In the year 1900 a little known French mathematician Louis Bachelier put forth the effort to eradicate risk involved with investing in financial markets. While his work was lost for 60 years, his original contribution to pricing options (more importantly, pricing volatility of a given asset) will become the cornerstone in what is today most widely used formula in finance; Black-Scholes-Merton formula for pricing options.
Until Bachelier, little effort was given to correctly price the assets which traded on exchanges. Bachelier, building on the efforts of physicists, decided to use arithmetic Brownian Motion to describe price movements in any given asset. Without questioning this, one of main contributors to BS option pricing model, Robert Merton assumed (taking a page directly from Bachelier's book) that price movements could be correctly described by using BM.
But there is a problem with using BM. In a recent paper, Oliveira and Mendes described the shortcomings of GB as following:
"Geometric Brownian motion (GBM) models the absence of linear correlations, but otherwise has some serious shortcomings. It does not reproduce the empirical leptokurtosis nor does it explain why nonlinear functions of the returns exhibit signi?cant positive autocorrelation.
For example, there is volatility clustering, with large returns expected to be followed by large returns and small returns by small returns (of either sign). This, together with the fact that autocorrelations of volatility measures decline very slowly , ,  has the clear implication that long memory e?ects should somehow be represented in the process and this is not included in the geometric Brownian motion hypothesis. The existence of an essential memory component is also clear from the failure of reconstruction of a Gibbs measure and the need to use chains with complete connections in the phenomenological reconstruction of the market process .
As pointed out by Engle , when the future is uncertain investors are lesslikely to invest. Therefore uncertainty (volatility) would have to be changing over time. The conclusion is that a dynamical model for volatility is needed and σ in Eq.(1), rather than being a constant, becomes itself a process. This idea led to many deterministic and stochastic models for the volatility (, and references therein).
The stochastic volatility models that were proposed described some partial features of the market data. For example leptokurtosis is easy to ?t but the long memory e?ects are much harder. On the other hand, and in contrast with GBM, some of the phenomenological ?ttings of historical volatility lack the kind of nice mathematical properties needed to develop the tools of mathematical ?nance. In an attempt to obtain a model that is both consistent with the data and mathematically sound, a new approach was developed in .
Starting only with some criteria of mathematical simplicity, the basic idea was to let the data itself tell us what the processes should be." [Oliveria, Mendes; 2010]
In the process of building the final formula, Merton had to build on some [untested] assumptions, which will later be proved, via ultra-volatile short-term price movements (October 1987, Fall 1998, May 6 2010, to name a few) to be wrong.
First assumption Merton made was that of a log-normal distribution, which was soon proven wrong by Fama who analyzed price distributions for all DJIA constituents. Fama's empirical analysis showed that prices are far from being log-normally distributed. Fama's findings are today popularly called "fat tails" and numerous techniques were developed in order to hedge fat tail risk (Good paper against using BS option pricing model "Why we have never used Black-Scholes-Merton option pricing formula" [Taleb, Haug])
It is needles to say, Black-Scholes-Merton formula never took into consideration Fama's findings and continued to use log-normal distribution as mathematical description. That proved fatal in 1987 when newly adopted portfolio insurance (built directly upon mathematics used in Black-Scholes-Merton formula ) caused the Dow Jones Industrial Average Index to have it's largest 1-day decline in history (-508 points).
Without getting into technicalities of BSM formula; best way to describe it's inadequacy is to read the following paragraph from above-linked Taleb and Haug paper:
"Such argument requires strange far-fetched assumptions: some liquidity at the level of transactions, knowledge of the probabilities of future events (in a neoclassical Arrow-Debreu style)4, and, more critically, a certain mathematical structure that requires “thintails”, or mild randomness, on which, later. The entire argument is indeed, quite strange and rather inapplicable for someone clinically and observation drivenstanding outside conventional neoclassical economics.
Simply, the dynamic hedging argument is dangerous in practice as it subjects you to blowups; it makes no sense unless you are concerned with neoclassical economic theory. The Black-Scholes-Merton argument and equation flow a top-down general equilibrium theory, built upon the assumptions of operators working in full knowledge of the probability distribution of future outcomes –in addition to a collection of assumptions that, we will see, are highly invalid mathematically, the main one being the ability to cut the risks using continuous trading which only works in the very narrowly special case of thin-tailed distributions.
But it is not just these flaws that make it inapplicable: option traders do not “buy theories”, particularly speculative general equilibrium ones, which they find too risky for them and extremely lacking in standards of reliability. A normative theory is, simply,not good for decision-making under uncertainty (particularly if it is in chronic disagreement with empirical evidence). People may take decisions based on speculative theories, but avoid the fragility of theories in running their risks.
Yet professional traders, including the authors (and, alas, the Swedish Academy of Science) have operated under the illusion that it was the Black-Scholes-Merton“formula” they actually used –we were told so. This myth has been progressively reinforced in the literature and in business schools, as the original sources have been lost or frowned upon as “anecdotal”" [Taleb, Haug].
It is easy to deduce, from the above paragraph what is exactly wrong behind arguments of BSM. First; BSM creators assumed that the market will always be liquid enough and gravitate towards equilibrium. Second; that the market participants are fully rational and their decisions are based solely on prices. Meaning that for every seller, there will be a buyer, and vice versa, and that the state of market symmetry will push assets prices to their equilibrium. Third; that asset prices experience absolutely no "jumps" [proven wrong by later research, as well as numerous new models which pay much attention to "jumps"].
Hedge Fund LTCM was built around these assumptions, re-creating the bond-arbitrage strategy that netted Solomon Brothers billions while its desk was the only one using this strategy. Basic assumption behind bond-arbitrage strategy was that of-the-run Treasuries [of same tenor, yield and coupon] were unnecessarily lower in price than their on-the-run equivalents.
Market argued that off-the-run securities had lower prices since the market for off-the-run securities was less liquid than the market for on-the-run securities. Solomon's arbitrage desk, building upon the assumptions of BSM model, correctly perceived that to be irrational. While that strategy [of-the-run / on-the-run convergence trade] worked well for some time, ultimately the market became efficient enough to arbitrage any spreads between two, or more, bonds that had the same yield, coupon and tenor.
But that didn't stop LTCM to further pursue the convergence strategy, but in slightly different form. Since bond prices are not as volatile as equities, and price movements are usually just a few cents, LTCM levered it's balance sheet to astronomical levels. This approach guaranteed it above-average return on equity, but in it's best year LTCM's return on assets was only 2.45%.
Venturing into European equities, event-driven arbitrage, European bond arbitrage (similar to convergence trade, but with more macro-economic uncertainty) risk profile of LTCM's balance sheet changed drastically, but it's VaR remained the same as it did when LTCM was involved only in convergence trade. They have blindly followed their models, without questioning the assumption behind those models. Something that would be repeated in the current crisis (in a slightly different form of pricing structured products and arguments behind high ratings given to those structured products).
Soon LTCM's positions grew so large that the markets wouldn't have enough liquidity if LTCM had to liquidate them. But the models showed large dis-equilibrium and LTCM' traders added more to their positions believing that no matter how large their position, market would accommodate potential unwind with necessary liquidity. Shorting macro-volatility across assets, LTCM's risk profile grew by the day, and as more markets became over-crowded, LTCM applied it's models to such exotics as Russian short-term bonds.
Then Russia defaulted and volatility shot up. Most of LTCM' positions were illiquid, and LTCM soon lost all of it's equity.
This is just one of the examples where financial modeling went wrong (there are more recent cases such as: AIG swap portfolio valuation, valuation of structured products, quant wipeout in 2007 which was very similar to LTCM fiasco etc etc).
In 2008 Emanuel Derman and Paul Wilmott (two most famous names among quants) wrote the following in the article published in Business Week:
"Financial markets are alive. A model, however beautiful, is an artifice. To confuse the model with the world is to embrace a future disaster in the belief that humans obey mathematical principles.
How can we get our fellow modelers to give up their fantasy of perfection? We propose, not entirely in jest, a model makers' Hippocratic Oath:
• I will remember that I didn't make the world and that it doesn't satisfy my equations.
• Though I will use models boldly to estimate value, I will not be overly impressed by mathematics.
• I will never sacrifice reality for elegance without explaining why I have done so. Nor will I give the people who use my model false comfort about its accuracy. Instead, I will make explicit its assumptions and oversights.
• I understand that my work may have enormous effects on society and the economy, many of them beyond my comprehension. "
The general consensus, with which I agree, is that the introduction of mathematical knowledge had vastly improved financial markets. That improvement translated into less risk for all market participants and into multi-year growth. But over-reliance on models, and models alone, no matter what the assumptions underlying the mathematics of those models may be, caused greater and greater systemic shocks.
What we must remember is; models are only representations of beliefs, not definitive statements about how the World operates. Risk modeling, and modeling in general was meant to be perceived as a set of tools, of guidelines to help market participants reduce their risk, not to eliminate risk in its totality.
When models went from being perceived as representations of belief, to statements about how the World operates, when "Human factor" was reduced to the minimum, finance stepped over the boundary of "scientific" into the area of dogmatic.
There are no fundamental laws of finance, there are no axioms of finance, only conjectures and beliefs of finance. And in that difference lies the problem. A system which is only based on probabilities (of any kind) is unable to produce true statements about itself, only valid statements, which validity needs always to be tested and questioned by observing empirical phenomena that underlay it's most basic assumptions.
We can not blindly rely on mathematical models to measure risk in financial world. There is no proof theory devoted to finance, there is no logic devoted to finance, only computations.
In conclusion. The state of financial markets is in no better shape today, than it was before the emergence of this crisis. Most of the basic assumptions are still considered true, most of basic modeling techniques are still used same as before.
Until that is changed, we continue to dance on the verge of a cliff with no safety net protecting us from the consequences if we fall.