QUANTITATIVE ANALYSIS

Harnessing The (Mis)Behavior Of Markets
by Rick Martinelli

Do market prices vary due to large numbers of random effects such as the whims of individual traders?

In 1900, Louis Bachelier was awarded a doctorate from the University of Paris following his defense of a dissertation titled "Théorie de la Spéculation," an event that marked the first time a serious academic paper addressed the behavior of the financial markets. In his dissertation, Bachelier proposed that market prices vary due to large numbers of random effects, such as the whims of individual traders, and hence can be modeled as Brownian motion. Slowly, the financial community adopted his ideas, which are now the foundation of modern financial engineering.

THE BROWNIAN MODEL

Three critical assumptions underlie the Brownian model, namely:

1. Price changes are statistically independent
2. Price changes are normally distributed, and
3. Price-change statistics do not vary over time.

The first assumption means that price changes behave like coin tosses, where the current change was not influenced by past changes and has no influence on future changes.

The second assumption says that the changes follow a bell-shaped curve. This assumption is relevant whenever random behavior is due to many small influences. It provides a distribution function characterized by only two parameters, the mean and standard deviation, and implies a certain "contained" behavior of the changes.

The third assumption states that the mean and standard deviation do not change with time.

Knowledgeable investors might take exception to one or all of these assumptions. In fact, there is ample evidence that these assumptions simply do not apply in the real markets. The recent book by Benoit Mandelbrot and Richard Hudson, The (mis)Behavior Of Markets, documents many of these violations. For example, it discusses the 1987 stock market crash, where there was a price change in the Dow Jones Industrial Average (DJIA) equal to about 18 standard deviations, an event with a probability of about one in 10 to the 17th power if the second assumption is true. A glance at price changes for many of the more volatile stocks over a long-enough time span suggests that the third assumption is often violated as well.

The violation of either assumption 2 or 3 may produce similar effects -- namely, larger than usual excursions in price. Are these excursions accompanied by precursors -- smaller changes but in the same direction, like earthquakes? If so, the changes are locally correlated in violation of assumptions 1 and 3, and it may be possible to harness those correlations by means of a simple linear predictor and some statistical data. I will describe some results from my attempt to do just that. I focus on the statistical behavior of the stock charts and ignore real-world issues such as commissions on trades.

...Continued in the June issue of Technical Analysis of STOCKS & COMMODITIES

Excerpted from an article originally published in the June 2006 issue of Technical Analysis of STOCKS & COMMODITIES magazine. All rights reserved. © Copyright 2006, Technical Analysis, Inc.