You probably heard about random walk model of the stock market. This model assumes that price change of any stock is random and does not depend on the price history. If so, no one can  predict the stock price behavior and there is no trading strategy which can beat the buy and hold strategy which exploits long-term market growth related to developing the economy.

However the situation is not so pessimistic. Many traders successfully use market trends or contrarian strategies to obtain good profits. They beat the market regularly and their strategies are statistically proved. It means the market is not random. Many academicians and other lovers of the random walk theory now agree with nonrandom effects.

In this short paper we will describe some nonrandom market behavior using historical prices of the Dow Jones Industrial Average for the period from 1900 to 2001.

The best known effect of nonrandom market behavior is the tail effect of the distribution of the market returns. The j day return R(j) can be written as

R(j) = [ P(i +j) - P(i) ] / P(i) *100%

where P(i) is the closing price on the i-th trading day. If j = 1 then the equation describes one day return. The one day return R(1) can be presented as a sum of one minute returns. If one minute returns are independent then one day returns must have Gaussian distribution.

The distribution of R(1) is shown on the left figures. The upper panel present this distribution in linear scale, the bottom panel presents the distribution in the log scale. The blue lines are the best Gaussian fittings. One can see that the tails of the distribution are much above the Gaussian curve. It is easily can be seen from the log scaled distribution.

The tail effects means that the probability of finding large price moves is higher than the corresponding probability for the random walk model. The possible reason is positive correlation's of intraday one minute returns. The sum of these returns generate one day return R(1) and large positive or negative R(1) are observed more often than it can be expected from the model of random walk.

Therefore, distribution of R(1) returns is not Gaussian and many day returns R(j),  j > 1 are also nongaussian. It is better to describe using time dependencies of the return variances

var(R(j)) = <[R(j) - Rav(j)]^2>

For the random walk model these variances must be linear function of j

var(R(j)) = const * j

The next figure shows the dependence of R(j) up to j = 100 trading days. One can easily see that calculated function is nonlinear and it is higher than the corresponding function for the random walk model (red line on the figure). It also confirms that one can find large returns more often than it is expected from this model.  

To make quantitative estimation we plotted var(R(j)) as a function of j using log-log scales (the right panel of the figure).   For the random walk the slope of this line must be equal to 1. From our calculation it is equal to 1.036. It indicates that prices of the Dow can be described by so called   fractional Brownian motion. This motion is a process when

var(R(j)) = const * j ^ 2H

where H is a Hurst exponent. For the random walk H = 0.5. For the Dow the values of H is equal to 0.5018. The values of H > 0.5   always indicate at a positive correlation within increments, i.e. returns for smaller time scale (Peitgen et al., 1992).

Is it possible to exploite the described effect to obtain better returns than the return of the buy and hold strategy? We will discuss this in our next publications.

References
1. H.-O. Peitgen, H. Juergens., D. Saupe. Chaos and Fractals, Springer, 1992.