Wired on the Gaussian copula

Because this article is spamming the internet today, I decided to read Li’s paper and learn what the heck is this Gaussian copula.

For five years, Li’s formula, known as a Gaussian copula function, looked like an unambiguously positive breakthrough, a piece of financial technology that allowed hugely complex risks to be modeled with more ease and accuracy than ever before. With his brilliant spark of mathematical legerdemain, Li made it possible for traders to sell vast quantities of new securities, expanding financial markets to unimaginable levels.

And anyway, here is the paper referenced in the article.

Firstly, so much for the sensationalism: so far as I can tell, the paper doesn’t say anything worthy of a Nobel Prize — but still it is mildly interesting. In fact, the whole point of the paper appears to be to introduce to the finance community an already known method for solving the inverse problem of distribution marginalization, that is, (non-uniquely) go from marginal distributions back to the joint distribution, by specifying a mediating copula that captures marginal-invariant joint structure. The technology is very straightforward, and Li didn’t invent it.

That aside, I did wonder, why the heck go through the motion of constructing a Gaussian copula (as in the article) if you assume your marginals and joint are all Gaussian to begin with and all you wanted to capture is the covariance matrix; you can just specify the joint Gaussian explicitly. It seems like a totally pointless exercise. After reading the paper though, I see that wasn’t really Li’s entire suggestion at all. He’s being descriptive rather than prescriptive of what his firm already did by casting it in the language of copulas, an interpretive generalization that allows for potentially more accurate modeling (of non-Gaussian marginals and complicated joint structure if so desired).

Now on to the accusations. The article says that Li tried to “model default correlation” using credit default swaps rather than ratings agency data. It turns out that wasn’t even a problem being solved in this paper. He suggested to use CDS market data to get implied marginal distribution, an established practice. As for how correlation is obtained from limited data, you’d have to blame one Greg Gupton:

Having chosen a copula function, we need to compute the pairwise correlation of survival times. Using the CreditMetrics (Gupton et al. [1997]) asset correlation approach, we can obtain the default correlation of two discrete events over one year period.

However, it is true that there is something funny going on with the concept of using market pricing to price other market instruments, when the only novel input for all of them must be what little information is collected from actual due diligence. A classic case of Garbage In Garbage Out in statistical modeling.

As somebody elsewhere wrote, this sort of thing would not pass muster in “real” engineering design. We’ve seen that dichotomy before between the absolutely error-free stricture of “hardware” design (chips and bridges) vs. the more lax attitude toward “software” design (operating systems and capital market systems). Maybe this dichotomy needs to go away.


  1. David Crow
    November 11th, 2010 | 2:08

    Great post. Li did not invent the Gaussian copula, he merely applied it to an important finance problem.

    I haven’t read the article (yet), but I think the value of Gaussian copulas resides in their ability to model bi- or multi-variate relationships when each of the marginal distributions is Gaussian, but their joint distribution is not. One example of this is “tail dependency”, when extreme events influence one another more than a bivariate normal distribution (with the relationship described by the linear correlation parameter) would predict. This is presumably the case with credit defaults.

  2. December 8th, 2012 | 14:10

    Only wanna input on few general things, The website design and style is perfect, the subject matter is real fantastic : D.

Leave a reply