Analysis of fractal properties on Polish share market

Tomasz Kałuziak
Prace naukowe Akademii Ekonomicznej we Wrocławiu Nr 1189 / 2007
Summary (according to author)
„This report is the attempt of the verification of a long-term memory of series consisting of logarithmic rate of return expanded by the research of the statistical similarity of time series on the Polish share market. It also tries to answer the questions connected with reasons of receiving dubious and ambiguous results. Showing the existence of real fractals would require a modification of currently accepted hypotheses, with the hypothesis of the effective market at the top of it, and would justify the use of opportunities, coming from the chaos theory in financial analysis. (…)
The research tries to estimate the Hurst coefficient (rescaled range analysis described by E. Peters) as well as the coefficient of statistical similarity (suggested by M. Zwolankowska).
Looking at the Polish share market one can clearly see the difference between the research of opportunities of the use of deterministic chaos in exact science, and economy which is a set of many currents and disciplines of science. (…) Some estimated values and signals may prove the potential existence of fractal properties but others contradicts it.”
Commentary
Methodologically the research seems to be very correct. We can hardly find any strong and hidden assumptions regarding studied share market. On the contrary, the main purpose of the research is to test whether the market can be considered as fractal structure and consequently investigated by the methods appropriate to such a structure. Fractals were indentified for the first time by Mandelbrot who tried to develop the theory on them. Hurst, while studying the behavior of Nil river throughout several centuries, discovered that changes in its level do not follows the normal, statistical distribution. He worked out the mathematical tool, which can be used in order to verify whether the particular time series follow such a normal distribution (what points on its purely accidental character) or tend to prefer some of the results (what suggest that their occurrence are not accidental, and that some other hidden regularity may lie behind the pattern). One of regularity, that may be taken into account is a fractal structure. If we establish that Hurst’s exponent exceeds certain value (0,5) then we may assume that investigated series are not purely accidental. However obtained results are dubious and ambiguous.

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