Pricing a CDO - Not only Bad Math, Bad Computation too

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 \mbox worthless crap assets = \sqrt{\mbox total assets}, 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 \mbox{n-sized subsets of} N, 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 P \not= NP 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

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Comments

missing info for buyer amplified further

To compute, run a mathematical model to create a CDO, it is required to leave out some information. Else, one cannot crank the numbers, in some, at all in others, not in real time, i.e. not in a day or two. Think about encoding your home movie to make a DVD in 1997. Think about how that took 8 hours to do. Then raise that time concept you have in your head by an exponential amount. In other words, you will be Rip Van Winkle waiting around for some of these things to finish processing on your computer if you do not constrain the data and model before letting it rip on your PC.

So, automatically the ones creating the CDO already know the missing information. They had to limit it to even create a CDO on their computers. They had to add constraints to their mathematical models.

This is before intentionally stuffing a CDO with one standard deviation, assuming a random Gaussian distribution, of toxic crap waste to pretty much guarantee that CDO will blow up and default.

Now with synthetic CDOs, I went over how modelers were using CDSes, evaluated on a mark-to-market, at the close of each business day. The problem is those CDSes are not 1:1, i.e. there isn't one CDS issue per home which pays out when that mortgage goes into foreclosure. It's not a 1 to 1 relationship. That violates the mathematical properties of a Copula (a), which is the "mathematical black box" which they are using to calculate some synthetic CDOs, and (b) assuming they reflect true market value and the probability of default....well, that's some serious statistical fiction, since they haven't even been around long enough to prove CDSes are correlated over time to actual default rates.

I also am wondering if they reason the Mathematics and Science communities did not do some deep investigative work previously into structured finance....

Could it be that Goldman Sachs pays these guys $200k+ per year while the Scientific and Technical traditional jobs are busy trying to get rid of Americans with these backgrounds and offshore outsource their job?

Who would not want to get some real buckos for one's skill and brain, even if it is spinning some serious statistical fiction.

...and they worried about those Russians in 1998, now out of a job and broke....selling Nuclear materials and secrets....

of course Nuclear Scientists weren't gonna sell those secrets....how could they make the mortgage if they were then dead? Hell no, they went to the Mafia and rigged up many a statistical scam and a hell of a lot of online spam.

More to the point:

oopsy, I guess they missed all of the Americans from 2000.....out of a job and broke.....selling their mathematical souls to pay the mortgage!

We need a new billboard, made along the Got Milk? ads...

Got Geeks? They ain't for offshore outsourcing their jobs anymore.

(China btw is front loaded with Geeks on their economic strategy teams).

(Geeks being slang for STEM occupations)

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Most of modern financial theory has been called into question

by this crisis. FYI - I just listened to an interview by Prof. Steve Keen where he talks about engineer capitalism - check the middle row under Debtwatch.

RebelCapitalist.com - Financial Information for the Rest of Us.

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I hate podcasts, as well as audio interviews

for some reason, which is strange because I'm highly geared towards sounds...

It takes so much damn time, it's linear, I guess I'm just a hyperlink, 3 minute youtube type of person these days. :0

If it's something useful, I usually want to reread it or some sections a couple of times, so good old fashioned paper works best for me (blogs being also paper, i.e. stuff on a sheet). (Maybe it's because I review so many "so called" documentaries everywhere to find some good juice so I'm burnt out, I don't know)....

Anywho, what's the gist? When you say "theory" do you mean derivatives, structured finance? That would be no surprise but in terms of all economic theory, I haven't seen anything "break" here, I've seen a lot of stuff proclaimed as something when....it isn't because it violates the actual theory. (i.e. the Stimulus is Keynesian....not so, I don't have the breakdown but because there is no "hire Americans, buy exclusively America" imposed in at least the direct spending, which isn't even half of the $787B, I think it was ?? $30B for infrastructure? anywho, all of that's not true Keynesian.

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No, I mean basic modern finance

Much of financial theory assumes rational investors and efficient market hypothesis and other crazy assumption like everybody can borrow money at risk-free rate. Modern portfolio theory - CAPM - Security Market Line and basic normal distribution assumptions have been shaken to their core.

RebelCapitalist.com - Financial Information for the Rest of Us.

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From quant to quack is only a matter of C and K

The general assumption is that we look after our own interest thus creating less risk via derivation.
Building ways to hedge the risk via the derivation of an asset, is the prevalent view at decision making level, educational institutions and government policy level.
The missing link here is, those who can create the risk reduction are the same ones creating higher risks, and able to cover-up using rating agencies as clown in the financial farce. The wheel of fortune continues.......
Professor Shiller(Case-Shiller index) will argue, derivatives will reduce risk if well placed in the asset risk time value.
The problem of CDO CDS's we are dealing is as much faulty quantification, faulty derivation, and dishonest valuation from the parties who are trusted to operate as self-regulated entities, as it is faulty acceptance of the present financial fallacies instilled in people.
The rating agencies allowed their profits to surpass their chartered mandate. At this point we are better off without chartered rating agencies.
It is my view, self-interest prevailed with the purchase of optimism from products that could have had added cushion if the dishonest ways in which the system operates will no always fail in the backs of tax payers to "bail out" the incompetents.
Keep up the cover up at the tune of mambo jambo!!!

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will do!

If someone else sees a finance/mathematics paper that implies something fictional, but they do not have the STEM background to read it, or understand the math contained within, fire it over here and we'll take a look at it and try to decipher to English.

I think we've got a huge problem via the concept of derivatives vs. what they are actually made of...
I don't see how writing a bunch of gobblygook that isn't sound science helps anyone mitigate anything beyond pocketing some nice short term profits.

It would be different if the particular derivative itself was a sound, verifiable model, then one could argue the function of it....but nonsensical functions themselves? How can they defend this?

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