Sunday, May 5, 2013

Benoît Mandelbrot's contribution to Finance

  Other mathematicians of probability like Kolmogorov may be more academic or progressive but Mandelbrot was unique he proved that mathematicians actually understand randomness, he is called by Nassim Nicholas Taleb as 'poet of randomness'. Before Nassim Taleb, Black Swans were dealt by him in a philosophical and aesthetic way.Mandelbrot was initially a probability guy but later went into other fields of maths and made his name in other fields.In 1960s Mandelbrot presented his ideas on prices of commodity and stock prices and made a contribution on mathematics of randomness in economic theory.Mandelbrot also knew the pitfalls in Louis Bachelier's model.Mandelbrot linked randomness to geometry and made randomness a more natural science.If stock markets were Gaussian then stock market crashed would have happen once in a Billion years. Mandelbrot's randomness methods make the statistics methods look useless. 

Benoît Mandelbrot , the late Sterling Professor of Mathematical Sciences at Yale University

The first formal model for security price changes was put forward by Louis Bachelier (1900). His price difference process in essence sets out the mathematics of Brownian Motion before Einstein and Wiener rediscovered his results in 1905 and 1923 in the context of physical particles, and in particular generates a Normal (i.e. Gaussian) distribution where variance increases proportionally with time. A crucial assumption of Bachelier’s approach is that successive price changes are independent. His dissertation, which was awarded only a “mention honorable” rather than the “mention très honorable” that was essential for recognition in the academic world, remained unknown to the financial world until M.F. M. Osborne , who made no reference to Bachelier’s work, rediscovered Brownian Motion as a plausible model for security price changes.

But in 1963 the famous mathematician Mandelbrot produced a paper pointing out that the tails of security price distributions are far fatter than those of normal distributions (what he called the “Noah effect” in reference to the deluge in biblical times) and recommending instead a class of independent and identically distributed “alpha-stable” Paretian distributions with infinite variance. Towards the end of the paper Mandelbrot observes that the independence assumption in his suggested model does not fully reflect reality in that “on closer inspection … large changes tend to be followed by large changes – of either sign – and small changes tend to be followed by small changes.” Mandelbrot later called this the “Joseph effect” in reference to the biblical account of seven years of plentiful harvests in Egypt followed by seven years of famine. Such a sequence of events would have had an exceptionally low probability of taking place if harvest yields in successive years were independent. While considering how best to model this dependence effect, Mandelbrot came across the work of Hurst (1951, 1955) which dealt with a very strong dependence in natural events such as river flows (particularly in the case of the Nile) from one year to another and developed the Hurst exponent H as a robust statistical measure of dependence. Mandelbrot’s new model of Fractional Brownian Motion, which is described in detail in Mandelbrot & van Ness (1968), is defined by an equation which incorporates the Hurst exponent H. Many financial economists, particularly Cootner (1964), were highly critical of Mandelbrot’s work, mainly because – if he was correct about normal distributions being seriously inconsistent with reality – most of their earlier statistical work, particularly in tests of the Capital Asset Pricing Model and the Efficient Market Hypothesis, would be invalid. Indeed, in his seminal review work on stockmarket efficiency, Fama (1970) describes how non-normal stable distributions of precisely the type advocated by Mandelbrot are more realistic than standard distributions .

Partly because of estimation problems with alpha-stable Paretian distributions and the mathematical complexity of Fractional Brownian Motion, and partly because of the conclusion in  Andrew Lo's work (1991) that standard distributions might give an adequate representation of reality, Mandelbrot’s two suggested new models failed to make a major impact on finance theory, and he essentially left the financial scene to pursue other interests such as fractal geometry. However, in his “Fractal Geometry of Nature”, Mandelbrot (1982) commented on what he regarded as the “suicidal” statistical methodologies that were standard in finance theory: “Faced with a statistical test that rejects the Brownian hypothesis that price changes are Gaussian, the economist can try one modification after another until the test is fooled. A popular fix is censorship, hypocritically called ‘rejection of outliers’. One distinguishes the ordinary ‘small’ price changes from the large changes that defeat Alexander’s filters. The former are viewed as random and Gaussian, and treasures of ingenuity are devoted to them .The latter are handled separately, as ‘nonstochastic’."

Shortly after the “Noah effect” manifested itself with extreme severity in the collapse of Long-Term Capital Management, Mandelbrot (1999) produced a brief article, the cover story of the February 1999 issue of “Scientific American”, in which he used nautical analogies to highlight the foolhardy nature of standard risk models that assumed independent normal distributions. He also pointed out that a more realistic depiction of market fluctuations, namely Fractional Brownian Motion in multifractal trading time, already existed.

Fractals are linked with power laws, Mandelbrot worked on it and applied it to randomness. Mandelbrot designed the mathematical object called "Mandelbrot set" and later worked on shapes and fractals of maths and also worked on Chaos Theory. These objects play an important role on aesthetics , music , architecture , poetry , gestures and tones are derived from fractals . Mandelbrot's book "Fractal Geometry of Nature" it made a fame in arts , visual arts and every artistic circle. Many artists used to call Mandelbrot "The Rock Star of Mathematics". Mandelbrot became famous because of the number of applications of mathematics in our society.

Mandelbrot used his fractal theory to explain the presence of extreme events in Wall Street.

In fact he was one of the pioneers in studying the variation of financial prices even before Bchelier's Brownian model became widely accepted in academia and Mandelbrot also knew the pitfalls in Bachelier's model.For this reason many call him as the "father of Quantitative Finance".Mandelbrot has been best known since the early 1960s as one of the pioneers in studying the variation of financial prices.He pointed out that two features of Bachelier's model are unacceptable (in 1960s when Bachelier's model got accepted by academia and financial world). These flaws were based on power-law distributions and so Mandelbrot scaled these both by fractal theory and thus correcting the errors and flaws.Since then scaling by use of fractal theory has become important in finance and as well as in Physics.In fact Nassim Nicholas Taleb's "Black Swan Theory" is inspired by work of Mandelbrot as Mandelbrot was much concerned about high-risk rare events (Black Swans).Nassim and Mandelbrot collaborated in  research projects related to risk and randomness.

Mandelbrot's contribution in finance fall into three main stages:

He was the first to stress the essential importance, even in a first approximation, of large variations that may occur as sudden price discontinuities. The Brownian model is unjustified in neglecting them. They are not “outliers” one can safely disregard or study separately. To the contrary, their distribution is much more important than that of the "background noise" constituted by the small changes of Brownian motion. He followed this critique in by showing in 1963 that the big discontinuities and the small "noise" fall on a single power-law distribution and represented them by a scenario based on Levy stable distributions. He and Taylor introduced in 1967 the new notion of intrinsic "trading time." In recent years, fractal trading time and his 1963 model have gained wide acceptance.

Secondly, Mandelbrot tackled the fact that the “background noise” of small price changes is of variable “volatility.” This feature was ordinarily viewed as a symptom of non-stationality that must be studied separately. To the contrary, Mandelbrot interpreted this variability as indicating that price changes are far from being statistically independent. In fact, for all practical purposes, their interdependence should be viewed as continuing to an infinitely long term. In particular, it is not limited to the short term that is studied by Markov processes and more recently ARCH and its variants. In fact, it too follows a power-law side of dependence. He followed this critique and illustrated long-dependence by introducing in 1965  a process called fractional Brownian motion which has become very widely used.

Thirdly, he introduced the new notion of multifractality that combines long power-law tails and long power-law dependence. Early on, his work was motivated by the context of turbulence, but he immediately observed and pointed out that in 1972 the same ideas also apply to finance. After a long hiatus while he was developing other aspects of fractal geometry, he returned to finance in the mid-1990s and developed the multifractal scenario theory in detail in his 1997 book "Fractals and Scaling in Finance". The concept of scaling invariance used by Mandelbrot started by being perceived as suspect, because at that time other fields did not use it. However the period after 1972 also saw the growth of a new subfield of statistical physics concerned with “criticality.” The concepts used in that field are similar to those Mandelbrot had been using in finance.In "The Misbehavior of Markets", another popular book by Mandelbrot,he argues that the Gaussian models for financial risk used by economists like William Sharpe and Harry Markowitz should be discarded, since these models do not reflect reality. Mandelbrot argues that fractal techniques may provide a more powerful way to analyze risk.

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