for any helpful answer.

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Aaron
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Sun Mar 06, 05 03:19 AM

I thought you got two helpful answers already.

Maybe it's simpler to think about specifically. The Gaussian copula is popular because it is easy to generate Normal random variates with a specified correlation matrix. It's also dangerous, the multivariate Normal distribution is very special in that the pairwise correlations tell you everything about the joint distribution. For realistic multivariate distributions, the pairwise correlations tell you very little beyond the bivariate distributions.

Anyway, the copula technique allows you to take advantage of the easy-to-generate property of the multivariate Normal, while using any univariate marginal distributions you want. Once you change the marginal distributions you get a different correlation matrix, of course, but sometimes people don't care about that.

The Student-t copula is also popular. It's not so easy to generate the variates for an arbitrary correlation matrix, but it's easy for constant pairwise correlation. In some applications we don't care about (or don't know) much about individual pairwise correlations, the average value is good enough. Student-t makes simultaneous extreme deviations of many variates more likely at low correlations than a Normal distribution would suggest. Again, you can get this dependence property for any marginal distributions.

I don't know which techniques Li is talking about. All I know is some special cases, and some ad hoc numerical and analytic techniques that are not very efficient. I don't think there is any efficient general-purpose technique generating multivariate samples with specified marginal distributions and correlation matrix.

-------------------------
Aaron Brown

Edited: Sun Mar 06, 05 at 03:20 AM by Aaron

Posts: 36
Joined: Jul 2004

Sun Mar 06, 05 03:07 PM

Thanks very much, I know I did get several helpful answers.

I will make clarification on my understanding of copula's advantage. Assume that we know (or want to build ) some multivariate distribution ( fro example multinormal one), by using the gussian copula as a bridge , and take advantage of copula's properties which can use any univariate distribution to generate gussian copula , then we get the multivatiate normal whose marginal may not come from normal one. For example, while doing empirial anlysis( like basket default swaps). we can easily to located a univariate distribution(may not be normal distribution) for default time, and then use above copula's advantage to get the multivariate distribution which you want to build.

I know somehow I repeat your guys' explanations. But anyway what I want to make sure, this is what copula's advantage is.

Thanks a lot again!

Posts: 45
Joined: Jul 2002

Mon Mar 07, 05 12:06 PM

As already noted copula is a function which couples given marginal distributions to form a joint distribution function. Besides introducing a whole new array of dependence structures, the interesting aspect of a copula is that it makes it possible to separate the marginal distribution from the dependence structure, thus splitting the model specification in two. From a mathematical viewpoint the copula approach introduces nothing new. From a statistical viewpoint, however the selecting and estimation process are eased, since we can pick the marginal distribution function independently from each other and from the dependence structure. Of course we still have to select a copula function and in practical application here lies the challenge. The use of a copula function introduces greater flexibility but it is an open question whether selecting a copula function is easier than selecting the multivariate distribution function directly. I doubt it. Theories seldom dictate which copula is the right one and you have to choose the copula that fits data best by some measure. In my (limited) experience I have found that choosing a copula function based on Akaikes Information Criteria (AIC) is generally an acceptable and relatively easy approach. That is choosing the copula with the lowest AIC from a list of copula functions. This separation between the dependence structure and the marginal distributions are also used in estimation. A method known as Canonical Maximum Likelihood estimation (CML) allows you to estimate the copula parameters without knowing the distributional form of the marginals. Instead one is using the empirical distribution function for each of the marginals.

I have been working on a Value-at-Risk model based on a copula/extreme value approach. As already mentioned a multivariate Gaussian distribution is seldom an adequate model with regard to VaR. This is especial true in finance. Financial data are characterized as being leptokurtic, skewed and heteroskedastic and often one observes more extreme observations in financial data than can be explained by the Gaussian distribution. Another point is that the Gaussian distribution is strictly symmetrical thus making it unsuitable for skewed empirical distribution. Even though the marginal distribution aren?t Gaussian ? and thereby making it impossible for the joint distribution function to be Gaussian ? the Gaussian dependence structure can still be the right one. The use of copula function makes it possible to join non-gaussian marginals with a Gaussian dependence structure. The same argument can be applied to the multivariate t-distribution. The multivariate t-distribution is characterized by ?degree of freedom? parameter and a correlation matrix. But by using a multivariate t-distribution you implicitly accept that all you marginals have the same degree of freedom. This is not realistic. This problem can be overcome by using a t-copula as dependence structure and t-distributions for each of the marginals all with different DoF. The resulting joint distribution function is not a multivariate t-distribution and is not necessarily even elliptical anymore. But ideally you have a model that is closer to the true distribution of your actual data.

Hope this helps.

/Jonas

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Without order nothing can exist - without chaos nothing can evolve

Posts: 36
Joined: Jul 2004

Mon Mar 07, 05 10:23 PM

Excellent answer for my question. BTW, for model selcetion, why do you think AIC is acceptable and relatively easy one?

Thanks

Posts: 45
Joined: Jul 2002

Tue Mar 08, 05 11:50 AM

I have performed a Monte Carlo simulation study. It must be mentioned that this was a small study including only 3 different bivariate copula functions: Gaussian, student-t and the Gumbel copula. The simulations were performed with different parameters for each copula function. For marginal distributions I used two t-distributions with DoF 6 and 8 respectively. But since I use CML for estimation the distributional form of the marginals aren?t important from a theoretical point of view. In practise this have to be checked and it is on my to-do-list. Most often the success rate by AIC were over 85%. That is the right copula was chosen by AIC 85 times out of a 100 simulations trials. The worst case overall was 65% success. For the gumbel copula the worst case were 92%. In each of the 100 trials, 1000 random drawings from the copula in question were generated and AIC were calculated for each of the 3 copula functions. I would be happy to send the results to you but it?s in Danish.

In short AIC is calculated via the likelihood function by adjusting the value of the likelihood function at the ML estimate with the number of parameters. So if you have the likelihood function you are able to calculate AIC. And you need the likelihood function for practical application

/Jonas

-------------------------
Without order nothing can exist - without chaos nothing can evolve

Edited: Tue Mar 08, 05 at 11:52 AM by jhlm

Posts: 45
Joined: Jul 2002

Tue Mar 15, 05 01:41 PM

Quantworm, if you want me to elaborate I'll be happy to. I will not though be able to respond right away since I?ll be in the States until 2. April.

/Jonas

-------------------------
Without order nothing can exist - without chaos nothing can evolve

Edited: Tue Mar 15, 05 at 01:43 PM by jhlm

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Siberian
Senior Member

Posts: 298
Joined: Jun 2003

Sun Mar 20, 05 08:19 PM

A quick question for Aaron, I was asked to do a presentation for the Risk Management class, something they have not covered, and I narrowed it down to two topics, Realized volatility, a paper by Andersen, Bollerslev and Diebold, (Parametric and Nonparametric volatility measurement), and Copula methods in VaR. It just seems to me that estimation using copulas is getting more and more popular. Any comment on that and perhaps a suggestions towards the chpice of topic?
I would really appreciate any remarks,
Thanks,

Siberian

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Siberian
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Sun Mar 20, 05 08:27 PM

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Siberian
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Sun Mar 20, 05 08:27 PM

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Aaron
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Joined: Jul 2001

Mon Mar 21, 05 09:35 PM

Copula methods have become more common in the literature, in fact they've become a standard part of the toolbox. Most of the time the use is pretty superficial.

I can't really advise you on which topic is better, there are interesting things to say about both.

-------------------------
Aaron Brown

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Siberian
Senior Member

Posts: 298
Joined: Jun 2003

Tue Mar 22, 05 04:03 AM

Thank you Aaron,

p.s.: sorry everybody for multiple postings, smth wrong with computers at the lab

Posts: 21
Joined: Dec 2004

Fri Mar 25, 05 10:44 AM

Hi all,

Was just wondering if there are any assumptions we have to make when using copula functions, for example Gaussian copula approach...

Read some papers that uses copula functions to value the credit derivatives, all of them use hazard rate in the simulation of default times and assumed that the hazard rate is constant throughout for all the obligors in a basket..

1) Does the Gaussian copula approach work if hazard rate is not constant and different for each obligor?
2) Using default probability density instead of hazard rate -> is there a great difference between the results obtained using hazard rate?

Thanks.

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Aaron
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Joined: Jul 2001

Fri Mar 25, 05 03:58 PM

(1) The great advantage of the copula is it works with any combination of distributions. It's the Gaussian part that imposes some restrictions, but the restrictions are on the dependence structure, not the marginal distributions.

(2) A hazard rate is a default probability. I think you are asking whether the correlation dynamics matter. If two issuers are correlated, we could model that as correlated default probabilities, correlated default times or correlation of default events (or any combination, or many other ways). This assumption matters quite a lot in theory, but in practice if you stick to simple instruments and calibrate to good market data, and don't go too far into the tail, you can get pretty good prices and hedges with any of the three approaches.

-------------------------
Aaron Brown

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