Mandelbrot’s “The (mis)Behavior of Markets” is a bit annoying as it implies that Mandelbrot invented fractals. The 1904 Koch curve was probably the first fractal well known to mathematicians. His introduction to fractals on page 116 seems accurate however.

Mandelbrot revived interest in fractals, named them and first showed how they might be profitably applied as a mathematical tool. He added substantially to their theory. He gave us the magnificent Mandelbrot set, but he writes with a chip on his shoulder. Mathematics is indeed sometimes adversarial and attribution is sometimes lacking. It is unfortunate that one cannot go for more than a few pages without being continually reminded of this. The book improves in this regard in later pages.

Modeling markets is not like modeling most physical systems: To understand the market and apply that understanding is to change the market! Perhaps there is a fix-point—perhaps not. It is slightly analogous to quantum mechanics where merely to observe a system is to change it. In both cases a successful theory must recognize this. Game theory explicitly recognizes that both players understand the game. Game theory and market theories must implicitly include a theory of mind. (Some think that an interpretation of QM must also.)

I like his example of the Cauchy distribution in contrast with the Gaussian distribution. I suspect that Cauchy’s example serves best to dislodge those with a religious devotion to the normal distribution which may result from the extreme beauty of properties of that distribution—several graduate courses full.

I suppose Google will have to do as an appendix to this book that suffers from too few equations. Indeed there are many notions that remain hazy for lack of equations. The term “H” is not adequately defined and Google does not help with such names. There are a few equations in the appendix but even those are mostly inadequate.

As of page 54 I am surprised that he has not yet noted that if some simple pattern recognition were to predict ‘movement on the exchange’, that that pattern would have been discovered, exploited and thus extinguished. In short—pattern arbitrage. To quote Beunza, Hardie & MacKenzie:

The best that one can hope for is to occasionally find an ‘arbitragable’ pattern and milk if briefly. (It is only brief because some set of traders who cause the pattern will start losing and either stop causing the pattern or leave the market when they lose their shirts.) Usually one class of trader will unwittingly cause a pattern as when ‘day traders’ liquidate their holdings over a weekend. This causes a pattern of weekend lows that an arbitrageur can exploit by buying on Friday and selling on Monday. Such patterns are easily enough found that the game or arbitrage is a commodity service and yields only commodity profits—barely worth the effort. Such remaining patterns will become ever less frequent, especially with the advent of computers seeking such patterns. Such a ‘resource’ is thus soon depleted. Such pattern seekers have no motivation to reveal themselves, unless, of course, they fail and want to ‘sell the services of their methods’ instead of making a profit trading.

This point is covered at the end of page 55.

Mandelbrot’s work is much in the same fashion as Wolframs’s “A New Kind of Science”. They are both descriptive and try to use our ability to see similar geometric patters in mathematical and real-world phenomena. These are both signs of an early phase of science. I think they are both promising but both still tentative.