In the recent economic crisis, many of the failures of the system have been attributed to the systematic trading of increasingly elaborate "derivatives"(which are sorts of bets on the future value of an asset), a practise which is made possible by the use of complex mathematical tools, such as the infamous "Black-Scholes" equation, but which has also led to a general lack of transparency in the financial market, and precipitated the crisis. Hence, traders and mathematicians alike have been forced to reexamine the new ties between abstract mathematics and trading, asking the question: where did things go wrong?
Take the Black -Scholes equation, used to estimate the value of a derivative: it is actually no more than a partial differential equation of the financial derivative's value, as a function of four variables, including time and "volatility" of the underlying asset (the derivative being a 'bet' on the future value of the asset). Differential equations are well-known to physicists, since such fundamental properties of nature as the wave equation or Schrodinger's equation for quantum mechanics are given in the form of differential equations, and in physics their solutions seem to be very reliable: so why is this not always the case in finance?
Einstein once declared that "as far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality", meaning that when modelling mathematically any real system, one must make a certain number of 'reasonable assumptions', which must be tested empirically. But this cannot always be done in finance: for instance, one of the conditions for the Black-Scholes equation to hold, is that the price of the underlying asset of the derivative has constant 'volatility', meaning that one can estimate precisely, once and for all, how erratically its price will fluctuate with time. This holds if the market in general experiences steady growth, but will fail otherwise, and all the more dramatically as the equation cannot quantify 'panic effects', whereby traders all start to sell massively at the same time, inducing overnight crises.
However, these flaws in the mathematical tools used by traders were clearly stated by their creators, so that the crisis was not provoked by mere unawareness of the risks, but by more or less deliberately ignoring them, with traders under pressure to make more money by speculating wildly over increasingly complex and risky derivatives, and losing all notion of the economic reality supporting the financial bubble, until it burst, and all were forced to recognise the failure of the equations to predict this.
Reactions went from condemning altogether this brand of "mercenary mathematics, working for the sole benefit of international financial institutions"(Denis Guedj, mathematician), to simply deploring the traders' lack of 'common sense' in their use of the mathematics they were taught (this was the stance of Nicole El Karoui, French 'star' of financial maths teaching, whose pupils include 800 'quantitative analysts').
But others argue that 'common sense' is not the only thing missing: more appropriate mathematics, that take in account phaenomena such as "herd instinct" or the occurence of "rare" events that radically change market conditions ("black swans") could be one way of avoiding future disasters, and demanding more transparency from traders (regulating the trading of overly complex derivatives, for instance, since these have often proved to be financial time bombs), would probably do no harm.Suggest a correction