Like the scene from the The Return of the Secaucus 7, earlier I was asking for details on the actual mathematics upon which derivatives, CDOs (Collateralized debt obligations) are based.

Wired Magazine has answered the call in the article Recipe for Disaster. This article outlines the actual mathematical formula, a Gaussian copula, upon which so many derivatives are based.

In 2000, while working at JPMorgan Chase, Li published a paper in The Journal of Fixed Income titled "On Default Correlation: A Copula Function Approach." (In statistics, a copula is used to couple the behavior of two or more variables.) Using some relatively simple math—by Wall Street standards, anyway—Li came up with an ingenious way to model default correlation without even looking at historical default data. Instead, he used market data about the prices of instruments known as credit default swaps.

You must read the entire article, yes they mention mathematics, but they are explaining it all in layman's terms.

One thing I did not know, pointed out in the article, is that there are no limits on the number of CDS (credit default swaps) that can be issued against one borrower. CDSes are literally unconstrained by are subject to mark-to-market.

Back to the formula it appears that it was not based at all on historical data of actual mortgage defaults. It is purely based on the assumption that credit default swaps are correctly priced and accurate.

In other words, credit default swaps are little insurance policies on the possibility of default. One can issue multiple little insurance policies on just one data point (i.e. one borrower). So, this formula based it's risk assessment by claiming the final total price on another variable, i.e. a CDS, which has no bounds on issuance and is traded in an open market, is in perfect equilibrium. In other words, the gaussian copula used assumed the CDS is correctly valued.

Pause for a moment. See a major assumption flaw yet? Now, in our bubble world does anyone believe anywhere, the open market is correctly priced to the actual real values or risk? Maybe in a 11-dimensional universe during a delta function.

Even worse, CDSes are a new vehicle as well, only in existence about a decade. Nice historical track record, empirical data there too!

So where is the bad math? The bad math is trying to boil down the world as having a scalar, or just one correlation multiple. It just don't work that way, the world is not static and of course one cannot derive correlation from yet another vehicle, which of itself is not proved to historical data or even correlated in quantity to the actual underlying variable. Call it a mathematicians pipe dream or look at it as trying to reduce the universe to a 1 dimensional time invariant point on a pin.

The actual paper is here. The abstract:

This paper studies the problem of default correlation. We first introduce a random variable called "time-until- default" to denote the survival time of each defaultable entity or financial instrument, and define the default correlation between two credit risks as the correlation coefficient between their survival times. Then we argue why a copula function approach should be used to specify the joint distribution of survival times after marginal distributions of survival times are derived from market information, such as risky bond prices or asset swap spreads. The definition and some basic properties of copula functions are given. We show that the current CreditMetrics approach to default correlation through asset correlation is equivalent to using a normal copula function. Finally, we give some numerical examples to illustrate the use of copula functions in the valuation of some credit derivatives, such as credit default swaps and first-to-default contracts.

Can't be blamed? The formula creator, David Li, ran back to China and won't can't talk to the press. As I wade through the mathematics for myself, I'll probably post more later. Other math geeks reading this, please post your insights.

There is also an audio podcast on the mathematics, but with the same Wired author.