  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.
