Fractals, the stock market and our very chancy world

Bell curves describe much of the world. The average major league baseball player hits about .265 over a season, and most players cluster around that number. More people hit .275 than .305; no one hits .650. Similar limits apply to outdoor temperature, freeway speeds, human height and much more. It’ll be a strange day when you see a 40-foot man driving a car 400 mph in the 400-degree heat. In statistics, bell curves tellingly fit “normal distributions.”

For a long time, people thought stocks followed the same rules. Prices would go up and down, but daily changes wouldn’t lurch to extremes. The most common mathematical models used to build portfolios and value stock options assumed that prices move relatively smoothly.

Benoit Mandelbrot puts this core assumption in a car and drives it off a bridge at 400 mph. The mathematician, who gained fame for his study of fractals as described in James Gleick’s 1987 bestseller “Chaos: Making a New Science,” believes that stock variation is better described by what are called power distributions than by bell curves. Sometimes -- much more frequently than standard theory predicts -- individual stocks and markets as a whole lurch wildly.

In his new book, “The (Mis)Behavior of Markets,” Mandelbrot describes the bucking stock market of August 1998: “The standard theories, as taught in business schools around the world, would estimate the odds of that final, August 31, collapse at one in 20 million -- an event that, if you traded daily for nearly 100,000 years, you would not expect to see even once. The odds of three such declines in the same month were even more minute: about one in 500 billion.”

We all know now that the stock market is capricious. But Mandelbrot meticulously explains why assuming that stock prices operated on a bell curve led to fundamental flaws in the mathematical tools used to value market risk. He gleefully notes that two of the original modelers of a key formula for valuing stock options lost their shirts in the 1998 collapse of their company, Long Term Capital Management.

Mandelbrot offers several reasons for why markets are turbulent and follow power laws, not bell curves. For example, prices aren’t continuous, as many people assume. They “leap, not glide,” depending on the actions of buyers or sellers in the stock market at a given time. But readers, he writes, need to know only that markets are vastly riskier than people assume. “To drive a car, you do not need to know how it goes; similarly to invest in markets, you do not need to know why they behave the way they do.”

Mandelbrot credits Richard L. Hudson, a former Wall Street Journal editor and reporter, as coauthor. But the book is written mainly in the first person and much of the narrative traces Mandelbrot’s life story. Thus, it shouldn’t be surprising when he connects his market analysis to the field in which he made his name: fractals.

The word “fractal” comes from a Latin word meaning “to break” and describes a phenomenon in which each part of a thing precisely echoes the whole. Rip a floret off a cauliflower head, and the piece will look like the head. Pull a smaller chunk from that piece and the same is true.

Similar patterns characterize seacoasts, tree branches and solar wind turbulence. One of Mandelbrot’s greatest contributions to mathematics and science is showing that fractals are found just about everywhere -- particularly in phenomena that otherwise appear chaotic.

Mandelbrot first saw fractals in stock tables while poring over cotton price charts. He later showed the fractal structure of the changing exchange rate of the U.S. dollar and the deutsche mark between 1973 and 1996. In such markets, the variation within one day looks the same as the variation over a week or over a minute.

Fractal modeling falls apart only when the units measured are less than two minutes or longer than 180 days, a breakdown he compares to the collapse of the normal laws of physics at the tiny quantum or the gigantic cosmic level.

Mandelbrot describes fractals wonderfully and fills the book with the pictures that have made them popular on T-shirts and screen savers. But he doesn’t fully explain why fractals connect so closely to the laws of chaos and power. He treats them more as an anthropologist talking about what exists than as a scientist detailing why.

Nor does Mandelbrot offer advice on how to beat the market. “This book will not make you rich,” he declares early on. This is refreshing. No one needs another book on how to get rich in 90 days, and his math is more trustworthy because he doesn’t have snake oil to sell. He concludes passively, recommending that major market regulatory agencies support fundamental mathematical research.

But if this conclusion demonstrates Mandelbrot’s humility, his prose does not. Saying that he toots his own horn is like saying that Dizzy Gillespie played the trumpet. Here’s a typical paragraph: “I developed these ideas over many decades of intellectual wanderings -- pulling together many stray, forgotten, under-explored, and seemingly unrelated artifacts and issues of the mathematical past, extending them in every direction, and creating a new, coherent body of mathematics.”

Still, as Mandelbrot surely knows, it’s better to be interesting and arrogant than humble and bland. The same goes for a book. *