Financial Market Dynamics: Superdiffusive or not?

Abstract

The behavior of stock market returns over a period of 1-60 days has been investigated for S&P 500 and Nasdaq within the framework of nonextensive Tsallis statistics. Even for such long terms, the distributions of the returns are non-Gaussian. They have fat tails indicating that the stock returns do not follow a random walk model. In this work, a good fit to a Tsallis q-Gaussian distribution is obtained for the distributions of all the returns using the method of Maximum Likelihood Estimate. For all the regions of data considered, the values of the scaling parameter q, estimated from one day returns, lie in the range 1.4 to 1.65. The estimated inverse mean square deviations (beta) show a power law behavior in time with exponent values between -0.91 and -1.1 indicating normal to mildly subdiffusive behavior. Quite often, the dynamics of market return distributions is modelled by a Fokker-Plank (FP) equation either with a linear drift and a nonlinear diffusion term or with just a nonlinear diffusion term. Both of these cases support a q-Gaussian distribution as a solution. The distributions obtained from current estimated parameters are compared with the solutions of the FP equations. For negligible drift term, the inverse mean square deviations (betaFP) from the FP model follow a power law with exponent values between -1.25 and -1.48 indicating superdiffusion. When the drift term is non-negligible, the corresponding betaFP do not follow a power law and become stationary after certain characteristic times that depend on the values of the drift parameter and q. Neither of these behaviors is supported by the results of the empirical fit.

arXiv commentThe paper has gone through considerable revisions after referees' comments. The major changes are, The old introduction has been split into Section 1: Introduction and Section 2: A brief review of the Theory. The Results section now has 4 subsections. Section 5 (Conclusions) is reworded but the conclusions remain the same. Some additional references have been added. 29 pages. 11 Figures

References in corpus (9)

• Power-law distributions in empirical data
• Anomalous diffusion in the presence of external forces: exact time-dependent solutions and entropy
• Inverse Cubic Law for the Probability Distribution of Stock Price Variations
• Critical Market Crashes
• Nonextensive statistical mechanics and economics
• Scaling in the distribution of intertrade durations of Chinese stocks
• Statistical-thermodynamical foundations of anomalous diffusion
• Maximum Likelihood Estimation for q-Exponential (Tsallis) Distributions
• Non-extensive Behavior of a Stock Market Index at Microscopic Time Scales

Cited by in corpus (6)

• Open Problems within Nonextensive Statistical Mechanics
• Asymmetric Tsallis distributions for modelling financial market dynamics
• Financial Portfolios based on Tsallis Relative Entropy as the Risk Measure
• A 2D Levy-flight model for the complex dynamics of real-life financial markets
• Selected Methods for non-Gaussian Data Analysis
• Evolving ab initio trading strategies in heterogeneous environments