Statistics  |  JUN 1996

Statistics Of Chaotic Markets by Hans Hannula, Ph.D., C.T.A.

For those of us who have had ""chaotic"" days - when everything was topsy-turvy and nothing went the way it should have - maybe the problem isn't in those days; maybe it's in what we expect. - by Hans Hannula, Ph.D., C.T.A. Have you ever had a truly chaotic day? One in which nothing was normal? Probably so. In trading and life in general, ample evidence exists that chaos does occur. There are times when things happen far outside the range of what we expect. But we expect things to be ""statistically"" normal. Maybe there's a problem in what we expect. Our research and modeling of markets clearly indicate that they are not statistically normal. Yet for decades, economists have held that markets are efficient and therefore follow the statistics of a random walkY´. This can be mathematically proven to be untrue. One way, presented elegantly by Edgar Peters in Chaos and Order in Capital Markets , is to compute the Hurst exponent, which measures the nature of any time series, such as a market. This exponent is 0.5 for a random series, which has no correlation between past and future values. When this exponent is greater than 0.5, we have a trending time series, one with fractal noise. Peters shows that the Standard & Poor's 500 index has a Hurst exponent of 0.78, proving that this market does not follow the normal distribution taught in statistics courses and used in many option evaluation programs. The normal distribution is part of a type of statistics called Gaussian statistics. What Peters refers to as fractal noise is perhaps better referred to as fractal patterns . These patterns have an order to them and are described by fractal geometry. Fractal geometry was brought to public scrutiny by mathematician Benoit B. Mandelbrot, who pointed out that Gaussian statistics only describe time series with continuous derivatives, meaning there are no vertical gaps. So when he looked at cotton prices, which do have vertical gaps, Mandelbrot pointed out that the gaps meant that Gaussian statistics could not be used in that situation; a branch of statistics called Paretian statistics had to be used instead. He went on to show how these statistics, rather than the Gaussian, could be used to describe cotton prices and their fractal nature.

by Hans Hannula, Ph.D., RSA, CTA