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Oleg
Member

Posts: 42
Joined: Jun 2002

Mon Jan 13, 03 06:01 PM

Copulas are funtions which join one-dimensional marginal distributions into the joined multivariate distribution:
Having Copula C=C(u,v) and one-dimensional marginal distributions and F(x) and G(y) one receives the joint d.f. H=H(x,y) by H(x,y=C(F(x),G(y)). This fact and its inverse are known as Sklar's Theorem and are central in the
theory of copulas.
The standard (and very pleasant to read) reference is R. B. Nelsen "An Introduction to copulas".
Copulas are used in finance for estimating the probabilities of common defaults or generally to estimate dependen risks e.g. An exmaple of such aplication may be found e.g in "Modelling Dependence with Copulas
and Applications to Risk Management" by Paul Embrechts, Filip Lindskog and Alexander McNeil.

Posts: 647
Joined: Jul 2002

pb273
Senior Member

Mon Jan 20, 03 06:04 AM

Copulas can be used to model joint distribution of multiple securities. Their most commonly reported academic usage is for risk-management in credit derivatives, insurance and event risk areas - however it doesn't do justification to the uses that it can be put to. It can be used to pair trading and statistical arbitrage modeling. Two of the most common types of copulas are the archimedian copulas and the elliptical copulas. For stocks in sectors where is a linkage or significant degree of co-movement (eg. software, semiconductor etc), elliptical copulas can be used to model joint movement. In sectors, where there is a lot of event risk of individual stocks, ex. say pharma companies which can rise or fall depending on whether it got some drug approval or won some court case on a pending patent litigation etc, archimedian copulas can be used. Now the question that can arise is "okay! i modelled the joint movement by a copula, now how do i make money out of them?" - one way which can be explored ... is to define a boundary within which say 90% of the time the joint movement lies, for instance an ellipse in the case of an elliptical copula and then whenever the joint-movement of the securities is outside the ellipse one can use options+futures (not pure futures as they are linear) and possibly other non-linear derivatives (barriers etc) that will make money if the joint-movement travels within the ellipse after some time etc .

Posts: 478
Joined: Sep 2002

mrbadguy
Senior Member

Fri Feb 14, 03 01:55 PM

Copula functions are particular functions linking other multivariate distributions. If we define n random variables:
N1 N2 N3��Nn we find a joint distribution function or copula D
D(n1,n2��,nn,sigma)=Pr [N1�Ün1,N2�Ün2��,Nn�Ünn]
First build a multivariate function from given univariate functions Fi named a copula function
Cop[f1(x1),f2(x2)��Fn(xn)]=F(x1,x2,..xn)
Risk managers use copulas to calculate joint default probability t for two or more correlated companies and related securities, once defined a default correlation parameter sigma, they try to calculate default correlation over specified and continous time intervals[0,t].
When buying a CDSwap the buyer of protection the exact default risk correlation between seller (counterparty) and underlying reference asset of CDS; my doubts are referred to fitted choices of correlation sigma and their implications for pricing CDSW, thanks for any answer.

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mrbadguy

Posts: 452
Joined: Nov 2002

newton
Senior Member

Sun Feb 16, 03 01:22 AM

Apparently copulas are useful when linear correlation (normalized covariance) gives incorrect results
as with non-normal distributions.

Why not use the covariation (Miller, 1981) instead?? For symmetric sub-gaussian distributions, the conditional expectation is just a linear span of X_t.

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newton

Posts: 59
Joined: Mar 2003

Mon Apr 14, 03 11:04 AM

Has anyone used copulas in monte carlo simulation (say of spread options)?

I work on an energy trading desk. We trade swaps and options on electricity in a number of interconnected regions. Thus we are dealing with highly non-normal underlying distributions. I thought that coplulas would be very useful to model the value of spread options between the regions, rather than using a a mean reverting jump diffusion model and using linear correlation between the regions.

Also can you get closed form solutions out this sort of modelling?

Cheers,

Yurtle

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A decision was wise, even though it led to disasterous consequences, if the evidence at hand indicated it was the best one to make; and a decision was foolish, even though it led to the happiest possible consequences, if it was unreasonable to expect those consequences.

Herodotus 500BC

Posts: 79
Joined: May 2003

mrmelchi
Member

Posts: 25
Joined: Jul 2002

Tue Apr 22, 03 05:59 PM

Hi,

I have designed VBA codes that perform the Bivariate Archimedean Copula, Trivariate Gaussian and T one, if you wish them just e-mail me. They are available for anybody else.

The spreadsheets were meant to be a simplified version of the models to explain how they work.

I hope my English is not too bad and hope to hear soon from you.

Mario R. Melchiori
mrmelchi@grupobica.com.ar

Tue May 20, 03 04:19 PM

hem, i'm not sure about that, but it seems to me that one should never use copulas for modelling (i mean that you cannot use it in order to get a closed form/analytical solution for your pricing issue). indeed, this would imply that the variables you're working with can be interpreted. Pbm : how do you interprete the standardized variables U = F_X ^(-1) (X), V = F_Y ^(-1) (Y) ????
nevertheless, you can use it as you want when it comes to simulating, calibrating and estimating. but modelling ... hum

Posts: 16
Joined: Jul 2002

amali
Junior Member

Thu May 22, 03 04:33 PM

Hi - i would be very pleased to have this spreadsheet. my email address is: amali@blueyonder.co.uk thnks and regards,

Posts: 23
Joined: Oct 2001

Fri Jul 04, 03 06:42 AM

Mrmelchi,

Is it possible that I also got your spreadsheet?
My email address is augut@adb.org
Many thanks.

Posts: 9
Joined: Feb 2002

Tigor
Junior Member

Fri Jul 04, 03 08:43 AM

Hi Mr Melchiori,
I've seen your message in contingencyanalysis.com which says that you have spreadsheets for bootstrapping and interpolating yield curve using linear and spline. I've tried to contact you at this email address: mrmelchi@ssdfe.com.ar, however it seems that the server doesn't recognise your email. Could you send me those spreadsheets, please?

Thanks.
stigor@bi.go.id
tigor@bloomberg.net

CB
Senior Member

Posts: 437
Joined: Jan 2003

Sun Jul 06, 03 05:26 PM

Any other alternatives are much better than copulas?

Posts: 7
Joined: Jul 2003

Gracely
Junior Member

Fri Jul 11, 03 02:13 PM

Any idea how we define the boundary within which say 90% of the time the joint movement lies? Thanks.

mrmelchi
Member

Posts: 25
Joined: Jul 2002

Mon Jul 14, 03 08:39 PM

In the Gumbel bivariate copula case:

Ctheta(u,v)=exp(-((- ln u)^theta + (- ln v)^theta)^(1/theta))

Example:

Set C(u,v)=0.90
Set u>C(u,v)= 0.91
Set theta=1.85 equivalent to Kendal�s tau=0.46

exp(-((- ln 0.91)^1.85+ (- ln v)^1.85)^(1/1.85)) = 0.90

Solve v using:

v=exp(-((-ln 0.90)^1.85 - (- ln 0.91)^1.85)^(1/1.85))=0.958523

so:

CGumbel1.85(0.91 , 0.958523)=0.90

Posts: 7
Joined: Jul 2003

Gracely
Junior Member

Tue Jul 15, 03 11:17 AM

Thanks.

Let's see if I get it correct - Assume Gumbel copula with theta 1.85, there is a 90% probability that v < 0.958523 conditional on u = 0.91.

But why set u > C(u,v) = 0.91?

If I want to build a contour plot for, say Gumbel copula, how can I do it? Correct me if I am wrong, but my original idea was to identify the 90% contour curve, ie f(x,y) = 0.9, as the boundary... For some reason I cannot get this bit right...

Posts: 7
Joined: Jul 2003

Gracely
Junior Member

Tue Jul 15, 03 12:22 PM

Going back to CGumble1.85(0.91,0.958523)=0.90. This guarantees that we will always choose the points on the upper tail, where both u and v are close to 1. How about lower tail? Or other points that are unusual while one, either u or v, is usual? I am really confused about this 90% boundary issue...

How do we reconcile what we see in the contour plot of a joint distribution, where contour curves are oval-like (not necessarily an oval), with the level curves of a cumulative distribution function of a copula?

Posts: 112
Joined: Sep 2003

mrmelchi
Member

Posts: 25
Joined: Jul 2002

Tue Jul 15, 03 08:26 PM

You wrote:

Let's see if I get it correct - Assume Gumbel copula with theta 1.85, there is a 90% probability that v < 0.958523 conditional on u = 0.91.

Assume Gumbel copula with theta 1.85, there is a 90% probability that v <= 0.958523 conditional on u <= 0.91.

You wrote:

But why set u > C(u,v) = 0.91?

When u tends to Copula�s value v tends to 1 and vice versa.

Tue Sep 23, 03 08:37 PM

Copula's are also used to calculate VAR ....aggregating across different risk distributions

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Bash on Regardless

Posts: 48
Joined: Feb 2003

taeglich
Junior Member

Posts: 1
Joined: Oct 2003

Sat Oct 25, 03 07:20 AM

Hello to all!

read about this wonderfull spreadsheet! ... is it still available?

Thanks a lot in advance!

Thomas (email: h9750735@wu-wien.ac.at)

Edited: Sat Oct 25, 03 at 11:00 AM by taeglich

Mon Nov 03, 03 10:22 AM

I have a spread sheet for Gumbel copula.
Do any body have spreadsheet for Archimedian, T-copula ....etc?
I can have a swap deal with him/her.