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Dynamical model and nonextensive statistical mechanics of a market index on large time windows

Marcel Ausloos·Kristinka Ivanova
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Abstract

The shape and tails of partial distribution functions (PDF) for a financial signal, i.e. the S&P500 and the turbulent nature of the markets are linked through a model encompassing Tsallis nonextensive statistics and leading to evolution equations of the Langevin and Fokker-Planck type. A model originally proposed to describe the intermittent behavior of turbulent flows describes the behavior of normalized log-returns for such a financial market index, for small and large time windows, both for small and large log-returns. These turbulent market volatility (of normalized log-returns) distributions can be sufficiently well fitted with a $χ^2$-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 $ca.$ a 200 day time lag. The intermittency exponent ($κ$) 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. The large value of $κ$ 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 ''no restoring force''. A comparison is made between normalized log-returns and mere price increments.

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Marcel Ausloos, Kristinka Ivanova. 2003-08-05. Dynamical model and nonextensive statistical mechanics of a market index on large time windows. https://doi.org/10.1103/physreve.68.046122

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