Mandelbrot, Benoit and Hudson, Richard L. “The (mis)Behavior of Markets: A Fractal View of Risk, Ruin, and Reward,” New York: Perseus Books Group, 2004, pp. 328, Cloth. US$27.50.
Most literate humans have by now heard of “chaos theory.” There has even been a Hollywood movie, “the Butterfly Effect,” intended to illustrate the great clich? of the field, the notion that the flap of a butterfly’s wings in Brazil can stimulate a tornado in Texas.
We owe that illustration to chaos pioneer Edward Lorenz, a meteorologist at the Massachusetts Institute of Technology. He concluded in a 1962 paper that there are limits to weather prediction–since it is impossible to know how every “seagull” everywhere will flap its wings. (In later speeches and essays he substituted the more picturesque butterfly.)
In its study of such limits of the knowable, chaos theory has come to incorporate fractal geometry, the study of figures with a kind of complexity that proves independent of scale. In a fractal figure, an attempt to break down a whole into simpler parts is foiled when the parts turn out to be just as complex as the whole and to be themselves composed of parts equally complex–and so on indefinitely. Fractal geometry turns its practitioners into Seussian elephants, looking for tiny but complete worlds on each speck of dust, in the expectation that those worlds will contain their own specks of dusts, with their own still smaller worlds.
The best-known fractal figure is the “Mandelbrot set,” which takes its name from the author of this book, Polish (French-educated) mathematician Benoit Mandelbrot. It’s called a “set” because it’s defined by the points on a plane that satisfy an equation. The equation itself, although relatively simple, contains a feedback loop that results in the intricacies of this shape–with its arms made out of swirling galaxies that themselves prove to contain arms made out of swirling galaxies.
In this book, Mr. Mandelbrot addresses the “modern house of finance.” He believes that the building is cracked, and the problems go to the foundations. Fractal geometry, he writes (with the assistance of Richard Hudson, former managing editor of the Wall Street Journal’s European edition) will repour those foundations.
It is a common observation by now that bell curves in finance always have “fat tails,” i.e. the odds of an extraordinary event are far greater than one would expect were every tick in asset price a step in a random walk, on the model proposed by Louis Bachelier more than a century ago. Finance theorists haven’t done anything fundamental about these fat tails. They have conceded their existence and sought to tack them on as amendments to the underlying random-walk model, the very model that makes them an anomaly.
By way of repouring the foundation, though, Messrs. Mandelbrot and Hudson propose, first, that we should regard the volatility of any market as an instance of a “power law,” i.e. a correlation in which the size of a price change varies with a power of the frequency of the change. The bell curve would be a special case, a power law in which the power equals -2. Far greater volatility exists in assets that show what is known as a Cauchy distribution, in which the power equals -1. The absence of any correlation between size and frequency of change (an asset manager’s nightmare in which a crash is just as likely at any moment as a one-penny tick) would be a power of zero. Empirical research shows that the actual power (which these authors call alpha and usually state without the negative sign, which can be taken as implied) varies between 2 and 1, between “mild” and “wild.” In the case of cotton prices, for example, alpha is 1.7.