Sunday, April 18, 2010

No, Maths was not Responsible!

In Gillian Tett’s recent piece in The Financial Times ('What Happens to Markets When Numbers Don’t add Up’, April 16, p. 18) the inference was made that mathematics (or ‘maths’ as the Brits refer to it) was at least indirectly responsible for the 2008 financial meltdown. She quotes a British academic, Tim Johnson, has saying “there is a sense of bewilderment among mathematicians that mathematics was somehow responsible for the crisis”.

Ms. Tett goes on to write:

“In the three years since the financial crisis has exploded, financial mathematics has come in the line of fire, with ‘quants’ and models blamed for fueling the banking woes”

But is there really such a thing as “financial mathematics” and can it be inextricably wed to mathematics as the queen of sciences? I don’t think so, though Ms. Tett argues:

“ the world of finance has certainly borrowed from disciplines such as maths and science and some of this plagiarism has produced disastrous results”

But, the question is, has finance developed an authentic math unique to itself? As opposed to simply copying equations or math entities, plugging them into expressions and extrapolating from these in an indiscriminate way. To given an illustration, Astrologers regularly plagiarize the math of celestial mechanics (used to locate the future positions of planets) but the product is useless gibberish with no meaning at all. (A “horoscope” which purports to embody the individual’s personality and/or future)

Is maths then responsible if an astrologer’s personal horoscope informs a person that the 15th of May will be his lucky day, and he then buys 1000 lotto tickets – but wins nothing? Not at all. It isn’t maths’ fault that a group of pseudo-scientific morons hijacks a subset of math from astronomy to confect bogus ideations. Nor is it maths’ fault that over-eager “quants” (the name for the bozos that devised these financial derivatives) used a form of statistical math to conjure up their nonsense. (Whch any legit mathematician would spot instantly)

Let’s look at the key equation, the Gaussian copula, invented by David X. Li while working at JP Morgan Chase and articulated in his (2000) paper: ‘On Default Correlation: A Copula Function Approach” (For those unaware, a “copula” is a statistical device employed to couple the behavior of two or more variables) and blamed for “wrecking your 401k” according to an article ('A Formula for Disaster’) appearing in WIRED a year ago.

First, examine the Gaussian copula equation itself (see image inset above) for which I have pointed out the key defects showing why this isn't a true mathematical equation, and why it is more of an intellectual Frankenstein monster that never should have seen the light of day any more than a four-headed baby with a pointed tail.

In normative mathematics, equations are derived from a self-consistent basis and a pre-existing modus operandi and history. For example, the Riemann Zeta function, or the Euler Gamma function.

In Ms. Tett's piece she states that “Dr. Johnson’s vision of maths sounds more akin to the financial version of quantum mechanics” –but this is in no way true.

Consider, for example, the probability density equation for locating a particle between x1 and x2 (Figure 2a) for which the quantities to the right of the integral sign are the wave function (psi) and its complex conjugate (psi*). This can be compared with Li’s Gaussian copula probability expression earlier. Now – what allows or enables us to pose an equal sign between the probability density(P) and the integral of the two wave functions? (After all, it is illegitimate to simply toss and equal sign into an equation as a fait accompli with no rational basis!)

What allows us to do this is the foreknowledge that the wave function is normalized so that effectively the probability that a given particle is somewhere along the x-axis is 1. (Fig. 2b)

I submit no such similar normalization applied to the Pr[T_A, T_B] of Li’s copula formula. More over, his misuse of the distribution functions (F_A(1)) and (F_B(1)) would appall any genuine mathematician. Each is actually based upon significant uncertainties via survival law distributions which can vary enormously. There is no way to normalize any probability based on (T_A, T_B so there is no way to equate Pr[T_A, T_B] to anything on the left side. The equal sign is dangerous recklessness masquerading as math. And the copula (PHI2: first variable on the right side) makes it even more an abomination, since ostensible survival law uncertainties are “coupled” to try to arrive at a single number (usually the correlation parameter). This is insanity!

The hideous Gaussian copula equation has then mixed chalk and cheese to new and hitherto unbelievable levels. Functions from human actuarial tables, generic survival distribution laws, a “probability” function based on coupling two financial entities as “humans” in an actuarial table – then tossing in a correlation coefficient which nearly every quant would likely be orgiastic over. (Since he can pin it down to one number, simplifying his financial analysis routines).

Can you now imagine, a quantum physicist performing an analogous desecration of the probability density equation by incorporating standard Gaussian distributions (not normalized) and mixing electrons with macroscopic entities like marbles – then deeming the wave function is capable of capturing the survival probabilities of both marbles and electrons? And – to add injury to insult, fabricating a “correlation coefficient” to relate (couple) survivable marbles to electrons? That quantum physicist would be run out of his lab so quickly he wouldn’t know what happened!

So, no – maths isn’t to blame for what unfolded. Rather, it was attributable to a quant who got too clever by half, and in his own mind, creating a pseudo-mathematical fabrication that never should have emerged in the real world- and certainly not to operate on CDOs, CDS or anything else.

My suggestion for what it’s worth? We need to get all these “quants” out of their finance ivory towers and back to doing REAL physics and mathematics for a change. Our financial futures may well depend on it, especially if rigid regulation doesn’t emerge to control their semi-quantified phantasmagorias.

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