Scaling in the Norwegian stock market

TLDR: In this paper, the authors investigated the validity of the much-used assumptions that stock market returns follow a random walk and are normally distributed and applied the concepts of chaos theory and fractals to examine price variations in the Norwegian and US stock markets.

Abstract: The main objective of this paper is to investigate the validity of the much-used assumptions that stock market returns follow a random walk and are normally distributed. For this purpose the concepts of chaos theory and fractals are applied. Two independent models are used to examine price variations in the Norwegian and US stock markets. The first model used is the range over standard deviation or R/S statistic which tests for persistence or antipersistence in the time series. Both the Norwegian and US stock markets show significant persistence caused by long-run “memory” components in the series. In addition, an average non-periodic cycle of four years is found for the US stock market. These results are not consistent with the random walk assumption. The second model investigates the distributional scaling behaviour of the high-frequency price variations in the Norwegian stock market. The results show a remarkable constant scaling behaviour between different time intervals. This means that there is no intrinsic time scale for the dynamics of stock price variations. The relationship can be expressed through a scaling exponent, describing the development of the distributions as the time scale changes. This description may be important when constructing or improving pricing models such that they coincide more closely with the observed market behaviour. The empirical distributions of high-frequency price variations for the Norwegian stock market is then compared to the Levy stable distribution with the relevant scaling exponent found by using the R/S- and distributional scaling analysis. Good agreement is found between the Levy profile and the empirical distribution for price variations less than ±6 standard deviations, covering almost three orders of magnitude in the data. For probabilities larger than ±6 standard deviations, there seem to be an exponential fall-off from the Levy profile in the tails which indicates that the second-moment may be finite.