**Tsallis nonextensive statistical mechanics and Beck
turbulence model for S&P500 in large time windows**

*M. Ausloos and K. Ivanova*

Abstract

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.