### Oct

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# Mandelbrot, Fractals and Chaos Theory, by Dr. Phil McDonnell

October 10, 2006 |

In response to Professor Haave's query: *"Mandelbrot does a good job attacking modern finance theory, and he does a good job explaining what the rest of us call "fat tails". But otherwise, well, is it the same merit as Elliot Waves?"*

There are several things wrong with Fractal and Chaos Theory:

1. The world view is fundamentally and fatally pessimistic. Benoit Mandelbrot argues that the variance must be infinite. He drew that conclusion 40+ years ago based on the fact that cotton prices had changing volatilities over time. Based on that slim evidence he jumped to the conclusion that it must be infinite. I have yet to see a real world price of infinity. Despite the lack of even a single infinite data point, Mandelbrot completely dismisses the modern GARCH models as being too complex. In my opinion, dealing with the intractability of infinite variance is far more complicated.

2. It is non-predictive. Generally speaking, randomly generated numbers (fractals) are produced and usually graphed. Then the pretty pictures are compared to real world phenomenon in the past. The pictures do seem to resemble some real world patterns to the human eye. In my opinion that is because the eye wants to see patterns in such things as snowflakes and turbulence swirls.

3. Lack of rigorous definition. Admittedly there are some valid mathematical proofs which have come out of this area. However in general the field is completely devoid of basic metrics. For example how does one define "similar to" or "close to" for a fractal? This lack of basic metrics comes out of the fundamental pessimism in point 1. The philosophy is that the real world has infinite variance so there is no point in measuring how far we are from something because the next event could be infinitely far away.

4. Lack of goodness of fit, statistics and feedback as to how well this theory fits the real world. You will never see a statistic of any kind in a paper on fractals or Chaos theory — no estimate of probability, no R squared, nothing. In his book, Didier Sornette performs numerous non-linear fits of various market crashes and yet never presents a single probability estimate or R squared value. It is always presented as "see how nicely the chart of the model overlays onto the actual market chart".

5. The theory is non-scientific. To be scientific a discipline must make testable falsifiable predictions. These theories are not predictive and therefore not testable nor falsifiable.

Chaos theory extends these fundamental issues one step further. Most mathematical definitions of a 'critical point' involve a term something like: 1 / ( t0 - t ) , where t is the current time and t0 (t zero) is the time of the critical event. At t approaches t0 the difference goes to zero. So at the critical juncture we are dividing by zero. The entire expression goes to infinity at that point. (Strictly speaking it is undefined not infinite). The point is that even if BM is wrong about infinite variance the models of Chaos Theory create their own asymptotic behavior which means these models *do exhibit* infinite variance even if the real world data does not!

About the only hope I have for these theories is that some of the older generation of thinkers will pass on and a new crop will come in and invent metrics and ways to measure goodness of fit which can turn this field into a predictive and testable science.

**Russell Sears responds:**

Could someone explain how theses theories lead to the doomsday scenario which Mandelbrot and the Derivatives Expert are so fond of? In my mind, I have worked out the following, tell me if I am on the right track:

First, the chaos part. You can often find "meltdowns" in markets. Points in time that markets "jump" and are discontinuous, or at least "non-normal" for brief periods. A butterfly flaps its wings and you have a thunderstorm, for 15 min. or perhaps even a hurricane for a couple days. Say a 13 sigma event for a day in Oct 87.

Second the fractals. The market is pattern that can be repeated, and scaled up simply by time, say perhaps by the square root of time. Combining these, if a 13 sigma event happened in the past for a day, it's only a matter of time until it happens for say a month, or a year. Likewise if we seen a 13 sigma, it is only a matter of time until we have a 1000 sigma event.

However, to "prove this" fractal they do the opposite. They take long periods of time and scale down. Of course these long time periods don't have the "chaos" pattern *yet*.

The problem is: these longer distribution are pretty much continuous. Therefore, they follow a random walk pretty closely. Of course when you take a distribution that is stable, you see these "patterns". It is the same distribution after all.

The real assumption concerning fractals is the "energy" of the markets, i.e. that people can go infinitely into craziness, an unlimited nuclear chain reaction, fusion versus fission. They simply reject all evidence of extremes as being contained fission, by saying fusion *will* happen.

As you said, it's not science, but it uses a lot of math terms. It's a belief system.

**Dr. Phil McDonnell responds:**

News can cause jumps. This may induce something like the Merton jump diffusion model. The idea is that markets are generally lognormal but a few times a year some big news happens to cause a few outlier events per year. Also remember that if the market follows a jump diffusion model that the 1987 crash should be counted in the model.

The modern GARCH and EGARCH models are better because they take into account shifts in the volatility regime. The 1987 event was say a 13 sigma event taken over a 50 year average volatility. But when you look at it in the time frame of that week's actual volatility it was only about a three or four sigma event.

The concept of self similar behavior at both smaller scales and larger scales is probably one of the more interesting aspects of fractal theory. Lo has found that the square-root scaling of volatility over time does not quite hold but it is close. He derived his own test statistic specifically to test for this in markets in a non-parametric way.

How much time? There is considerable evidence that large negative jumps are mean reverting. This is a violation of the idea of self similar. As an aside even the normal distribution is self similar regardless of scale. The normal distribution for, say, 20 trading days can be decomposed into two periods of 10 days, four periods of five days, all of which are normal and scaled proportionally to the square root of the time. This has been known for something like 200 years.

Generally speaking a philosophy of pessimism pervades the study of fractals and chaos. I believe it is direct consequence of the assumption that the variance is infinite. This requires the ineluctable conclusion that the big one is *out there* — hence the predilection for doomsday scenarios.

**David Wren-Hardin adds:**

I went to a talk Benny Mandelbrot gave at NYU, soon after his book came out. While it was interesting, it had very little relevance anyone trading on a daily basis to actually make money, or to value financial instruments. He was completely uninterested in questions of underlying mechanisms or market behaviors that might underlie his models. Attempts to get from him information on how to plug in actual market pricing into his models to give a predicted price were met by seemingly sincere shrugs of "the picture is enough, why make it more complicated?"

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