Mathematical models in finance and embedded risk underestimation

The statistical assumption of normal (log-normal) distribution of stock returns (prices) is not that strong and tail events’ occurrence is largely undervalued

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Misfortune and almost 15 years of financial markets’ history show that this modeling framework has been widely used in three main fields: trading books’ risk assessment and their supposedly adequate hedging; options pricing; asset management

Risk assessment

The search for simplicity and speed of computation has always been a priority. Simplicity since all financial instruments depend on risk factors that supposedly follow a statistical normal distribution (log-normal, precisely), and fast computation because the log-normality assumption of risk factors enables straightforward computation of Value At Risk (VAR)

I recently read in a magazine that to be a good financial market professional in research, trading and structuring, you had to be: a good mathematician, a good physicist and computer-savvy. Being able to be all at once is perfect while this does not prevent from deepening knowledge in the field of behavioral finance.
Mory Doré

Derivatives pricing

Mainly first generation derivatives, based on Fisher Black and Myron Scholes’ model. At the time of deregulation and disintermediation of financial markets (early 1970’s in the USA, early 1980’s in the UK, and during the mid-1980’s in France) a new concept appeared: volatility. Exchange rates became volatile on August 15, 1971 with the end of Bretton Woods - probably the most important date in modern monetary history - when U.S. president Richard Nixon terminated convertibility of the dollar to gold. Interest rates also became volatile when monetary policies started to be applied to control money supply (under leadership of Paul Volcker, federal reserve chairman from 1979 to 1987) and the questioning of administered economy in continental Europe during the 1980’s (i.e France with the end of credit control in 1985)

From those factors will emerge the need of new financial techniques and products known as derivatives in order to hedge exchange rates and interest rates variations. The Black and Scholes formula designed to price some of these products will provide framework with scientific legitimacy. Simplicity and computation’s speed are once again almost irrefutable, justifying arguments: mathematical simplicity – in a mathematician’s view – as knowing how to integrate a differential equation is enough (apologies to non-math-savvys); efficient computer capabilities not weighing too heavily on bank operating ratios

Asset Management

Third and final field relying on the normality assumption of returns and risk’s distribution: modern portfolio theory and the work of Markowitz and Sharpe. Once again, asset management’s modern activities find here a scientific legitimacy, with the everlasting efficient frontier target, where the risk-reward is, supposedly, always optimized and asset allocations as wall. Proving to have all virtues in the world of modern finance on its side, it was thus impossible to refute the "scientific" thrust of this Gaussian framework.

It will never be possible to model fear, imitation, and further less the impact of regulatory, prudential and accounting rules on investors’ behavior
Mory Doré

Indeed, all was for the best in the best of all possible worlds. First, the Corporate and Investment Banks (CIB) were supposed to perfectly monitor their market risks thanks to a "magic" tool called Value at Risk. Then, the asset management subsidiaries of these banks were supposed to manage private and institutional investors’ excess savings within an optimal framework. Finally, the economic agents were supposed to hedge their financial risks using derivatives accurately priced through CIBs.


Benoît Mandelbrot

- In 1961, he developed a valuation model of stock prices based on fractal geometry. This practice, which had the advantage to better detect the occurrence of extreme variation was ignored until the mid-1990s (officially due to complexity but unofficially most certainly because it was not part of the official dogmas of finance). Fortunately, recent events in the history of markets helped it being re-studied for precise purposes.

- In 1997, Mandelbrot set up new models that incorporate the long-term memory in asset returns. It defines the multifractal time to better understand the coexistence of turmoil and calm periods in the financial markets. The size of variations can indeed remain independent from one session to another, but it can also be correlated over very long periods of time, which is not taken into account by the Black and Scholes assumptions.

- In 2004, he wrote "The (mis)behavior of markets" in which he shows the inadequacy of most of the mathematical tools used in finance.

So, why have we never heard about top scientists’ warnings related to the improper use of models in financial risk management and asset pricing? We actually did, the mathematician Benoit Mandelbrot, died in October 2010, being one of them. One could rea here and there the rich biography of this genius scholar forgotten by the masses. An oversight which might be justified by the fact he disturbingly challenged scientism and unanimity of the gaussian nature of financial markets.

The many market events that have occurred since the 2007 crisis proved Mandelbrot was right in its findings on extreme risks underestimation.

Looking 4 years back, one can judge for himself
-Was it credible to guess that some dynamic money market funds would suddenly be forced to suspend their net asset values during the summer of 2007, and freeze the investors’ assets?

-Who would have believed back then that complex structured issues rated "AAA" in April 2007 could be downgraded six or seven notches within few weeks or even default six months after launch (the famous mean-reversion of some credit spreads disappeared!)

-How unthinkable was it to bet on the collapse of the major U.S "investment banks": bankruptcy (Lehman), bail out/merger (Bear Stearns and Merrill Lynch)

-And what about the rescue of three of the most prestigious "AAA" on the planet by US Treasury: two unquestioned mortgage companies, Fannie Mae and Freddie Mac, and the world’s largest insurer, AIG.


First there is the well-known technical explanation: the log-normality assumption of risk factors distribution is not really valid. This approach does not suit to products whose prices are not a linear function of risk factors and this include nowadays a very large set of products: first generation plain vanilla, exotic derivatives, structured notes with implied call options (indexed on any asset class), structured securities with implied put options (callable, reverse floaters or reverse convertible), synthetic securitizations…

But we can not be satisfied with such explanations and one must in fact understand the underlying reasons that have led to domination of some form of modeling on the financial markets. There are mainly, in our view, three reasons

1. For political and economic leaders around the world, this framework is ideal because the math (at least within a certain use) justifies the fact that occurrence of major disasters is almost impossible. The theory is pleasant and attractive, but also impossible. It is at least reassuring to everyone, voters, clients, employees, shareholders. Some say it is legitimate as the engine of growth and progress would be trust (trust in who and what?). In fact, the true engine of progress would be the access and use of tools that assess extreme risks and implement strategies and adaptative policies that would be required should they occur.

2. The need to keep on funding waste and excessive debt leads to bubbles which occurrence is deliberately minimized. Overvaluation of assets is clearly ignored as well as the risks of imitation and illiquidity of some held financial assets. Behind this, three motivations

  • Some heavily commissioned institutions’ greed (1999-2000 technological bubble, 2006-2007 subprime crisis)
  • Sometimes the need to justify the funding of public deficits (overvaluation of supposedly safer sovereign bonds issued by the US, the UK, Germany and France is considered manageable, yet it is not)
  • Finally the rationale for diversion of household savings to most unproductive uses: public debt financing, including those of most insolvent such as PIIGS’. In this context, the risk level of investments such as life insurance is clearly underestimated (not necessarily a loss in capital, but the risk of a significant decline in future returns)

3. Third reason, a risk minimization designed to legitimize a development model with two pillars:

a/ Excessively high standards of profitability, compared to economic fundamentals, that favor short-term profitability often poorly reinvested rather than balanced, long-term development (in both micro and macro levels);

b/ Banks financial engineering aiming to transfer risk to private economic agents in order to save capital and improve again and again the return on capital employed

  • This led to significantly underestimate the risks related to excessive leverage in some complex structured products
  • This also contributed to spread the market efficiency dogma as well as the idea of good risk pooling (the subprime crisis showed how dangerous this pooling could be, as the risk could not be identified or localized properly)


-First, we must continue to invest in mathematical research to improve modeling (which will always include deficiencies). This calls to deepen Mandelbrot work, but also to establish true stress scenarios to take into account discontinuities in the evolution of asset prices. Better modeling of extreme values (fine tuning of distribution tails) that are poorly taken into account in the calculation of VAR is also needed.

-Most of all we need to better understand the markets. I recently read in a magazine that to be a good financial market professional in research, trading and structuring, one had to be: a good mathematician, a good physicist and computer-savvy. Being able to be all at once is perfect while this does not prevent from deepening knowledge in the field of behavioral finance. For sure, it will never be possible to model fear, imitation, much less the impact of regulatory, prudential and accounting on investor behavior

The recent history of financial market crises (1998-2011) cannot be fully understood if we do not include the dynamics of contagion between financial assets and fire-sale for commercial, regulatory and accounting reasons
Mory Doré.

How many times have I been asked why a financial asset price fall abruptly even though the fundamentals of the asset were perfectly healthy?
I often said that crises and periods of stress on the markets usually go along with fire sales, regardless of assets fundamentals. Let’us review a serie of examples that I mentioned in recent articles
Example 1: My portfolio holds an Asset A that has become junk and illiquid, and I need cash for various reasons (compliance with regulatory ratios, anticipated clients’ redemptions...). I’ll be then forced to sell an asset B, whose fundamentals are rather strong, or an even higher quality asset C.
Example 2: The crisis that affects a given asset X of my portfolio will stress the monitoring of my annual report because of the use of fair value pricing (according to IFRS standards) and mark-to-market that will negatively impact my income statement. Here again, and for other reasons, I am forced to sell good assets: an asset Y which still show potential upside and perhaps an even higher quality asset Z with even better gains expectations.

The recent history of financial market crises (1998-2011) cannot be fully understood if we do not include the dynamics of contagion between financial assets and fire-sale for commercial, regulatory and accounting reasons. Another behavior often observed is the use of risk aversion, instead of fundamentals, to explain directional trends.


-On the foreign exchange market, the Swiss franc and yen tend to sharply appreciate. It is certainly not due to superiority of Swiss and Japanese economies over other countries, but rather to investors’ memory who link risk aversion to the settlement of carry trades initiated between 2004 and 2007 at the expense of CHF and JPY. We remember that, at the time, the lack of risk aversion led to the following game: cash borrowing in CHF and JPY at very low rates, Fx spot trades to sell these currencies against USD and other high-interest paying currencies to instantaneously get positive carry; the return of risk aversion led the hedge funds suffering from important losses and holding carry trade positions to settle their positions by going massively naked long on JPY and CHF; this is what happened between July 2007, late 2008 and today. Rightly or wrongly, any resurgence of risk aversion comes very often with appreciation of JPY and CHF.

-On the debt markets, there is usually massive flight-to-quality to U.S. treasuries, including short and long-term papers. The rationale behind is the idea that these are the most liquid assets in the world, therefore still tradable even in case of severe liquidity crisis.

-In the euro zone, during risk aversion periods, even when it is not related to the peripheral areas of the zone, investors find safe havens in public debts of the "core" Europe at the expense of what are known today as the PIIGS

-Finally, gold is a natural safe investment even though fundamental reasons to overweight this asset are obvious (excessive monetization of public debts and early loss of confidence in major currencies, high inflationary risks on the long run and therefore extended period of negative real interest rates)

Gold put aside, it is clear that these safe havens are often irrational: money market rates around 0% in CHF and JPY; contribution to a bubble on U.S. debt and Core Europe’s debt as well

Mory Doré May 2011

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