Thursday, 16 February 2012

The mathematical equation that caused the banks to crash (?!?)

Claiming that the Black-Scholes equation had anything to do with the Credit Crisis of 2007-2009 is a bit like claiming Daimler-Maybach were responsible for bombing Hiroshima. Sure the planes used the internal combustion engine, but the causal relationship is being stretched beyond reason. 

The claim that Black-Scholes was involved in the Credit Crisis has been made by Ian Stewart in a piece, apparently promoting his new book, Seventeen Equations that Changed the World, in The Observer.   

Prof Stewart is the most important promoter of mathematics in the United Kingdom, writing his first book in the early 1970s, Concepts of Modern Mathematics, as an exposition of Bourbaki mathematics. In the 1990s, Bourbaki and ‘New Math’ had faded/failed and Prof Stewart turned his attention tho the more pressing question of introducing mathematics into industry, as explained in the Preface to the Dover Publications edition. 

Given this track-record I am at something of a loss to understand what Prof Stewart was trying to achieve in his Observer article, beyond generating publicity for his book. It is unfortunate that he has chosen to approach a critically important issue, the use of mathematics in finance, in such an unthoughtful way as he has. 

His article is little more than a string of factually incorrect statements, my guess is they are culled from the BBC’s Midas Formula which in turn is based on Lowenstein’s When Genius Failed. For example, the piece begins with “It was the holy grail of investors.”. At its heart is the rather depressing statement, from an investors perspective, that profits should equal the riskless rate, that's more a poison chalice than a holy grail - as one student asked in lectures last week “what's the point of that then”.  

What is more, if it was an esoteric secret, why was the paper initially rejected by the Journal of Political Economy on the basis that there was not enough economics in it. Peter Bernstein, in Against the Gods reports that Black felt it was because he was a mathematician and had no qualifications in economics. The paper then went to the Review of Economics and Statistics, again without success. Bernstein reports that the paper was only published by the Journal of Political Economy after the intervention of influential Chicago academics. 

A more thoughtful assessment is that Black-Scholes enabled the CME and CBOT to justify the introduction of financial options trading, and this is touched upon. But there is no accompanying explanation of the collapse of Bretton-Woods, destroying fixed exchange rate and broadly flat bond yields, meaning that volatility suddenly became an issue for the markets. 

The article, then links Black-Scholes with the sub-prime crisis, a gross mis-representation since Black-Scholes played pretty much no (pricing) role in the collapse of LTCM, let alone in the credit crisis. What Stewart fails miserably to understand is that, in typical practice since the 1987 crash, Black-Scholes has not taken volatility as an input and produced prices, but taken prices to imply volatility. The failure is miserable, because this is the genius of the equation, a consistent measurement tool of the markets’ assessment of risk in the future. 

The piece goes on to claim “The idea behind many financial models goes back to Louis Bachelier in 1900”. Well, Prof Stewart should know better. It would be better to write that “The idea behind many mathematical models goes back to the financial mathematician, Fibonacci in 1202” or “The origins of the Black-Scholes formula lie in the Pascal–Fermat solution to the Problem of Points in 1654. The generally accepted origin of mathematical probability”. These are far more interesting points. 

The article continues
The Black-Scholes equation was based on arbitrage pricing theory, in which both drift and volatility are constant. This assumption is common in financial theory, but it is often false for real markets.
This is nonsense, arbitrage pricing is a concept, constant drift and volatility are implementations. Stewart is happy to talk about the economic-physics Black-Scholes equation but seems ignorant of the mathematical Fundamental Theorem of Asset Pricing
A market is free from arbitrage if and only if a martingale measure exists.
A market is complete and free from arbitrage if and only if a unique martingale measure exists.
Why oh why didn’t Prof Stewart talk about this, admittedly not an equation, but profound mathematics that tells us something significant about markets. 

Mathematics, in my humble opinion, is not equations, it is concerned with ideas and understanding. Understanding the Fundamental Theorem requires an appreciation that probability is not relative frequency, of epistemology (Black Swans), through the idea of completeness, and ethics, being arbitrage free is about fairness, a martingale measure is about equality. This is all in Aristotle and Aquinas, as described by the mathematician James Franklin, and all in financial mathematics, where contemporary papers include words like ‘belief’ and ‘greed’ in their title. 

Stewart finishes his article by suggesting the solution lies in the mathematics of dynamical systems and complexity and adopting analogues from ecology. The Bank of England has followed this line of thinking building on work of Lord May, the biologist and past President of the Royal Society. The problem is, Lord May’s analysis is based on the model of ‘contagion’, the banking system is an ecology through which default spreads, in the same way that mad-cow disease spreads through farms. The response is to build firewalls, quarantine zones, around banks. The Bank of England seems reluctant to develop this line of thinking, possibly because they now realise that the model is just plain wrong. A more appropriate model for banking is the internet, the Credit Crisis was a result of linkages in a network collapsing not of some invisible infectious agent spreading throughout the network. The preferred model now seems to be electricity grids (the article was apparently prompted by a discussion with Andy Haldane). So the Bank is still using a physical analogue, but a better analogue. 

Stewart would see the solution in dynamical systems and complexity, because that is an area of mathematics that the British are strong in. But, dynamical systems are typically ergodic (in the 2003 Review, “dynamical systems and complexity” was labelled “dynamical systems and ergodic theory”) , and, unfortunately, the economy is not ergodic

Mathematicians must start taking the economy seriously, and not try and shoe-horn economic problems into a framework of the mathematics they understand. Ian Stewart is extraordinarily influential in defining what mathematics is and how it can be applied. Academics should not publicly discuss topics they are not expert on, that is the realm of journalism (or blogs). In this article, Prof Stewart has misrepresented mathematics, damaging its reputation. For this he should be ashamed.


  1. Simply put in, writers (even if highly-recognized mathematicians) and journalists need to simplify in order to stupefy and sell.
    Simplification does not require or imply correctness, though!
    This approach is as wrong as saying that the Schroedinger Eq lead to the Atomic bomb, or that a logistic function leads to birth control....

  2. Keynesianism was the cause. A recession is natural, due to money = debt since 1600. Keynesianism disturbs the natural occurring recessions, leading to ripples that lead to bigger and bigger super recessions. Basically Keynesianism switches dozens of naturally occurring smaller recession cycles for delayed bigger ones eventually leading to a very delayed big one (Second Great Depression), that is about to come. What pattern is better for us? Time will tell. Warren Buffett and baby-boomers don't mind, keynesianism worked out well for them in the end.

  3. Fortunately, dynamical systems theory is a wide field and the harm in suggesting it may have useful application to finance might only be in the failed tenure attempts of a small number of academics. My sense is that the benefits of dynamical systems and "chaos" will largely be the descriptive qualities journalists will use to communicate with an interested public. It probably won't catch on like catastrophe theory did in the 1970's.

    In the near term, mathematicians will continue to have at least one influence on economics and finance, in that they can debunk pseudo-mathematical arguments presented in the marketing of new but silly investing ideas. I wish they had spoken up sooner about Gaussian copulas back around 2005!

  4. I also saw this BBC documentary on LTCM which throws a bit of light on how BS model and its variations were misused and abused by traders. I think what that documentary was trying to say was that Dynamic Hedging was responsible for he crash ! also they gave examples how Black Scholes was used to hedge and price non-equity claims which was certainly neither suitable nor recommended by original model itself.
    However I agree that BS model was not direclty linked to the market crash we saw back in 2007. That had got to do more the US Sub prime credit market which later turned into a global credit crises.