Mathematicians predicted stock market volatility years ago

Crash, crash. Doom, doom. Panic on the Street. Wall Street yesterday suffered its worst decline since the Sept. 11 terrorist attacks – and the free fall suggests it may be time to consider burying the family stash of Krugerrands under the rose garden or tool shed.

The list of Chicken Little events was long:  investment house Lehman’s bankruptcy, the capital crunch for mega-insurer AIG, the sale of Merrill Lynch to Bank of America. But Stormy Monday (a weather system that appears to be carrying over to be lingering) probably came as no surprise to a cadre of mathematicians who specialize in surveying the dynamics of markets.

One of those long-time market watchers is fractal pioneer Benoit Mandelbrot. In 1999, Scientific American published an article by Mandelbrot that showed how fractal geometry can model market volatility, while revealing the intrinsic deficiencies of a cornerstone of finance called modern portfolio theory (for which there has been awarded more than one Nobel Prize in Economics).

Mandelbrot, 83, contends that portfolio theory, which tries to maximize return for a given level of risk, treats extreme events (like, say, yesterday's market shockers) with “benign neglect: it regards large market shifts as too unlikely to matter or as impossible to take into account.” The faulty assumption of modern portfolio theorists, in Mandelbrot’s view, is that price changes do not drift far from the mean when observing daily ups and downs—so extreme events are exceedingly rare. “Typhoons, in effect, are defined out of existence,” he wrote.

In place of modern portfolio theory’s reliance on the canonical Bell Curve, Mandelbrot drags in (surprise!) the fractal. A fractal is a geometric shape that can be divvied up into parts, each of which is a Mini-Me facsimile of the whole. If you look closely enough, you can see fractals everywhere. Besides monotonous screen savers, fractal patterns describe the distribution of galaxies and the shape of coastlines. Mandelbrot devised so-called multi-fractal generators that can use historical market data to simulate alternative scenarios of where stocks or other securities might be headed.

The experience of editing Mandelbrot a decade ago was both fun and exasperating. Trying to translate the complexities of multifractals into a finished text demanded a lot of neural gear grinding to say the least.
Dealing with this renowned creator of the Mandelbrot set was another challenge.

The conversation always seemed to drift back to his other favorite subject: how the rest of the world  did not appreciate the  value of his work. At one point, I asked  whether he believed he should receive a Nobel (highly unlikely for a mathematician).  One does not discuss such things, he said, as if it might jinx his chances.

We received a stack of letters and e-mails after the article was published, including a bundle from adherents of  the so-called Elliott Wave Theory alleging that Mandelbrot had lifted his ideas (on the multifractal nature of markets) from the work of Ralph Nelson Elliott in the 1930s. An insulted Mandelbrot dismissed the charges with withering disdain: “The idea is ancient, but his use and mine stand in absolute contrast,” he responded in the mag's letters column.

Perhaps the most telling criticism of Mandelbrot’s work comes from the markets themselves. In the decade or so since his article was published, the use of multifractal market analysis is still largely an academic endeavor. But Mandelbrot should not be judged too harshly. Multifractals may not be in routine use on the trading floors. But Mandelbrot’s work on market extremes has served to broadcast to the Street a notion that has been known forever on the street: Yes, Virginia, sh*t really does happen.

Photo of Benoit Mandelbrot, 2007, by Rama





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