The Gaussian copula and financial risk correlation

Felix Salmon has an excellent article; here is one excerpt:

"The corporate CDO world relied almost exclusively on this copula-based correlation model," says Darrell Duffie,
a Stanford University finance professor who served on Moody's Academic
Advisory Research Committee. The Gaussian copula soon became such a
universally accepted part of the world's financial vocabulary that
brokers started quoting prices for bond tranches based on their
correlations. "Correlation trading has spread through the psyche of the
financial markets like a highly infectious thought virus," wrote derivatives guru Janet Tavakoli in 2006.

Comments

not a bad article at all. ignores a few fine points -- for instance, a lot of calculations were done using a stochastic credit-rating change model rather than using implied default probabilities and correlation cds -- but the discussion of the model itself is correct. this is actually quite remarkable for a popular publication.) ignores the fact that ratings agencies, not models, created "AAA securities", and that ratings agencies were engaged in misrepresentation and fraud. (Why we haven't seen more prosecutions here is beyond me.) (this is not an exclusive list of what's ignored.)

The article overstates Li's intellectual contribution. Nobody has ever won a Nobel for applying a theory decades old to a specific problem.

The history of economics for the last 70 years is the history of the fallacy of mistaking a simple math function for real phenomena in the market -- the fallacy pointed out by Coase, Knight, Mises, Hayek, etc., etc. The fallacy found in the modeling of uncertainty, the economic agent, preferences, learning, equilibrium, unemployment, recession, growth, capital, etc., etc.

Got to love this!

Lucas critique

Credit derivative models presume (i) a market consistency that implies no gains from unsubsidized trading alone--an innocuous enough maintained hypothesis--but also (ii) an inability to deduce from that presumed consistency what the dynamically maintained hedge portfolio must be or even that there might exist such a portfolio (market incompleteness). The Merton-Black-Scholes world has both consistency and hedgeability, but it too suffers from lack of the latter when returns aren't Gaussian, e.g., when there are black swans/Mandelbrot effects (exponential Levy models).

Gaussian copulas are used for getting to the assumed consistency in order to have the pricing, but still leave the hedgeability issue formally untouched.

The soup (model pricing) tastes delicious (nice consistency with certain market prices by calibrating model parameters apropos) but there is no recipe on how to make it. These points sound "techy" but they drive to the heart of whether the model is producing a price for a contract that is derivative from market assets (redundant security in the terminology of standard texts) or for an entirely new security as unrelated to those already in the market as, say, stocks are unrelated to bonds.

Since the model framework itself is saying that the second is true, it is also saying that the modeling itself provides no means of directly benchmarking the model’s price against the “real world† of other assets for validation; it merely exists (with a presumptive consistency to the world only deducible from knowledge of the full model framework itself). So if the skeptic says “Show me†—i.e., “What’s the hedge?†—the modeler’s response is “Gaussian copulas, blah, blah†¦Trust me.†

Tyler, your linking is off a bit.

Here is the link to Janet's SEC submission.

http://www.sec.gov/comments/s7-04-07/s70407-1.pdf

(If you click on "wrote" in your link you get it, but if you click elsewhere you simply get Janet's bio.

I highly recommend reading the SEC submission, because it covers a number of recent fatal flaws, the reasons for them, and what to do about them.

Salmon's article is good, but Janet's is by far more comprehensive.

"I presume that you are claiming that there are mathematical defects with the IS-LM model, or that it is unrealistic. Precisely how?"

Bob -- there is a huge literature on this. I'm not going to summarize it here.

An interesting article, and correct that one should not blame Li, even if the
article kind of implies that maybe one should. The underlying math is old,
Sklar's Theorem having first appeared in 1959, and people like Paul Embrechts
were posing substantially more complicated copulas in the late 1990s, which
were picked up initially by the Swiss banks, multivariate non-Gaussian ones,
before the more simple-minded bivarariate Gaussian was cooked up by Li.

So, of course even the more sophisticated ones would ultimately face the same
sort of Talebesque critique that the simpler one did, just the widespread
adoption of one of them would have allowed a bit more slack than in the simple
one. As the article notes, it may have been its simplicity that allowed it to
get adopted so widely.

A further point is that underlying the problems of the CDO market was that they
were built on credit-default swaps themselves, many of which were leveraged to
30 to 1 and even worse ratios. Throwing Gaussian assumptions on top of such
stuff to cook up nicely tranched CDOs was a mess to begin to with. I suppose
Li should be blamed for that (and he does seem to be hiding out in Beijing now),
but I would agree that it was the other quants and especially the crooked ratings
agencies that are much more to blame here.

From the Laidler / Backhouse paper on the IS-LM model:

"In The General Theory, Keynes not only expounded his own ideas. He also re-wrote the
macroeconomics that had come before, and labelled the result “classical economics†. In the
process he presented a theoretical critique of propositions that were entirely static and atemporal.
In the absence of this distortion of what macroeconomics was like before the General Theory,
Keynes’ own contribution would have been perceived very differently, and it would have been
much more difficult for such early exponents of IS-LM as Harrod (1937) and Hicks (1937) to
present this model as capable of capturing the fundamental issues at stake between Keynes and his
predecessors. As it was, none of the functional relationships that lay at the heart of IS-LM, in
either its “Keynesian† or its “classical† form, explicitly modelled issues having to do with time,
nor did the framework itself permit time to be brought into the analysis of their interaction. In that
model the implications of economic activity taking place in time were all in the background, and
those whose intuitions were trained on its manipulation came to view the macroeconomics that had
existed before 1936 as being just as amenable to characterisation in comparative static terms as
Keynes’ own."

http://www.econ.ucdavis.edu/faculty/kdhoover/pdf/Hope/Backhouse-Laidler.pdf

Greg,

You seem to have rather gotten off the ship on this one. So, we have this goofy
formula that got way overused in an inappropriate way, leading to all sorts of
problems. You then compared this to a variety of more general economics models,
many of which indeed have their own problems, but not of the sort associated with
a very precise formula of this sort. Eventually you got focused in particular on
the ISLM model and moved on to more general denunciations of Keynes and the GT.
Hmmm.

Only two points. The first is that Keynes was never all that big a fan of the
ISLM model, although he never totally disowned or denounced it, so blaming him
for it is a bit of a stretch. The second is that with regard to the main point
here, this use of this very precise and flawed formula for pricing all sorts of
wacko CDOs in financial markets, Keynes would almost surely have been as critical
as either Hayek or their current fan, Nassim Taleb, who is quoted in the article
as being so. Do please keep in mind that Keynes was a deep student and advocate
of the concept of fundamental uncertainty and not someone who would have fallen
all over himself in praise of the sort of nonsense that went on in the financial
markets in these recent years, especially the avid use of such inane formuli to
excess. Please reset your target here.

So, this burgeoning literature on the problems of computer models, are the climate scientists paying attention? Just asking.

A funny math joke just destroyed half of everything.

Except Black and Scholes didn't win a Nobel Prize, they won The Sveriges Riksbank Prize in Economic Sciences in Memory of Alfred Nobel.

Allison,

The econophysicists are the ones currently insisting on power laws and fat tails
and all that. It is the garden variety financial economists who believe in Gauss.

Lorenzo,

Many associated with climate modeling are worrying about non-Gaussian matters,
although I suspect from what I read of you that you are more concerned that they
are overly concerned with such mattters. Is Martin Weitzman right that we should
pay huge economic sums to insure against a "tail event" of climatic catastrophe?

it sounds like we should be blaming this all on gauss. who is this guy and how can we nail him?

Isn't the fundamental problem with using the Gaussian copula to price mortgage backed securities the assumption that the behavior is random?

David Li has a Master's in actuarial science and he borrowed one concept from the actuarial problem of figuring when the second spouse will die from when the first spouse dies. It's reasonably safe to assume dying is fairly random. But what happened in, say, California wasn't an unlikely random event. It wasn't random at all. A whole lot of people noticed that previous people had made a lot of money by speculating on houses, so they kept pushing the system until it broke.

What's random about that?

"the problem is that there were PhD mathematicians and physicists developing models which were then being implemented by people without the mathematical sophistication to understand...": in my experience, modellers commonly have the defect of being interested in their models, not in reality. In part, it's a result of specialisation - theoretical physicists typically neither design nor perform experiments; it's someone else's job. And that attracts into theoretical physics people with a character and intellect well suited to ignoring the complexity of reality. They're not likely to change just because they join a financial firm and find themselves better paid. The man who hires such an 'idiot savant' has the duty not to aquire "mathematical sophistication" but to ask penetrating questions about the evidence that the mathematical toys are a usefully good representation of reality.

Anybody who thinks they can solve np-complete problems is a complete idiot.

A post by Steve Hsu from 2005 quotes a WSJ article from Sept 12 of that year by Mark Whitehouse on this very subject.
The revaluing of GM's debt in May, 2005 caused many hedge funds to lose a great deal of money. The article ascribes this, in part, to Li's formula.
Prof. Hsu pointed out that "The model itself is almost certainly too simple, but is (hopefully) improved in proprietary ways by sophisticated traders and researchers." No improvements were made by traders/researchers, sophisticated or otherwise.
I recommend Steve's blog, BTW.

thats what I look for..

Congratulations on your site I reviewed was beautiful†¦

These women have aged ..:))

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