A working paper, Computational complexity and informational asymmetry in financial products, Sanjeev Arora, Boaz Barak, Markus Brunnermeier, Rong Ge. sheds some light on the complex mathematical models upon which credit default obligations and other derivatives are based.

What Arora et al. prove is not only are many derivative mathematical models impossible to compute, never mind in real time, because they require more computing power than the world possesses, the missing information to run a mathematical model is a very good place to cheat with.

To understand what CDOs, derivatives are, see this post, complete with video tutorials. For some background on the mathematics behind derivatives, read We Want the Formula and this one on some of the probability functions.

Onto the paper. Firstly this quote:

One of our main results suggests that it may be computationally intractable to price derivatives even when buyers know almost all of the relevant information, and furthermore this is true even in very simple models of asset yields.

They ain't talking about your new PC cranking through these calculations, they are referring to massive supercomputers.

This result immediately posts a red flag about asymmetric information, since it implies that derivative contracts could contain information that is in plain view yet cannot be understood with any foreseeable amount of computational effort.

So, individual investors or even online brokerage firms can kiss it goodbye in verifying these values easily due to computational complexity of the algorithms themselves.

The practical downside of using derivatives is that they are complex assets that are difficult to price. Since their values depend on complex interaction of numerous attributes, the issuer can easily tamper derivatives without anybody being able to detect it within a reasonable amount of time.

The paper points out current variations in price can be 17% and they can give widely variable pricing evaluations, even within the same bank issuing the same tranch (little slices of rated assets) in a derivative.

Now here is the reason this paper is so mind boggling damning and I'm translating from computational research to Populist terms.

There is no friggin' way to crank these numbers in these models with typical processing power. There are not enough computers in the world. That means not only are many results invalid, but this:

Designers of financial products can rely on computational intractability to disguise their information via suitable “cherry picking.” They can generate extra profits from this hidden information, far beyond what would be possible in a fully rational setting

Translated to Populist blog speak: Derivatives are a way to scam and screw investors out of their dough through a lot of high fallutin' gobbledygook that sounds real technical.

How do sellers scam on CDOs? By taking a few of the ones they are peddling, a subset, and stuffing them with more toxic assets than the other ones. To load the derivative dice, one adds , to be precise. This puts that particular CDO at a much higher probability of default. So, instead of mitigating risk, one can increase risk! Supposedly one of the justifications of derivatives is to mitigate risk. Ho ho ho!

Now, because there are only some CDOs which are rigged, finding which subset of them is, in a sea of CDOs....computationally impossible. It wouldn't matter if you had gobs and gobs of super computers, and billions of years, you ain't gonna find them because one has to go through all sorts of permutations to calculate and determine them. To make matters worse, the CDO seller, can stuff CDOs with a subset of worthless assets in a way that even if one had all of the computers in the world and could crank through , it won't pop up in the detection algorithm anywho due to the probability spread. In Math geek, this is technically a NP-Complete problem.

In layman's terms, the equation simply means even with a huge bunch of honking fast computers, one cannot get a concrete result or answer.

Then asymmetric information means that one guy has more info that you do when making a transaction. Say the seller of a house knows it has termites, but you don't and buy the house thinking you got a great deal because it was below market value.

Surely there is a way to guarantee these derivatives are not tampered with right? Uh, no! If ya can't prove these things are rigged, how ya gonna guarantee they ain't? Even more interesting, let's say a patsy buyer gets wiped out and suspects he's been scammed, this plan is full-proof because there is no evidence, thus nothing the screwed over buyer can do to get their money back.

Is there anything that can be done to make derivatives a computationally bounded problem to make them legit? Indeed there is, say the authors. One is a logic statement, an exclusive OR, although I don't recall seeing such a thing in any probability or statistical formula...(yet, there is integer mathematics)....and then they define a more realistic bound, what is called a tree of majorities.

Now, on regulation, here on EP we've called for the regulation of the mathematical models themselves. How can one sell a product built on bad math, that is not even valid by the mathematical properties themselves? One could also incorporate the ability to validate a price computationally as part of a regulation requirement. The above type of derivatives outlined in the paper? Plain just ban them would be my druthers.

The rest of the paper is an exceptional read, but be warned, it does use many computer science theoretical terms, equations and advanced probability and statistical concepts. I've broken down a few key concepts above.

I'm personally thrilled to see some computer scientists look into financial derivatives! When we first reviewed them on this site, we were shocked that the Mathematical and Scientific community had not flagged many of these models for being theoretically flawed, from the mathematics themselves. Good work Arora et al.!

The authors have also put up a derivatives FAQ of the implications of their paper.

^{h/t PhysOrg & Washington's blog}