Fractals and Finance

Question: How can we understand financial market fluctuations in fractal terms?

Benoit Mandelbrot: Well, what I discovered quite a while ago in fact, that was my first major piece of work is that a model of price variation which everybody was adopting was very far from being applicable.  It’s a very curious story. 

In 1900, a Frenchman named Bachelier, who was a student of mathematics, wrote a thesis on the theory of speculation.  It was not at all an acceptable topic in pure mathematics and he had a very miserable life.  But his thesis was extraordinary.  Extraordinary in a very strange way.  It applied very well to Brownian motion, which is in physics.  So, Bachelier was a pioneer of a very marvelous essential theory in physics.  But to economics, it didn’t apply at all, it was very ingenious, but Bachelier had no data, in fact no data was available at that time in 1900, so he imagined an artificial market in which certain rules may apply.  Unfortunately, the theory which was developed by economists when computers came up was Bachelier’s theory.  It does not account for any of the major effects in economics.  For example, it assumes prices are continuous when everybody knows the prices are not continuous.  Some people say, well, all right, there are discontinuities but they are a different kind of economics that we are doing, not because certain discontinuities become too complicated and only will the **** look more or less continuous.  But it turns out that discontinuities are as important, or more important than the rest. 

Bachelier assumed that each price changes in compared of the preceding price change.  It’s a very beautiful assumption, but it’s completely incorrect because we know very well, especially today that for a long time prices may vary moderately and then suddenly they begin to vary a great deal.  So, even we’re saying that the theory changes or you say the theory which exists is not appropriate.  What I found that Bachelier’s theory was defective on both grounds.  That was in 1961, 1962, I forgot the exact dates and when the development of Bachelier became very, very rapid.  Since nobody wanted to listen to me, I did other things.  Many other things, but I was waiting because it was quite clear that my time would have to come.  And unfortunately, it has come, that is, the fluctuation of the economy, the stock market, and commodity markets today are about as they were in historical times.  There was no change which made the stock market different today than it was long ago.  And the lessons which are drawn from **** peers do represent today’s events very accurately.  But the situation is much more complicated than Bachelier had assumed.  Bachelier, again, was a genius, Bachelier had an excellent idea which happened to be very useful in physics, but economics, he just lacked data.  He did not have awareness of discontinuity which is essential in this context.  Not having an awareness of dependence, which is also essential in this context.  So, his theory is very, very different from what you observe in reality.

Recorded on February 17, 2010
Interviewed by Austin \r\nAllen

\r\n

The mathematician has long believed the traditional understanding of market fluctuations would need to be replaced with his fractal model. Unfortunately, he says, "my time…has come."

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