Financial Market Dynamics: Superdiffusive or not?

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 long range correlations persist. 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 β 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 deviation β_FP from the FP model follows a power law with exponent values between -1.25 and -1.48 indicating superdiffusion. When the drift term is non-negligible, the corresponding β_FP does not follow a power law and becomes stationary after a certain characteristic time that depends on the values of the drift parameter and q. Neither of these behaviors is supported by the results of the empirical fit.