In 1963 Mandelbrot published research into the distribution of cotton prices based on a very long time series which found that, contrary to the general assumption that these price movements were normally distributed, they instead followed a pareto-levy distribution. While on the surface these two distributions don’t appear to be terribly different, (many small movements, and a few large ones), the implications are significantly different, most notably the pareto-levy distribution has an infinite variance.
This implies that rather than extreme market moves being so unlikely that they make little contribution to the overall evolution, they instead come to have a very significant contribution. In a normally distributed market, crashes and booms are vanishingly rare, in a pareto-levy one crashes occur and are a significant component of the final outcome.
It has taken years for this to be taken seriously, and in the mean time financial theory has gone on using the assumption of normally distributed returns to derive such results as the Black-Scholes option pricing equation, ultimately winning an Nobel Prize in Economics for the discoverers Scholes and Merton (Black having already died), not to mention Modern Portfolio theory (also winning Nobels). That modern finance ignored Mandelbrot’s discovery and went onto honor those working under assumptions shown to be false has clearly annoyed Mandelbrot immensely and as mentioned previously he spends much of the book telling us of his prior discoveries and how he was ignored.
That Black-Scholes has significant short comings due to unrealistic assumptions is very well recognised. In the market there exists what is known as a volatility smile, options with strikes not near the current market (ie. out-of, or in-the-money options) are priced with different volatilities to at-the-money options. The very existence of such a thing is a contradiction of the basic assumptions of the B-S model and one of a number of ways the market in practice tries to compromise between using equations that roughly works in some circumstances and getting a fair price. For all known flaws of the Black-Scholes framework, no one has been able to figure out anything else that uses improved assumptions and enables calculation of real prices. GARCH models are an attempt to fix this but embed many of the same assumptions.
While the book shows some nice fractal schemes for generating much more realistic market models than are generated by a straight Brownian motion scheme and this is all very interesting. The discussion of how varying time scales may explain some of the observed behaviour is also good, the speed that trading occurs may be a better measure of market time than the clock is.
However in the end we are left at the end with a lot of criticism, a few good ideas but little to show for it. Attempts to try to tinker with calculations such as VAR to include estimates of fat tails are dismissed as being like the Ptolemaic system. I don’t disagree with this analogy, but essentially we are in the situation even worse than astronomy was after Copernicus. Until Kepler calculated the orbits to be elliptical, the predictions of the Copernican system were no better than the Ptolemaic system. Similarly while it is widely recognised that many of the assumptions of modern finance are wrong it does give us a framework to make calculations and will continue to be used until something better comes along.
In the end he calls for research to be done into better developing a theory to understand market behaviour, which is a good thing. In the interim though work in the old paradigm will continue with some recognition that there are flaws which will be dealt with in an ad hoc way. Continued railing against Efficient Markets, the normal distribution and the independence of returns this will not change this without some solid results.