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Mandelbrot Beats Economics in Fathoming Markets: Mark Buchanan

December 06, 2011

Dec. 6 (Bloomberg) -- The possible collapse of the European monetary union, at least in its current form, brings home the truth that there’s little in economics that is certain. We’re again “thinking the unthinkable,” as we were a few years ago when we suddenly realized that financial engineering hadn’t banished financial crises, and that 70 years of relative stability since the Great Depression didn’t guarantee a thing.

It seems that we’re complete suckers for the illusion of certainty and the seeming unlikelihood of the unthinkable, even though financial and economic history is one long string of crises. This time always seems different, until it turns out not to be.

Nothing in mainstream “neoclassical” finance theory explains these persistent crises. Almost without exception, economists since Adam Smith have viewed economic systems as being in balance or equilibrium, and as having a natural tendency to return there after any disturbance. In this view, crises can be understood only as anomalies, the consequences of unusual outside shocks.

All this makes for tidy and comforting theory, with simple mathematics, but it fails utterly to account for the most basic market dynamics. Most notably, large and violent events -- like the stock market crash of 1987 or the flash crash of last May -- happen far more frequently than equilibrium theories suggest. In fact, the pronounced frequency of market upheavals is precisely what’s most constant in economics.

Over the past 15 years or so, physicists have demonstrated this in mathematical studies of market volatility. Inspired by work of the mathematician Benoit Mandelbrot in the 1960s, these scientists have used enormous sets of historical data -- hundreds of millions of minute-by-minute prices stretching over more than a decade, and daily and monthly prices over half a century -- to show that large market movements, up or down, follow a single mathematical pattern.

Larger movements of, say, 10 percent to 15 percent, are less likely than movements of 3 percent to 5 percent. And the probability of a movement decreases in simple inverse proportion to the cube of its size: If moves of 5 percent or more have a certain likelihood, then moves of 10 percent or more are 8 (2 cubed) times less likely, and moves of 20 percent or more are in turn 8 times even less likely. But they still occur with some regularity.

This pattern, it turns out, can be seen in markets for stocks, foreign exchange and futures around the world. And it is every bit as regular as the statistical patterns physicists know for the movements of molecules in solids, liquids and gases, or for innumerable other physical phenomena. It’s too regular to be an accident, and it is as deserving of fundamental explanation as anything in the rest of science.

‘The Fat Tail’

There are practical implications, of course, as Nassim Taleb explored in his book “The Black Swan.” Among other things, this particular pattern -- scientists describe it as a “fat tail” in the probability distribution -- suggests that really big movements are much more common than we might suppose. Thinking of markets in terms of the usual statistics that apply to things like people’s heights or weights or test scores leads to gross underestimation of the risks of rare catastrophes.

A credible economic theory of markets -- something we do not yet have -- would explain why the distribution of market returns shows such a preponderance of large events. It would account for why the mathematical form of this distribution is so uniform in markets the world over.

And, importantly, it would explain why markets share the same pattern with many other natural systems. Look, for example, at the flow of traffic on the Internet, solar flare activity on the Sun, the course of biological evolution or earthquake frequency on the San Andreas fault, and you see patterns strikingly similar to those in financial markets. All these systems and many others exhibit a naturally irregular rhythm in which long periods of relative quiescence are sporadically broken by bursts of intense upheaval.

The mathematical similarity between markets and other natural systems goes deeper still. Consider volatility -- a measure of the momentary vigor of a market’s erratic movements up and down. Eugene Stanley, a physicist at Boston University, and his colleagues have shown in recent studies that bursts of high volatility in the stock market tend to cluster in time in the same way that earthquake aftershocks do. They follow a precise pattern known as Omori’s Law -- named for Fusakichi Omori, the Japanese seismologist -- which describes the likelihood of aftershocks in the days and weeks following an earthquake.

This similarity between market movements and earthquake aftershocks seems weird if you think of a financial market as a system in equilibrium that naturally balances people’s conflicting aims and desires. But it looks less weird and even quite natural if you think of a market as just another system that, like the Earth’s crust, is perpetually driven out of balance by the action of various forces, and responds to those forces in complex, dynamic ways.

It’s not so surprising that the best emerging models of markets look a lot like models of disequilibrium processes in other areas of science. They eschew the “rational agents” of neoclassical economic theory and instead view markets as systems involving interactions among people and companies with realistic characteristics -- prone to mistakes, always learning and adapting, and often copying what others do. These models see markets as driven by feedback and instability, as is the case with most other natural systems.

It’s fair to say these models don’t yet give us an adequate understanding of the basic patterns we see in markets, but they at least move in the right direction by taking the historical data seriously and trying to explain it. Nothing in mainstream economics seems as likely to succeed in this. “I urge students to read narrowly within economics, but widely in science,” Vernon Smith wrote in his 2002 Nobel Prize for economics lecture, because “within economics there is essentially only one model to be adapted to every application.”

Simple ideas of equilibrium and balance may flatter our Platonic prejudices, but they don’t fit the real world. Nothing in the mathematics and physical science of the past 30 years stands out so much as the increasing importance of the irregular, the chaotic and the disordered in every part of the natural world. In many cases, this disorder isn’t simply random, but rather contains important regularities. Finding the expected disorder in the marketplace, and understanding its origins, could give economics a much stronger scientific foundation.

(Mark Buchanan, a theoretical physicist and the author of “The Social Atom: Why the Rich Get Richer, Cheaters Get Caught and Your Neighbor Usually Looks Like You,” is a Bloomberg View columnist. The opinions expressed are his own.)

--Editors: Mary Duenwald, Brenda Batten.

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To contact the writer of this article: Mark Buchanan at buchanan.mark@gmail.com

To contact the editor responsible for this article: Mary Duenwald at mduenwald@bloomberg.net


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