Benoit Mandelbrot, the pioneer of fractals, died last year. I only found out months later. His death should have been announced as a great tragedy to the whole world. Mandelbrot I also only recently found out also made contributions to financial mathematics. I would like to discuss one aspect, which may not be original Mandelbrot, but which Taleb names after him.
The old
It has been well publicised that normal distributions do not work. At least they do not work for modelling financial markets. It is my impression that, despite knowing this, everybody continued to use them. Because the quants, the people who were supposed to ‘know’ said it was okay, the normal distribution became a kind of placebo.
These methods have, however, always had their critics. Taleb (who wrote Fooled by Randomness and The Black Swan) is one of them. The latest financial crisis proved them right. But still, the normal distribution is what everybody knows, and I think it will take a while before ‘better’ practices become widespread.
The new
One alternative to normal distributions that has been suggested, and which Taleb argues we should use, albeit with caution, is that of power laws. He calls such distributions Mandelbrotian, to challenge the conventional Gaussian. Power laws have, of course, been around for a very long time (they are hardly new). Zipf’s law is a specific example in the field of linguistics that was proposed in 1935 and the specific example we will be looking at was researched even earlier. I am not quite sure how much work on power laws has been done in terms of financial markets nor am I certain exactly what Mandelbrot’s contribution was.
Power laws are interesting creatures, showing themselves seemingly everywhere. They occur in nature where they give the frequency of different sizes of earthquakes, of the sizes of craters on the moon and the extinctions of species. These are some examples mentioned by John Gribbin in his book Deep Simplicity.
Things such as the distribution of city sizes and the disparity of wealth also seem to follow power law distributions. The ubiquity of this distribution not only in nature, but in human behaviour, makes it seem like some kind of universal law (which we cannot help but obey). Such an interpretation should be avoided of course. With interest in power laws high, you are more likely to hear about confirming evidence (this is what people will be looking for). Eventually we will find important examples that are not governed by power laws or we may find the power law distribution is a poorer estimate than some other law.
Definition
There is a general definition of a power law distribution, but I will merely define a special case, the Pareto distribution which has the distribution function
 = 1 – \left(\frac{k}{x}\right)^a)
with support x > k and a > 0. There is a more general definition for which some of the properties I discuss below would only hold asymptotically. The Pareto distribution is also the source of the infamous 80-20 or Pareto law (not really a law). I have yet to establish exactly how this was derived but watch this space.
Why power laws are cool
Self-similarity
They display self-similarity. Self-similarity is a property of fractals (remember that Mandelbrot is famous for his work on fractals). Taleb calls these distributions scalable (as opposed to the normal distribution, which is not scalable). It follows from the following property:
 = t^{-a} Pr(X > x).)
So, for example, if a = 1, doubling x always lowers the probability of seeing a higher value by half. This also demonstrated by the fact that if you fix a new minimum wealth level you get a Pareto distribution with the same exponent:
 = \left(\frac{x_m}{x}\right)^a)
More concretely, if X represented wealth levels, the inequality among the rich would be the same as that among the poor. So if your wealth is x_m, half the population will have twice that wealth, and half of them four times that and so on.
Infinite moments
For certain (common) cases they have infinite variance or even infinite mean. For a <= 2, the variance is infinite. For a <= 1, so is the mean. In these cases we will not be able to rely on the Central Limit Theorem (or the Law of Large numbers in the second case). This makes the maths harder and moves us irrevocably away from normality.
Fat tails
They have fat tails. This means that extreme values are more likely than for the normal distribution. The structure of the tails is given by the self similarity of the distribution as shown earlier. For the normal distribution doubling the wealth level would lower the probability of higher observations by much more than half and much more rapidly further up the tail we go.
Estimation difficulties
They are hard to fit. Determining the exponent to use is apparently not easy and the resulting distribution is very sensitive to use. Taleb gives a good illustration.
Can they help us?
The danger is that everyone will jump on the new bandwagon. I can see the headlines twenty years from now. Math whizzes cause greatest financial crash in history. Math models fail us again. These Mandelbrotian distributions are just another fraud. Taleb warns that we cannot model everything, that even appropriate mathematics is of but limited use. In any event we should use models with caution. I am very interested in how far we can take them. I am definitely not yet done with power laws. Expect more posts on this topic.
Some references
Gribbins links power laws and chaos theory (both linked to fractals):
- Gribbin, J., 2004. Deep Simplicity, New York: Random House, Inc.
Taleb gives some useful examples and a mathless discussion:
- Taleb, N.N., 2007. The Black Swan, Penguin.
From Wikipedia1:
- Wikipedia, 2011a. Pareto distribution. Wikipedia. Available at: http://en.wikipedia.org/wiki/Pareto_distribution.
- Wikipedia, 2011b. Pareto principle. Wikipedia.
- Wikipedia, 2011c. Power law. Wikipedia. Available at: http://en.wikipedia.org/wiki/Power_law_distribution .
- Wikipedia, 2011d. Zipf’s law. Wikipedia. Available at: http://en.wikipedia.org/wiki/Zipf’s_law.
1 Yes I am going to shamelessly reference Wikipedia. It is useful and usually accurate. This is a blog, not an academic article. I am aiming for clarity of thought rather than accuracy of assumptions.
Enjoyed the reading. Although I must add that "Mandelbrotian" randomness shouldn't be thought as an alternative to "Gaussian" randomness. I see the impossibility (or should I say difficulty) of prediction more in the tools at our disposal rather than the type of randomness we want to deal with. There is no such randomness of type one or two. Random is random. Or well, that's what it should be.
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