Wednesday, 15 February 2012

Mathematician thinks mathematicians are very important

Ian Stewart, writing in the Guardian, says that the Black-Scholes equation
opened up a new world of ever more complex investments, blossoming into a gigantic global industry. But when the sub-prime mortgage market turned sour, the darling of the financial markets became the Black Hole equation, sucking money out of the universe in an unending stream.
This is just wrong.  The Black-Scholes formula is used by major players in the derivatives market only as a calibration tool, to obtain "implied volatilities" which give one a measure of how far market option prices diverge from an equation all the players know to be a simplification.  And the pricing models used for the mortgage-backed securities that caused the financial crisis have got very little in common with the Black-Scholes equation.

Stewart is wrong too in what he says about Nassim Taleb's notion of "black swans".  In Taleb's conception a black swan is not just an extreme event, it's an event that couldn't possibly have been built into a model because the modeller couldn't reasonably have even considered the possibility of its happening.

Stewart opens his concluding paragraph by asserting that "Despite its supposed expertise, the financial sector performs no better than random guesswork."  He doesn't explain this remark, but it's not a fault of the financial sector if the Efficient Market Hypothesis is nearly true.

Having established to his own satisfaction that financial mathematics caused the financial crisis, Stewart concludes that what's needed is more, better, financial mathematics.  This isn't exactly wrong - who would be against doing things better?  But I am sceptical that we need mathematical advances before we can design a sufficiently stable financial system.

I have the advantage over Stewart of having done this stuff for a living.  And I say that:
- nothing can make financial markets follow any particular model, unless a tradeable arbitrage exists when the market deviates from the model.  (The Black-Scholes model is based on an arbitrage strategy of sorts, but one that operates only in an idealized world: the arbitrage is not tradeable in the sense I mean).  Therefore financial models are tools for understanding financial risk, not for eliminating it.
- financial mathematicians (quants) don't make ill-advised trades: traders do.  It's the duty of the quant and trader both to make sure that traders understand the limitations of the models.
- from a mathematical viewpoint, the financial crisis occurred because traders grossly underestimated the probability of correlated mortgage defaults.  That's got little to do with black swans - it was foreseeable and foreseen that the US housing market wouldn't go up forever.  And not very much to do with models - even the simplest models for mortgage-backed securities had a correlation input.  It's got a lot to do with the hubris of traders who chose to bet the house on their guesses of an unobservable and unhedgeable parameter, and the failures of understanding of the bankers and insurers who let them do it.

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