Dear Wilmotters',

I would like to come up with a solution to measure the likelihood of two economical scenarios arising simultaneously. For instance, how likely it is that a drop in the S&P 500 will cause a drop in the WTI price?

Of course, correlation is limited and would not fit for that purpose as it will only give me the linear dependency between these risk factors. Instead, I am more thinking about copulas.

Say I have n risk factors. I can graph their (historical) marginal distributions. Assuming a multivariate copula structure (which one???), I can then get the joint distribution of these n factors which is essentially what I am interested in, right?

The problem is, how to choose the copula structure? Is there other solutions, non-parametric, similar to these one? Indeed, if I am not mistaken this approach is parametric as you assume first a parametric (say Gaussian) copula structure that you then fit to your dataset.

Could someone enlighten me on this?
Many thanks in advance.    

Ive had good experience with fittinggaussian mixture models to the empirical density. Mixture model can exactly 'memorize' your samples by adding lots of mixture components (overfitting). To counter that the likelihood function of the mixture fit typically had a penalty term on the number of variables in the model. (number of mixture components). Bayesian information criterion (BIC) is popular. You can easily find implementations in R and Pyhton SciPy 

Another option is to use a Kernel density: replace all historical data point with little Gaussians. This method has a bandwidth parameter (the width of the Gaussians) you need to set, there are some heuristics for that, using cross validation is a very good one: use a subset of your data to model the density, use the remaining points to compute the likelihood of those points given the fitted density. Having a likelihood allows you to optimize it.

@outrun Thanks for your answer. Indeed I already had some exposure to Kernel density estimation. However if I am not wrong, this method and the ones presented above (BIC-based and Gaussian mixture) only focus on marginal distributions fitting. 

In my case, I would like the focus to be on the joint distribution so that I can then get some probability estimation on some scenarios like "Risk Factor 1 loses 10% while Risk Factor 2 loses 15%, simultaneously".

No, they fit the full distributions. Lots of examples might give 1d plots, but the methods can fit any dimensional density.

My bad, you are right, apologizes for that.

Indeed I have just come across a nice presentation about Multivariate Kernel density estimation (http://www.buch-kromann.dk/tine/nonpar/ ... ltidim.pdf) which would indeed correspond to what I am looking for!

However, I am still parsing the web to find some multivariate gaussian mixture models. For the latter however, I am a bit skeptical with the "Gaussian" term in the model name, implying that a "Gaussian" distribution is already assumed before the fitting, and is therefore not model/distribution free, is it?

The first link I gave had a slide where you see a picture of 3 2d elliptical gaussian.

These methods are flexible enough to approximate any (non gaussian) distribution with arbitrary precision.

What programming language are you using? Eg in the pyhton scikit learn package you can find all these merhids.

eg, http://scikit-learn.org/stable/modules/mixture.html

Great news, I am using Python

I found out more on the web on the Multivariate GMM, and indeed it is also fit for purpose!

Thanks a lot

PS : I wish Wilmott forum could have some ratings system / Thumbs up-down / +/-1 to be able to reward users, similar to stackoverflow... PW, are you listening?!

Exactly, that feature would be a lot of fun! I've also asked for it before