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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

Copyright © 1999 by John R. Schuyler. All rights reserved. Permission to copy with
reproduction of this notice.