The Misbehavior of Markets: A Fractal View of Financial Turbulence
Benoit Mandelbrot, Richard L. Hudson
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three—mild, slow, and wild—as if the realm of chance were a world in its own right, with its own peculiar laws of physics. Mild randomness, then, is like the solid phase of matter: low energies, stable structures, well-defined volume. It stays where you put it. Wild randomness is like the gaseous phase of matter: high energies, no structure, no volume. No telling what it can do, where it will go. Slow randomness is intermediate between the others, the liquid state. I first proposed some of my
analysis would reveal it to be so. Such is the power of fractals and chance working together: Simple rules build complex structures, and complex structures deconstruct into simple rules. Fractals in the physical world: clouds and cluster. With random processes added, we finally start to see the hand of nature. The top diagram is the work of a computer to illustrate the principle. It represents a completely artificial cloudy sky. The bottom diagram illustrates fractal growth starting from an
vanished—and in many cases had swung to a loss—when a more realistic price series was used. “The big, bold profits of Paper 1 must be replaced with rather puny ones,” he wrote. “I must admit that the fun has gone out of it somehow.” Alexander can be forgiven the mistake. Continuity is a common human assumption. If we see a man running at one moment here and a half-hour later there, we assume he has run a line covering all the ground in between. It does not occur to us that he may have stopped to
their own money to complete trades. Their function, according to the rules, is to “ensure the continuity of the market.” Lately, they have come into disrepute in the post-bubble scandals that have engulfed most of Wall Street. In the SEC study of the 1997 collapse mentioned earlier, the agency found specialists in the most tumultuous twenty-four minutes were powerful net buyers; the volume of their purchases exceeded their sales by a ratio of 2.06. These were good bets: Prices did recover.
has begun in finance. A first step is agreeing on a way to measure the intensity and path of a market crisis. The famous Richter Scale is the analogy most drawn upon. It measures the energy released by an earthquake on a logarithmic scale; for instance, a catastrophic quake of magnitude 7 packs ten times as much energy as a merely devastating quake of magnitude 6. What is a financial market’s analog to energy? Volatility, some have surmised. Thus, two University of Paris researchers recently