*Le Monde*or the online magazine Wired collectively flew into a rage over global maths. Le Monde accused French engineers with French-style education and training of pushing American finance into an abyss and, with it, global financial markets. How did this happen? By proposing mathematical models to the heads of financial institutions that claimed to counteract the risks inherent in financial products, but which ultimately failed to live up to their promises.

**Mathematicians plead not guilty**

Michel Crouhy, director of R&D at the French bank Natixis, an institution that was badly affected by the sub-prime mortgage crisis, and the founder of the first master’s course in mathematical finance in 1986, refutes this criticism outright. “The models worked perfectly well, even during the heart of the 2008 crisis. Each morning they predicted the amount that we would be losing during the day with great accuracy. The main issue here does not concern models, but has to do with the drying up of liquid assets, so there was no compensation whatsoever for our positions. Without liquidity, the market is bound to plummet whatever the model.”

Banks generally cover their risks because they are constantly involved in mark to market dealings; they know what their assets are worth and verify that what they pay for their protection is a sustainable price. If the covering price soars, the positions come down. In addition, banks make “model risk” provisions by holding in reserve a part of the profits contributed by the activity of these models in reserve. In the end, they make compromises between the precision of the models and the duration (and therefore the cost) of the calculation.Every banker knows there are no profits without risk. And mathematicians know very well that 100% coverage of risk is a myth. Bernard Lapeyre, mathematics professor at the Ecole Ponts ParisTech (ParisTech School of Civil Engineering), sums this up. “The incompleteness of markets disallows perfect replication (coverage). There will always be a residual risk.”

In the beginning of this century when the market moved from the share derivatives and rate derivatives of the 1980s to credit derivatives, we reached a stage where risk coverage was no longer, even theoretically possible. What we have seen in the late 2000s was in fact a direct result of the 1990s innovation of credit derivatives. Credit derivatives are securitized derivatives whose value is derived from the credit risk associated with the underlying loan, deal or debt. The aim of credit derivatives is to transfer the risks (and part of the revenue!) associated with the credit through “securitization” without transferring the asset itself.

Michel Crouhy |

**Risky formulas**

Up to mid-2007, credit derivatives (collateralized debt obligations (CDOs) and credit default swaps (CDSs)) were traded on the basis of the Gaussian copula, a formula devised by Chinese mathematician David X. Li. Li was born in the Chinese countryside in the 1960s, went to the University of Nankai (Tianjin), where he earned a masters, and then to the University of Ontario where he did a PhD in statistics. It was in 2000, while working for J. P. Morgan Chase that he created his equation. The formula measures the correlation between the default probabilities of two borrowers on the sole basis of the history of credit derivative prices, ignoring the history of the defaults themselves.There is something outrageously simplistic about Li’s formula (as is elegantly put in Wired magazine) and perhaps because of this it achieved spectacular success and was widely adopted. It made pricing risk seem easy, but in so doing it gave a deceiving sense of certainty of the real value of the underlying assets, which for the most part was real estate.

Because of this, can Li be held accountable for the bursting of the sub-prime mortgage bubble? Natixis’ Michel Crouhy does not think so. “This type of equation was useful for discussions among us (mathematician bankers) about the market. But to assess their own risk, banks used much more comprehensive models.” He has absolutely no doubts whatsoever where the blame lies. “Those responsible for the crash are the political and financial leaders who, for years on end, encouraged banks to make loans to insolvent borrowers, the sadly notorious Ninja (no income, no job, no asset) loans. Rating agencies also took part in the grand patch-up exercise. In 2006-2007, they took part in the structuring and rating of sub-prime CDOs using inaccurate information on securitized mortgages and inaccurate statistics. In a clear breach of ethics, Moody’s got half of its profits by distributing CDOs to which it had just assigned the AAA top rating. As for the regulators, they did little to stop the massive fraud in the distribution of sub-prime loans even when problems first came to light as far back as 2005.”No one wanted to see the storm coming. After two decades of expansion during which few borrowers on the American real-estate market defaulted (or if they did it was for specific, individual reasons), no one seems to have predicted that if the market went into decline, the subsequent numbers of defaulters would throw the Li equation off course. It would be similar to the crash of 1987, which highlighted the weaknesses of the Black-Scholes equation in pricing options.

**Theories and practices**

In the case against mathematics and mathematicians, there is one accused party who stands out: Nicole El Karoui and Marc Yor, both professor of applied mathematics and the persons in charge of the masters program in probabilities and finance at the Paris VI university and a research fellow at Ecole Polytechnique. Many of the graduates from Nicole masters program, or “quants” (quantitative analysts) ended up at in the top ranks of banks like Goldman Sachs, Lehman Brothers, BNP Paribas and Société Générale, as well as in rating agencies and hedge funds. Karoui puts the role of maths into perspective. “Maths is just a simple tool in decision making, just like computers. Each person must take his or her responsibility seriously. If you observe bank sociology carefully, you will see that mathematicians are not the ones who make decisions. All of us warned that the risk factor linked to credit derivatives (CDOs and CDSs) would increase in a non-linear fashion depending on the quantity of operations, but who listened to us? Faced with greed, it wasn’t the models that were lacking, but pragmatism and common sense.”

Nicole El Karoui |

At a certain stage, even common sense would have served no purpose. Andrew Lo, head of the financial engineering department at the Massachusetts Institute of Technology (MIT) explains: “Imagine it is 2005 and you are the risk manager at Lehman Brothers, and you know that the real estate market is going to burst and that your bank is in grave danger because of its exposure to sub-prime mortgages. If you advise the chairman, Dick Fuld, to withdraw from the sub-prime mortgage market, he will ignore your advice as he will not be willing to jeopardize traders’ bonuses and to trigger an exodus of his best people. And even if you were commissioned, theoretically speaking, to cover the bank against its questionable mortgage-backed products, you would have lost huge sums in 2005 and 2006 and the first six months of 2007. But you would already have been sacked long before that.”

Most of the time, pro Math arguments smack of false modesty and, in effect, hide behind technicalities. The models are, to borrow Nicole el Karoui’s definition, “simple reductions of the reality, which work under certain hypotheses and are derailed in volatile situations.” The anti-maths crowd claims that pure mathematicians are motivated only by the aesthetics of their solutions. They sacrifice pragmatism to elegant formulae and end up confusing reality with the models. In short, according to them, models are neutral and fragile and therefore blameless.The accusers claim that mathematicians injected to the financial markets the destructive illusion that risk is a known, measured and containable factor. When all traders, trained in the same theories, seek refuge behind the same simplistic equations, so as to maximize their employers’ profits and their own bonuses, no one wants to remember that free and efficient markets can sometimes be struck by sheer madness and come to a grinding halt. The herd instinct wreaks havoc and the sweet theory of rational agents generates irrational behavioral patterns through a widespread unwillingness of anyone to be held accountable. Furthermore the excessive technicality of the models leads to a de-linking between the decision makers and financial engineers within banks because they lack a common language.

*De facto*, this inability to communicate cancels out any possibility of arbitrage, and favors unexpected crashes.

In the eyes of Jean-Philippe Bouchaud, professor of physics at Ecole Polytechnique, given its theoretical leanings, the French school of mathematics, “is an integral part of the systemic risks” which threaten global finance. The models, adds Claude Bébéar, founder of the French insurer AXA, “are intrinsically incapable of taking major market factors into account, such as psychology, sensitivity, passion, enthusiasm, collective fears, panic, etc. One must understand that finance is not logic.” Bouchaud further suggests replacing financial mathematics by another approach. “Over a span of 20 years, tremendous changes have taken place. Computers and their storage capacity enable us to handle colossal amounts of data and thereby compare reality with theory. Experimentation must replace what might appear to be axiomatic in finance. The investigative methods used in physics, those used in fluid mechanics for instance, are better adapted to this work as they verify the plausibility of the results obtained at each point in time. “ In effect, one must start with reality, and be distrustful of pre-suppositions.

Jean-Philippe Bouchaud |

Financial mathematicians are also accused in certain circles of having blindly followed “Brownian” ideology and these arguments are set out in a recently published book, Virus B: Financial crisis and mathematics , by Christian Walter and Michel de Pracontal, This ideology holds that financial phenomena, such as share price fluctuations, obey Brownian laws of motion, a theory put forward by the British botanist Robert Brown (1773-1858) when he studied the behavior of pollen grains suspended in water. He theorized that behind apparent chaos, the grains actually gravitate around a balanced position from which they never move very far. This “wise” randomness, which seems to guard against or preclude extreme irregularities appealed to many economists who were trying to develop a model for financial markets using Gaussian distribution. In the beginning, the works of Brownians Jules Regnault (1834-1894) and subsequently Louis Bachelier (1870-1946), did not please Henri Poincaré (1854-1912), the world’s best known mathematician in his time, but they came back in full force around 1960 when the American economist Paul Samuelson was looking for a solution to the problem of pricing options. The Black-Scholes formula that we mentioned earlier is typically Brownian in nature.

Samuelson, Black and Scholes should have looked at the works of the French mathematicians Paul Levy and Norbert Wiener and the Japanese mathematician Kiyosi Itô. They had developed non-Brownian models because they had detected a worrisome inability in Brownian models to understand “rogue” (randomness) coincidences which they believed were typical of financial markets. For non-Brownians, the market is constantly shaken up by micro-crises and dangerous discontinuities. The French school of mathematics has produced great non-Brownian thinkers, such as Benoit Mandelbrot, founder of the “theory of fractals” in 1974. But their discontinuous models are harder to demonstrate, less reassuring and require large amounts of computer power and are therefore costly for banks to use. Consequently, in finance, these mathematicians are not so highly valued.

The argument is far from being purely intellectual in nature. In 2004, a group of “dissident” quants actually recalculated the ratings of Moody’s and instead of using the Brownian KMV model that the agency generally uses, they used a non-Brownian one. The AAA ratings of a few products were brought down in a ratio of one to five.

Paul Samuelson |

**The blame game**

The debate between those who are pro and those who are anti mathematics is far from over. The real question is whether the movements of financial market can be subjected to probability analysis (ie.predicted). Some post-Keynesians, such as Michel Aglietta and André Orléan, believe that there is a “radical uncertainty factor” in finance and that it would be foolish to apply probability models to markets. Free market economists fear that if probability can never be calculated, the state will use the void to step in with tougher financial regulation. They’d rather leave it to the “invisible hand” of Adam Smith. These are neo-liberals, representend in France in the “Cercle des économistes,” a group of 30 mostly French economists that share theories and ideas.

So has mathematical finance done its mea culpa yet? Not yet. That will take a lot of time. It is true that in 2010 there are slightly fewer applicants for Nicole El Karoui’s masters program but the models remain and the big clean-up operation of the toxic substances is still underway. Mathematicians are not giving up the fight so easily, and the boldest amongst them do not think twice about revisiting 2008. They maintain they were not listened to enough. They believe that if there had been more maths on the markets, the crisis would perhaps not have taken place at all. And, in spite of the carnage in the markets in 2008, Ben Bernanke, the president of the US Federal Reserve bank, continues to vigorously defend the contributions of financial innovation.

Then there are those who call for a certain kind of “moral” financial mathematics. In 2009 Paul Wilmott, a PhD from Oxford and a well-known expert in quantitative finance, wrote what he called an ethical charter manifesto for the inventors of financial models and it is fast becoming the reference manual for those in the field.