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Fat-Tailed Butterflies Invade the Finance Faculty Lounge

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Book Review

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.

Furthermore, asset prices exhibit a constant volatility across time scales. “A month looks like a day, one set of days like another,” just as a fractal geometer would expect.

The authors also propose that the assumption of the independence of jumps be abandoned. In this case, the commonsense of traders has always conflicted with finance theory. Traders routinely speak of “hot streaks,” or “momentum.” “All illusion!” the theorists have replied, in the spirit of Bachelier and the random walk theory. Each step a drunk takes as he staggers about a lamppost is independent of the step he took before, or the one he will take next.

Here, Messrs Mandelbrot and Hudson side with the traders, and they use illustrations that remind me a bit of that famous butterfly. “Think of a small country, like Sweden, where every big company does business, directly or indirectly, with every other one. Volvo does something that affects Saab–say, launches a new car model that steals market share. Saab comes back with a fancier car, making satellite-location services standard … so Ericsson starts selling more Global Positioning System receivers. And so it spins on, throughout the Swedish economy–and spilling gradually into neighboring Finland … and as far around the world in ever-diminishing ripples as we can measure it.” This mutual dependence of the commercial world will naturally be echoed in, and then feed back into that world from, financial markets, the authors suggest. Furthermore, the dependence of future upon past events does much work in creating those apparently unlikely events, cascading bad fortune of the sort that destroyed Long-Term Capital Management. The butterflies are fat tailed, one might say, in defiance of entomology.

But then, what is to be done? What does all this mean for, say, the chief financial officer of a corporation, trying to hedge his company’s currency position? What does it mean for the hedge funds or speculators who might be tempted to strike a deal with that CFO? What does it mean for the ordinary working stiff trying to invest some of his savings wisely in anticipation of retirement? It means very little, as yet. The authors acknowledge that their “multifractal model of asset returns” is still in its infancy. Indeed, although they acknowledge that fractals are now in vogue in some financial circles, they are wary of this, “as with any fashion there is often more show than substance.” They believe that much fundamental research is necessary before a new model can replace the old.

Since I am almost comically underqualified to pass judgments on the merits of this book, I convened (through cyberspace) a panel of experts, consisting of authors who have written well-received books on related subjects, and I asked my panels’ opinion. Perhaps the consensus opinion was best expressed by Salih Neftci, author of “An Introduction to the Mathematics of Financial Derivatives,” (1996).

Mr. Neftci agreed that financial data do exhibit more that just random walk behavior. He agreed, too, power laws “do not turn out to be integers, but fractions. This effect is fairly significant.” The fractal model, then, “is a useful avenue to pursue but it is also one of several possible approaches.”

Mr. Neftci’s views might be especially significant in that he, a professor at the Graduate School of the City University of New York, is a pillar of the establishment that Messrs. Mandelbrot and Hudson propose to overthrow. We can perhaps take it as a given, then, that this particular establishment, the masters of the house of modern finance, are well aware of the cracks to which their critics point and don’t even quarrel with the proposition that a new foundation might be necessary. They are, though, sensibly curious about what exactly the new contractor proposes to do and want to see at least a full blueprint before their existing abode is razed.

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Contact Bob Keane with questions or comments at: [email protected]</a.


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