Tsallis nonextensive statistical mechanics and Beck turbulence model for S&P500 in large time windows
M. Ausloos and K. Ivanova
It is often asked "What is the use of looking at the shape and tails of partial distribution functions (PDF), in particular for a financial signal?". The case of the S\&P500 and the turbulent nature of the markets are hereby examined and linked through a model encompassing Tsallis nonextensive statistics. In fine, this is leading to standard evolution equations of the Langevin and Fokker-Planck type. The corresponding turbulence model is one originally
proposed to describe the intermittent behavior of turbulent flows by Beck. In so doing the behavior of normalized log-returns PDF is well described for such a financial market index, for small and large time windows, both for small and large log-returns. The turbulent market volatility (of normalized log-returns) distributions can be well fitted with a $\chi^2$-distribution. The transition between the small time scale model of nonextensive, intermittent process and the large scale Gaussian extensive homogeneous fluctuation picture is found to be at a $ca.$ 200 day time lag. The intermittency exponent ($\kappa$) in the framework of the Kolmogorov log-normal model is found to be related to the scaling exponent of the PDF moments, -thereby giving weight to the model and method. The large value of $\kappa$ points to a large number of cascades in the turbulent process. The first Kramers-Moyal coefficient in the Fokker-Planck equation is almost equal to zero, indicating a surprising ''no restoring force'' and a $quasi$-purely diffusive sort of mechanism. A comparison is made between normalized log-returns and mere price increments.