Tuesday, October 19, 2010

In Memory of Benoit Mandelbrot

I would have posted on the death of mathematician Benoit Mandelbrot on October 14 sooner, but my mother died on Oct. 8, and I have been completely absorbed with that until now (she is buried and all that).

Anyway, I think he was enormously important and deserved the Nobel Prize in economics. While one can see it implicit in not empirically sound work of Pareto in the 1890s and in Lotka and Zipf on city size distributions in the 1940s, it was Mandelbrot writing on cotton prices in the early 1960s who first recognized the phenomenon of fat tails in asset markets, which are ubiquitous in fact, and also linked these with the concept of fractals and fractal dimensionality, which has since become enormously influential and widespread across many disciplines, although the first fractal set was the Cantor Set, discovered by Georg Cantor in 1883.

I met Mandelbrot on several occasions and found him to be enormously impressive. He did have an enormous ego (and also the largest head I ever saw on anybody), but he was one of those people who had some grounds for his egotism. As most probably know, his career was very unconventional, and he was only offered an academic position when he was quite old (a special professorship at Yale), after he worked for many years at IBM.

In terms of current issues and debates, Mandelbrot was one of those along with Taleb who was warning for some time that we were likely to have a major crash of the financial system. More recently, John Cochrane invoked him in an effort to defend the Chicago School people from charges that their conventional financial market models that assume Gaussian normal distributions were partly responsible for the problems were not all that they believed. He pointed out that his father-in-law, Eugene Fama, had once been a follower of Mandelbrot, and therefore anyone who studied with Fama knew about fat tails, although Fama had fallen out with Mandelbrot over a technical issue (do variances vanish asymptotically; Fama was right that they do not but then dropped the ball by failing to notice that fourth moments, kurtosis, that is fat tails, do appear to vanish that way, making Mandelbrot right in the big picture).

Of course, when I went to look at Cochrane's big grad textbook, Asset Pricing, fat tails, kurtosis, and so on, much less fractals or power law distributions. do not appear in it at all. So, maybe Cochrane knew about fat tails, but he was not about to let anybody else know about them.

In any case, I think Mandelbrot was one of the few genuine geniuses writing about economics, although it was not his primary field of interest. His death is a great loss.


Unknown said...

Mandlebrot was not the first to recognize the phenomenon of fat tails in asset prices. Fat tails were observed by Osborne (1959) and Alexander (1961) in the frequency distributions of returns of stocks before Mandlebrot wrote on the phenomenon in 1963.

Mandlebrot was, however, generally credited as being the first to discover that stock returns exhibit volatility clustering. Here is Mandlebrot writing in 1963 of what we now know as volatility clustering. "...large changes tend to be followed by large changes, of either sign, and small changes tend to be followed by small changes (of either sign)."

rosserjb@jmu.edu said...


OK. Which makes it all the weirder that Osborne was a signal influence on adopting a Gaussian random walk as the model for stock price movements, foreshadowing the somewhat similarly bizarre conduct of Fama.

Unknown said...

Yes, but "Mandlebrot (1963) used stable-Paretian distributions to explain the fat tails observed by Osborne (1959) and Alexander (1961) in the frequency distributions of returns of stocks. However, financial economists were reluctant to adapt this model primarily because they would have to give up variance as their favored measure of risk (since the variance of a stable-Paretian random variable is infinite). In the end, the stable-Paretian hypothesis proved a dead end, particularly as alternative finite-variance explanations of stock market returns were developed." From Mark Rubinstein’s book, A History of the Theory of Investments, pg 185-190.

In other words economists put most of their research efforts into volatility clustering (ARCH, GARCH, and their decedents) rather than fat tails, while Mandlebrot continued to try to address the fat tails directly. Mandlebrot's approach has shown very little progress over the last 40 years. Focusing on volatility clustering has resulted in major advances in the last 30 years.

There are several reasons economists have focused on volatility clustering rather than fat tails. While stock returns are random, stock volatility is not random. (Something that clusters cannot be random.) So they focused on the somewhat predictable aspect of the problem. Secondly, a significant amount of the fat tails in the distribution can be explained by the volatility clustering. So if the time varying volatility can be modeled we have also explained most of the fat tail. Thirdly, volatility is the primary measure of risk in financial economics. Mandlebrot's approach uses methods with infinite volatility. So volatility as a measure of risk goes out the window using Mandlebrot's approach.

rosserjb@jmu.edu said...


While I might disagree on some details, it is certainly correct that there has been a lot of effort put into modeling volatility clustering through ARCH, GARCH, and their variations. These certainly do "model" fat tails" to some extent and I would agree that such clustered episodes are the main source of the fat tails.

However, fundamentally there is no model. It all depends on random shocks that set off the clusters. Furthermore, the options pricing and other formuli do not even take into account this phenomenon. Garden variety Black-Scholes and CAPM and so on assume constant finite variances.

Eleanor said...

Barkley -- Condolences on the loss of your mother and thank you for posting on Mandelbrot. A fascinating guy.

PT-12599 said...

My condolences for the loss of your mother as well, Barkley.