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"A Multifractal Walk down Wall Street"

by Benoit B. Mandelbrot, Scientific American, Feb. 1999, pp. 70-73.

Portfolio theory is flawed. The customary theory holds that changes in prices follow a "random walk" that follows the normal distribution. [Or, I've seen elsewhere that differences in the log of prices is normal.]

Dr. Mandelbrot demonstrates convincingly that random deviations from a normal distribution do not characterize actual prices movements. With the customary theory, large fluctuations, e.g., greater than 10 standard deviations, would be extremely rare. "But in fact, one observes spikes on a regular basis—as often as every month—and their probability amounts to a few hundredths." Are financial markets "abnormal"? "They are what they are, and it is portfolio theory that is flawed."

Fractals patterns have been observed where designs in nature are self-replicated, regardless of scale. Coastlines, for example, appear to have the same geometry when viewed from space or afoot. Mandelbrot developed an elaboration of fractal geometry, "multifractals," that better characterizes variations in prices.

As I understand the process:

A trend line is projected.
The line segment is divided (interpolated) into three sections.
Step 2 is repeated until the time intervals are down to a scale of, say, the time between actual trades.

The ratios of the interpolated section lengths can be adjusted as a parameter. Moving the "multifractals" can model an increasingly volatile market. My explanation, without diagrams, cannot do this justice. What is most convincing is Maldelbrot's comparisons of actual price and exchange rate charts with some generated by his system.

I highly recommend this article as a starting point for persons wanting to develop real-to-life price forecasting simulation models.

Further reading (suggested at the article's end):

The Fractal Geometry of Nature by Benoit B. Mandelbrot, W. H. Freeman and Company, 1982.
Fractals and Scaling in Finance: Discontinuity, Concentration, Risk by Benoit B. Mandelbrot,Springer-Verlag, 1997.
Multifractals and 1/F Noise: Wild Self-Affinity in Physics by Benoit B. Mandelbrot,Springer-Verlag, 1999.

—John Schuyler, February 1999