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Benoit Mandelbrot and the wildness of financial markets


In a lecture at Columbia University this week, famed fractal pioneer Benoit Mandelbrot once again inveighed against traditional economic theories, returning at a time of financial malaise to many of the points he raised in a 1999 Scientific American feature. (In September 2008, as the U.S. economy began to shake, editor Gary Stix provided a brief recap of Mandelbrot's article and the ensuing response from readers in this blog post.)

Mandelbrot, 84, spoke at the Festival della Matematica, or Mathematics Festival, an event produced jointly in Rome and New York City by a consortium of Italian governmental and cultural agencies.

A persistent complaint levied by the Wolf Prize–winning French mathematician: many economic models ignore dramatic jumps, whether in a commodity's price or in an index such as the S&P 500, treating them as outliers. But real-life economic systems, Mandelbrot said, are "dominated by details"—the extreme cases, and specifically the outer 5 percent, are just as important as the rest of the data. To prove his point, Mandelbrot showed a graph of the S&P since 1985, overlaid with the same data minus the wild swings that constitute the outliers. The two graphs were completely different, implying that to ignore the extreme cases is to ignore reality. "I'm extremely visual," Mandelbrot said. "Often the pictures suggest the deeper truth underlying the formulas."

Mandelbrot also has beef with economists who model prices for shares or commodities using variations on so-called random walks, which assume that the price at any given moment depends on what it was the moment before. But prices, Mandelbrot noted, can be discontinuous, jumping instantly from one value to another without any graduated transition—more like a random teleportation. "Prices do not have any element of physical inertia," Mandelbrot said by way of illustrating the difference between economics and physical science, a difference that he said is all too often ignored. "A very large part of economic theory is just physical theory with the words changed," he said.

Theories grounded in the physical sciences, Mandelbrot said, presume that the markets harbor elements of randomness, but in a form that he calls "mild randomness." Mild randomness is embodied by the roulette wheel at a casino—each spin is random but over time the distribution of winning numbers averages out. (And, of course, over time the casino wins out.) He contends that more realistic models of economics—including, naturally, models based on fractals—are driven by "wild randomness," wherein things don't average out and individual freak occurrences matter. This wildness, he said, "imitates real phenomena in a very strong way."

Mandelbrot took care to note that economics is just one field that he has investigated in his decades of work on fractals and mathematical modeling. But it's certainly a timely one: a book on fractal-based financial risk management that he co-wrote in 2004 ("before things were bad"), called The (Mis)Behavior of Markets, has recently begun "selling like hotcakes," he said.

For more on the fractal pioneer, see this Scientific American article on the disputed provenance of the Mandelbrot set from the April 1990 issue—an issue, incidentally, that featured a back-page essay by a Tennessee senator named Al Gore.

Photo of Mandelbrot at Columbia © John Matson/Scientific American

The views expressed are those of the author and are not necessarily those of Scientific American.

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