Has anyone built a semi-analytic CDO using student-copula under a single factor model?I understand that for Monte-Carlo, the upgrade from Gaussian to Student-t is simple, involving the conversion of the latent variable to a student-t random variable and then using the student-t cdf to generate time-to-default. Easy.However, it seems that the upgrade of a semi-analytic Gaussian is not quite so straightforward. (By semi analytic I am referring to FFT or Recursion). I would have guessed that the Gaussian cdf and inverse cdf's are swtiched to student-t cdf and inverse cdf, but the literature I have read says this is not so. The literature I have encountered says that the problem beomes a double integral (one for latent variable, one for a chi-square variable). Has anyone dealt with this before?BM

For recurison methods the update is a bit more involved. You have to adjust the calculation of the conditional PDs and to integrate over one additional dimension because of the additional Chi^2 variable introduced via the Student-t copula. Andersen, Sidenius and Basu wrote all that down in "All your hedges in one basket". You will find it on www.defaultrisk.com.Regards

Thank for this DDoom. I followed up on the paper and you are perfectly right.I still have a conundrum to resolve: Hull & White's paper ("Valuation of a CDO and an n-th do default CDS without Monte Carlo Simulation") does not use double-integration. Instead, they proceed to substitute the PDF, CDF and inverse CDF's from Normal to student-t, adding the caveat that the community and idiosyncratic variables need to be rescaled to achieve unit variance. My first examination of this is that the model of Andersen et al is different from H&W's in that Andersen et al's community and idiosyncratic components are Normal and are united with a student- copula; H&W, on the other hand, permit Normal or student-t community variables and Normal or student-t idiosyncratic variables. H&W do not do any integration over a chi^2 variable.The H&W route is seemingly simpler but the two models, though both student-t, appear quite different.Anyone any thoughts on this?

It has been a while since I read the H&W paper you mentioned but I thought that the models are similar in spirit, at least for Gaussian random variables. H&W circumvent the need for numerical integration with the evaluation scheme they propose but the results should be the same, again under a Gaussian copula.H&W's model should be easier to implement, but Andersen et al's model is easier to upgrade (for example calculation of Greeks or random factor loadings).

Thanks again for your reply - I agree that, for Normal random variable, H&W and Andersen's models are similar in spirit.I re-read H&W and they do indeed use numerical integration over the community variable (M) for their implementations (both their implementation I and II). So the conundrum still stands in my mind: why do Andersen et al need to integrate over a chi^2 AND community variable, whereas H&W get away with integrating over community variable alone for a student-t CDO?Any views appreciated,BM

BM==========================================================So the conundrum still stands in my mind: why do Andersen et al need to integrate over a chi^2 AND community variable, whereas H&W get away with integrating over community variable alone for a student-t CDO?==========================================================I'm interested to know if you ever reconciled this point?Have you looked at:http://laurent.jeanpaul.free.fr/Credit_ ... 005.pdfJPL neatly summarises the 'Student-t' Copula on page 12 and the H&W 'double t' model on page 14. At a glance the 'Student-t' Copula assumes Gaussian RVs for both the common and firm specific variables with a Student-t to define the copula for generating the joint density across the obligors. As a result it has two conditioning factors that need to be integrated across.The H&W 'double-t' model uses Student-t densities to define the common and firm specific RVs and an unspecified distribution for the unconditional default probability at point in time 't'. What H&W assume (Gaussian?) for this distribution doesn't seem to be specified in their paper. This seemingly only has one conditioning factor to be integrated across.It might be a misnomer to refer to the H&W double-t as a Student-t copula model? Does the Student-t copula refer to the distributional assumption on the unconditional default probability with time? In JPL speak the distribution used to invert the Vi term.

vespaGL150,I believe you are spot on and the JPL reference answered the question.In short, ASB use Normal Common, Normal Idiosyncratic, student-t Copula.H&W 's "double-t" uses Student-t Common, student-t Idiosyncratic, student-t copula.I surmise that ASW's extra conditioning results from the need to perform the student-to-normal mapping, hence H&W is far simpler to implement (albeit it is a different model).I could not find and explicit statement in H&W's paper to their copula either, but JPL spell it out for the double-t.Thanks again for the post,BM

Right, thanks vespaGL150 for that post. Maybe I'm missing something here but in the double t-copula in H&W's paper, don't you have to determine the distribution function for Xi numerically (the distribution function termed H in JPLs slides)? It really boils down to what distribution the sum of two t(0,1) variables have. I've tried characteristic functions but that expands to really awful expressions. In addition, ASB's t-copula will give you a nice fit to the distribution you get from monte carlo. This is not the case for H&W's version according to my implementation.Cheers

Karwitz - agreed and it's bugging me.Neither JPL nor H&W specify what they assume for the distribution of the sum of the idiosyncratic and community independent student-t RVs with in general different dofs - Hi(vi) in JPL parlance and Fi(xi) in H&W parlance. This may well be, as you point out, because the true distribution is a function of the dofs chosen in each case.Conjecture here (dangerous!) but H&W may simply use/assume a Gaussian so that their model reverts to the single-factor Gaussian when t_M=inf and t_Zi=inf. Any views from those closer to this work? It may be worth contacting H&W / JPL to resolve the distribution issue.

In addition to the distribution issue discussed, I also found the H&W recursive method not as fast as I had expected. It requires numerical integration of the probabilities of loss in each bucket with respect to the common factor. Because the probability of the loss in each bucket is calculated recursively, the integration of the loss probability of each bucket involves the evaluation of the probabilities of all the other buckets as well.

snowwhite,Maybe I don't understand your concern, but here's what I suggest you do:For each discrete point in time:For each bucket you integrate over a common factor Y. For each Y=y you have to calculate the conditional probabilities for all buckets. Why don't you just cache the conditional probabilities for each bucket and Y=y? Then you only have to perform the bucket calculations the first time the integration enters with a new value of Y=y. Or is it something else bothering you?

QuoteOriginally posted by: vespaGL150BM==========================================================So the conundrum still stands in my mind: why do Andersen et al need to integrate over a chi^2 AND community variable, whereas H&W get away with integrating over community variable alone for a student-t CDO?==========================================================I'm interested to know if you ever reconciled this point?.If you take a N(0,1) r.v. X and divided by a chi-squared distributed (n-deg) r.v. Y, the ratio of these two is now student-t distributedwith n-deg of freedom. So both HW and ASB are doing the same thing?.

Hi,For the 'double t-copula' I am interested to know if you guys found out about the unspecified distribution for the unconditional default probability at time 't' ?I assumed Gaussian, but the results did not match with HW results... please let me know... any hints would be appreciated ... Thanks,Tarun

To my knowledge there is no closed form solution for the distribution of a sum of two students rv (especially since student rv are not stabe under convolution). An easy way to estimate it is to compute non paramterically the distribution of such sums and then use the empirical cdf that you found in your CDO pricing algorithm.QuoteOriginally posted by: tarunmakhijaHi,For the 'double t-copula' I am interested to know if you guys found out about the unspecified distribution for the unconditional default probability at time 't' ?I assumed Gaussian, but the results did not match with HW results... please let me know... any hints would be appreciated ... Thanks,Tarun

Thanks for the reply meteor.I went through 2 papers and both mentioned that the distribution function for the idiosyncratic factor needs to be "computed numerically"pardon my ignorance, but I am not sure how to go about doing this. I read that I can use Newton Raphson for the same. However, I do not know whats the non-linear equation that I am trying to solve in the very first place!!I have the following and I need to find the function F_inv to implement the model: P(tau < T | V) = CDF(F_inv(P(tau<T)) - rho*V)/sqrt(1-rho^2)) where: P(tau < T) = 1- exp(-Lambda * T) V is the common factor F is the distribution function of Vi Vi is the firm specific idiosyncratic factorCan you tell me the non-linear equation on which i will apply newton raphson? you mentioned about the "emperical CDF". How do I get such an emperical CDF... is this more in line with the Implied Copula/perfect copula paper of Hull and White? please guide me to any appropriate literature ... addressing numerical computation of a distribution function.thanks,tarun