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Mar 13, 2009 05:10 PM | 6
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
Tags:
fractal geometry,
risk management,
the misbehavior of markets,
Mandelbrot set,
Fractals,
applied mathematics
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6 Comments
Add CommentIt is difficult to see the link between fractals and markets. Fractals are totally regular patterns repeated at all scales, and so can be easily programmed using recursive code. Turbulences are also patterns which repeat at all scales, but they are not regular. Financial markets are turbulent, and obey chaos theory, where small differences in initial conditions can lead to wildly varying outcomes, because of the irrationality of human behaviour.
Reply | Report Abuse | Link to thisvery pretty,
Reply | Report Abuse | Link to thismakes some of this other stuff seem somehow less important, or at least less all-consumingly dangerous
@eco-steve "Fractals are totally regular patterns" Not at all. Fractals are indeed characterized by self similarity. But it can be a self similarity in highly non regular patterns. Just look at a stock chart. Take out the time legend. Zoom / Dezoom, You won't be able to tell where you are because you will see always the same patterns of rise and crashes on one day, one week, one month, one year, ten years ...
Reply | Report Abuse | Link to thisMoreover have a look at the Mandelbrot set which is a fractal. Visually you will see it is not about regularity. Fractals are deeply linked to chaos theory, basically they are the visualization of chaos.
http://en.wikipedia.org/wiki/Mandelbrot_set
Finally, He is from a minority. He is American, lives in the US but has a dual culture. He wants to change the economy and he has a plan. Guess who he is ;-) http://harryseldon.thinkosphere.com/2009/02/23/yes-he-can-save-the-world
Harry_Seldon : Indeed Chaos Theory can be programmed using random number generators. But can you quote me a single computer program that can generate irregular 'fractals' such as turbulence patterns or market charts? Markets charts are not self-similar, but truly chaotic, because of the irrational nature of human behaviour. From the same perspective, patterns of random dots may appear similar, but they have no internal structure to relate them as being fractal...
Reply | Report Abuse | Link to thisTo understand the relationship of fractals to markets you have to understand or define what a market is. A market is a buyer and a seller who agree upon the current price of something and disagree about its future value. That being said, buyers and sellers are people. People are natural systems. Their emotions are also natural. Natural systems such as the flow of rivers, the shape of clouds, blood flow etc. do not obey linear mathematics. They flow or are formed according to the theory of chaos and therefore lend themselves to fractal analysis.
Reply | Report Abuse | Link to thisOne of my heroes, J.K. Galbraith, wrote: "The only function of economic forecasting is to make astrology
Reply | Report Abuse | Link to thislook respectable." -- It's still true. Mandelbrot confirms this.