Fractional Brownian Motion in Multifractal Trading Time

Since the book came out, it’s been a real treat trying to explain to friends and family what a fractal is.  My grandma says she understood it one afternoon a few weeks ago, but can’t remember how it made sense at the time.  Most people just assume it has something to do with fractions and leave it at that.  The truth is, I’m not even remotely a mathematician (although it might have been my best subject in school), and don’t understand enough about the interactions between computers and modern mathematics to really understand fractal geometry, either.  I find fractals fascinating on a metaphorical, poetic level — the concept of recursive symmetry, the interplay of order and chaos — but if you asked me to create a Mandelbrot set, I wouldn’t know where to start.

As I’m forced to talk about the book more and more, I feel like I really should become more comfortable with the geometry itself, so whenever I come across a link to anything fractal-related, I read it.  I found two yesterday.  The first is really just a teaser for the longer article, reprinted from a 1999 issue of Scientific American, which the journal is hoping you’ll find relevant today.

How Fractals Can Explain What’s Wrong with Wall Street” was written 10 years ago by Benoit Mandelbrot himself.  I’ll summarize, so you don’t have to read:  Traditional models for maximizing portfolio growth ignore the chaos inherent in the market, because it is, after all, chaotic — you can’t predict it, so why bother?  Mandelbrot recognizes that the market time-lines — those line-graphed cliffs in the price indexes we’ve been seeing on the news every day — display fractal geometry.  Just looking at a graph, you can’t tell if the time scale is an hour, a day, or a year — no matter how close you look, or how far you stand back, the fluctuations look the same.  So he proposes using fractal models to shock-test your portfolio.  You can’t actually predict what’s going to happen, but you can run your money though a chaos generator over and over again and see what the odds are of it spontaneously combusting.

Here’s a metaphor:  Your stock portfolio is a ship on the ocean.  You know it can handle 6 foot waves, because that’s all you’re used to seeing.  But how big could the waves get, and how fierce would they have to be for your boat to sink?  Traditional portfolio theory can never answer that question — but Mandelbrot can run the numbers through his algorhythms, and tell you how risky your voyage really is.

Apparently no one actually did this 10 years ago, or else we wouldn’t be in this mess, right?

This has been your monthly lesson in the uses of fractal geometry. Tune in next time to see how we can graphically model strike rates at a bowling alley…

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