Power law distribution in high frequency financial data? An econometric analysis

Power law distributions are very common in natural sciences. We analyze high frequency financial data from XETRA and the NYSE using maximum likelihood estimation and the Kolmogorov–Smirnov statistic to test whether the power law hypothesis holds also for these data. We find that the universality and scale invariance properties of the power law are violated. Furthermore, the returns of Daimler Chrysler and SAP traded simultaneously on both exchanges follow a power law at one exchange, but not at the other. These results raise some questions about the no-arbitrage condition. Finally, we find that an exponential function provides a better fit for the tails of the sample distributions than a power law function.

[1]  P. V. (Sundar) Balakrishnan,et al.  Power Law and Evolutionary Trends in Stock Markets , 2006 .

[2]  M. E. J. Newman,et al.  Power laws, Pareto distributions and Zipf's law , 2005 .

[3]  R. Cont.  Empirical properties of asset returns: stylized facts and statistical issues , 2001 .

[4]  J. Kwapień,et al.  Nonextensive statistical features of the Polish stock market fluctuations , 2007 .

[5]  Nabendu Pal,et al.  Handbook of Exponential and Related Distributions for Engineers and Scientists , 2005 .

[6]  V. Plerou,et al.  Scaling of the distribution of fluctuations of financial market indices. , 1999, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[7]  H. Eugene Stanley,et al.  Inverse Cubic Law for the Probability Distribution of Stock Price Variations , 1998 .

[8]  M. Dacorogna,et al.  Statistical study of foreign exchange rates, empirical evidence of a price change scaling law, and intraday analysis , 1990 .

[9]  A. Barabasi,et al.  Functional and topological characterization of protein interaction networks , 2004, Proteomics.

[10]  H. Eugene Stanley,et al.  Scale-Dependent Price Fluctuations for the Indian Stock Market , 2004 .

[11]  H. F. Coronel-Brizio,et al.  Asymptotic behavior of the daily increment distribution of the IPC, the mexican stock market index , 2003 .

[12]  H. Bauke.  Parameter estimation for power-law distributions by maximum likelihood methods , 2007, 0704.1867.

[13]  R. Gencay,et al.  An Introduc-tion to High-Frequency Finance , 2001 .

[14]  Inverse-cubic law of index fluctuation distribution in Indian markets , 2006, physics/0607014.

[15]  A. Lo,et al.  THE ECONOMETRICS OF FINANCIAL MARKETS , 1996, Macroeconomic Dynamics.

[16]  Michel L. Goldstein,et al.  Problems with fitting to the power-law distribution , 2004, cond-mat/0402322.

[17]  Mark E. J. Newman,et al.  Power-Law Distributions in Empirical Data , 2007, SIAM Rev..

[18]  Taisei Kaizoji,et al.  A precursor of market crashes: Empirical laws of Japan's internet bubble , 2005, physics/0510055.

[19]  Saibal Mitra,et al.  Power Law Versus Exponential Law in Characterizing Stock Market Returns , 2008 .

[20]  P. Gopikrishnan,et al.  Inverse cubic law for the distribution of stock price variations , 1998, cond-mat/9803374.

[21]  Stephen Kinsella,et al.  MAXIMUM LIKELIHOOD ESTIMATION OF STABLE PARETIAN DISTRIBUTIONS APPLIED TO INDEX AND OPTION DATA , 2009 .

[22]  V. Plerou,et al.  A theory of power-law distributions in financial market fluctuations , 2003, Nature.

[23]  The Power (Law) of Indian Markets: Analysing NSE and BSE Trading Statistics , 2006, physics/0605247.

[24]  Thomas Lux,et al.  The stable Paretian hypothesis and the frequency of large returns: an examination of major German stocks , 1996 .

[25]  M. Handcock,et al.  An assessment of preferential attachment as a mechanism for human sexual network formation , 2003, Proceedings of the Royal Society of London. Series B: Biological Sciences.

[26]  V. Plerou,et al.  Scaling of the distribution of price fluctuations of individual companies. , 1999, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.