The Misbehavior of Markets by Benoit Mandelbrot and Richard Hudson
The Misbehavior of Markets by Benoit Mandelbrot and Richard Hudson
Mandelbrot stuck all the math in the notes at the end of the book. Unfortunately, it just skims an intro to power laws and his work. Even worse, he spends most of the book on a history and critique of finance based on the normal distribution. All that required was the first quote below (observed not matching the predicted).
Looking at the distribution of outputs has its uses, in that you can bound the functions that generate the output. Unfortunately, Mandelbrot demonstrates an unwillingness to explore how we get our outputs (see the second quote below). Perhaps he's just leaving the problem to the reader. ;)
From 1917 to 2003, the daily index movements of the Dow Jones Industrial Average do not spread out on graph paper like a simple bell curve. The far edges flare too high: too many big changes. Theory suggest that over that time, there should be fifty-eight days when the Dow moved more than 3.4 percent; in fat, there were 1,001. Theory predicts six days fo index swings beyond 4.5 percent; in fact, there were 366. And index swings of more than 7 percent should come once every 300,000 years; in fact, the twentieth century saw forty-eight such days. Truly, a calamitous era that insists on flaunting all predictions. Or, perhaps, our assumptions are wrong.
Sure, after the fact, with enough time and effort, we can piece together a tolerable cause-and-effect story of why a price moved the way it did. But who cares? It is too late by then. Fortunes have been gained and lost. Before the fact, in the real world of fast markets, veiled motives, and uncertain outcomes, probability is the only tool at our disposal.