In the previous two posts (here and here) I wrote about mathematical risk models and how they contributed to the financial crisis of 2008. This post contains references to more information.
I recommend the following three books, all written before last year's financial crisis:
Against the Gods: The Remarkable Story of Risk, by Peter Bernstein (1996). This is a very readable history of the mathematics of risk. It is surprising how much of this history is from the study of gambling. This book is the source of the graph of monthly changes in the S&P 500 stock index in this post. The illustration on the cover of the book is Rembrandt’s painting of “The Storm on the Sea of Galilee.” Now there’s an example of risk!
The Black Swan: The Impact of the Highly Improbable, by Nassim Nicholas Taleb (2007). Dr. Taleb was a practitioner of mathematical finance, and gives an inside view of the profession. This book was a New York Times bestseller. Dr. Taleb calls the normal distribution the "GIF": the Great Intellectual Fraud (chapter 15).
Dr. Taleb dedicated The Black Swan to the author of the third book that I recommend: "To Benoit Mandelbrot, a Greek among Romans."
The (mis)Behavior of Markets: A Fractal View of Financial Turbulence, by Benoit Mandelbrot and Richard Hudson (2004). Benoit Mandelbrot is a famous mathematician, known for his work with fractals and the Mandelbrot set. He was not originally known for his work with the mathematics of risk. In a twist of fate that we in Farm Credit can appreciate, Dr. Mandelbrot became interested in the mathematics of risk when studying the volatility of cotton prices. He tells that story in Chapter VIII "The Mystery of Cotton."
Dr. Mandelbrot writes that there is a spectrum of risk, from "mild" to "wild." He says that the normal distribution is sufficient for analyzing mild risk, but that there is no mathematics currently available that is adequate for analyzing wild risk. If a mathematical tool is ever to be developed that will help us analyze wild risk, Dr. Mandelbrot believes that it will be found in fractal mathematics.
Of the three books, The (mis)Behavior of Markets is the most insightful. But The Black Swan has the most delightful quote—on p. 225 Dr. Taleb says: "Likewise, the government-sponsored enterprise Fanny Mae [sic], when I look at their risks, seems to be sitting on a barrel of dynamite, vulnerable to the slightest hiccup. But not to worry: their large staff of scientists deemed these events 'unlikely.' " Dr. Taleb wrote that in 2007. Here's what happened in 2008.
I also recommend the following newspaper and magazine articles, all published during and after the financial crisis:
“The 1% Panic,” by L. Gordon Crovitz, Wall Street Journal, 10/13/08 (may require subscription to read). Discusses two of the three books mentioned above.
“The End,” by Michael Lewis, portfolio.com, 11/11/08. No math, but an interesting discussion of people.
“Risk Mismanagement,” by Joe Nocera, New York Times, 1/4/09. Discusses one of the three books mentioned above.
“Recipe for Disaster: The Formula That Killed Wall Street,” by Felix Salmon, Wired Magazine, 2/23/09. Discusses more esoteric math than I have written about on this blog.
If you wish to play around with the normal distribution, there is an excellent Excel file that you can download here. I used this file to generate the graph of the normal distribution in this post.