"Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line".
"Being a language, mathematics may be used not only to inform but also, among other things, to seduce".
It does not matter whether you have heard of the name of the author of the above two quotes, but it matters if you are dealing with finance and financial markets.
Benoit Mandelbrot, author of the above two quotes, has had a remarkable career which includes seminal work in theoretical and applied mathematics. Mandelbrot might be largely unknown to the wider world, but for the beautiful pictures that can be produced on a computer using the fractal equations he popularized (including the famous Mandelbrot set, named after its discoverer).
Most people who work in mathematics write for an audience of their colleagues. The majority of Mandelbrot's writing falls into this category. He recently published Fractals and Chaos (Springer-Verlag, 2004), which is a collection of some of his papers from 1979 onward. Mandelbrot's papers can be difficult reading for anyone who is not a skilled mathematician. Mandelbrot has also written for a more general audience. His book The Fractal Geometry of Nature (Freeman, 1982) can be read by anyone who has a solid high school math background and patience with an academic writing style.
But his latest book, selling like hot cake, The (Mis)behavior of Markets Mandelbrot is writing for the general reader, who usually has no tolerance for mathematical equations.
The roots of the book The (Mis)behavior of Markets go back to 1961 when Mandelbrot was a new researcher at IBM. Among other things, he was working on using computers to analyze the distribution of income in a society. Mandelbrot's work echoed the work of Vilfredo Pare to and showed that many economic factors, including wealth, are distributed according to an inverse power law. Most of the economists have claimed that the change in market prices followed a Gaussian distribution. This distribution describes many natural features, like height, weight and intelligence among people. The Gaussian distribution is one of the foundations of modern statistics. If economic features followed a Gaussian distribution, a range of mathematical techniques could be applied in economics.
The behavior of markets reflects a complex system and fractal mathematics. In The (Mis)behavior of Markets Mandelbrot 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.
But can Fractals Explain What's Wrong with Wall Street?
Individual investors and professional stock and currency traders know better than ever that prices quoted in any financial market often change with heart-stopping swiftness. Fortunes are made and lost in sudden bursts of activity when the market seems to speed up and the volatility soars.
According to Benoit Mandelbrot, the classical financial models used for most of this century predict that such precipitous events should never happen. A cornerstone of finance is modern portfolio theory, which tries to maximize returns for a given level of risk. The mathematics underlying portfolio theory handles extreme situations with benign neglect: it regards large market shifts as too unlikely to matter or as impossible to take into account. It is true that portfolio theory may account for what occurs 95 percent of the time in the market. But the picture it presents does not reflect reality, if one agrees that major events are part of the remaining 5 percent. An inescapable analogy is that of a sailor at sea. If the weather is moderate 95 percent of the time, can the mariner afford to ignore the possibility of a typhoon?
The risk-reducing formulas behind portfolio theory rely on a number of demanding and ultimately unfounded premises. First, they suggest that price changes are statistically independent of one another: for example, that today's price has no influence on the changes between the current price and tomorrow's. As a result, predictions of future market movements become impossible. The second presumption is that all price changes are distributed in a pattern that conforms to the standard bell curve. The width of the bell shape (as measured by its sigma or standard deviation) depicts how far price changes diverge from the mean; events at the extremes are considered extremely rare. Typhoons are, in effect, defined out of existence.
Do financial data neatly conform to such assumptions? Of course, they never do. Charts of stock or currency changes over time do reveal a constant background of small up and down price movements-but not as uniform as one would expect if price changes fit the bell curve.
According to portfolio theory, the probability of these large fluctuations would be a few millionths of a millionth of a millionth of a millionth. (The fluctuations are greater than 10 standard deviations.) But in fact, one observes spikes on a regular basis-as often as every month-and their probability amounts to a few hundredths.
Modern portfolio theory poses a danger to those who believe in it too strongly and is a powerful challenge for the theoretician. Though sometimes acknowledging faults in the present body of thinking, its adherents suggest that no other premises can be handled through mathematical modelling. This contention leads to the question of whether a rigorous quantitative description of at least some features of major financial upheavals can be developed. The bearish answer is that large market swings are anomalies, individual "acts of God" that present no conceivable regularity. Revisionists correct the questionable premises of modern portfolio theory through small fixes that lack any guiding principle and do not improve matters sufficiently.
Mandelbrot claims that variations in financial prices can be accounted for by a model derived from his work in fractal geometry. Fractals-or their later elaboration, called multifractals-do not purport to predict the future with certainty. But they do create a more realistic picture of market risks.
A fractal is a geometric shape that can be separated into parts, each of which is a reduced-scale version of the whole. In finance, this concept is not a rootless abstraction but a theoretical reformulation of a down-to-earth bit of market folklore- namely, that movements of a stock or currency all look alike when a market chart is enlarged or reduced so that it fits the same time and price scale. An observer then cannot tell which of the data concern prices that change from week to week, day to day or hour to hour. This quality defines the charts as fractal curves and makes available many powerful tools of mathematical and computer analysis.
A more specific technical term for the resemblance between the parts and the whole is self-affinity. This property is related to the better-known concept of fractals called self-similarity, in which every feature of a picture is reduced or blown up by the same ratio-a process familiar to anyone who has ever ordered a photographic enlargement. Financial market charts, however, are far from being self-similar. In a detail of a graphic in which the features are higher than they are wide-as are the individual up-and-down price ticks of a stock-the transformation from the whole to a part must reduce the horizontal axis more than the vertical one. For a price chart, this transformation must shrink the timescale (the horizontal axis) more than the price scale (the vertical axis). The geometric relation of the whole to its parts is said to be one of self-affinity.
