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New to this forum & hope someone can point me in the right direction.I think I understand various different copula structures & their uses in correlation modeling. So, in the case of credit modeling where t1, t2,... are default times for different names we have that the probability P(t1,t2,...) = C(P(t1),P(t2),...,params) where C is the copula. In the case of a Gaussian copula, we have P(t1,t2,...) = Normdist(Norminv(P(t1)), Norminv(P(t2)),...,params)I think I also understand the following argument for modeling default correlations:Define X1 = Norminv(P(t1)) which is therefore a normally distributed variable, similarly X2,etc. By inverting we have ti as a function of XiThen model Xi as a sum of normal variables Xi = Ai*V + Bi*Vi. This captures correlation between the Xi because the stochastic variable V is common to all the Xi. Then work out (various methods) P(X1, X2,...) and we can generate P(t1,t2,...)What I can't work out is the algebraic relationship between these two approaches:1) P(t1,t2,...) = Normdist(X1,X2,....,covariance matrix)2) P(t1,t2,...) = integral over V(product of P(Xi|V))Can anyone point me in the direction of a paper that carries out the algebraic transformations to get from 1 to 2? How does the covariance matrix in 1 get transformed into the appropriate values of Ai and Bi in 2?Thanks,MrH

Let me try to shed some light...Ususally in the 1 factoer setting With V and Vi are iid N(0,1) so that Xi is also a N(0,1) and cor(Xi,Xj)=rho_i*rho_j. You can clearly see from that line that the model is equivalent to your model 1. The trick to get rho_i is to use the Choleski decomposition of the X covariance matrix.Note that to find the unconditional joint default proba P(t1,t2,...) you do: integral over V sum( P(ti|V) ) .Where P(ti|V)= with F_i(t) the default proba of i at time t.For a good intro you should check the paper of Gibson (Understanding the risk of CDO) for more complex thing check Brutshell, Gregory and Laurent ("A Comparative Analysis of CDO Pricing Models") and if you want to have some more info about copula check the book of Nelsen ("An Introduction to Copulas").hope it helps

Thanks, meteor. I guess what you're telling me is that there's no substitute for hacking through the algebra myself to see how rho_i is related to the X covariance matrix. But thanks for the Cholesky pointer - I'll work my way through at some point.And the derivation of P(ti|V) involves partial differentiating the gaussian copula function, right?I'm just surprised that none of the papers or books that I've seen ever demonstrates all of that algebra or references anywhere that does.

Sorry, I realise the first paragraph of my previous post is nonsense since the relationship is obvious. That was the step I was missing - thanks again meteor.

QuoteAnd the derivation of P(ti|V) involves partial differentiating the gaussian copula function, right?You can see it that way (as P(u|V=v)=dC(u,v)/dV evaluated at v) but in your case the computation of P(ti|V) doesn't require any partial derivatives since everything is known in this expression.By the way you could check also the Schonbucher book as it has a chapter which provides a good overview on the copulas.