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.