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

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 ${\chi }^2$-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 $ca.$ a 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. 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 ''no restoring force''. A comparison is made between normalized log-returns and mere price increments.

40 pages, 14 figures; accepted for publication in Phys Rev E

References in corpus (12)

• Effect of Trends on Detrended Fluctuation Analysis
• Interpretation of the nonextensitivity parameter $q$ in some applications of Tsallis statistics and Lévy distributions
• Anomalous diffusion in the presence of external forces: exact time-dependent solutions and entropy
• Dynamical foundations of nonextensive statistical mechanics
• Option Pricing Formulas based on a non-Gaussian Stock Price Model
• How to quantify deterministic and random influences on the statistics of the foreign exchange market
• Nonextensive statistical mechanics and economics
• Measuring non-extensitivity parameters in a turbulent Couette-Taylor flow
• Markov properties of high frequency exchange rate data
• Non-Gaussian PDFs from Maximum-Entropy-principle considerations
• Derivation of Tsallis statistics from dynamical equations for a granular gas
• Deterministic and stochastic influences on Japan and US stock and foreign exchange markets. A Fokker-Planck approach

Cited by in corpus (45)

• Topological Data Analysis of Financial Time Series: Landscapes of Crashes
• From time series to superstatistics
• Superstatistics, thermodynamics, and fluctuations
• Asymptotics of superstatistics
• Modelling train delays with q-exponential functions
• Superstatistics in high energy physics: Application to cosmic ray energy spectra and e+e- annihilation
• Superstatistical fluctuations in time series: Applications to share-price dynamics and turbulence
• Stretched exponentials from superstatistics
• Generalized statistical mechanics for superstatistical systems
• On the distribution of high-frequency stock market traded volume: a dynamical scenario
• Superstatistical distributions from a maximum entropy principle
• Stochastic analysis of different rough surfaces
• Stochastic Opinion Formation in Scale-Free Networks
• Stochastic Cellular Automata Model for Stock Market Dynamics
• Model for Non-Gaussian Intraday Stock Returns
• Multiscale reconstruction of time series
• Spectral fluctuations of billiards with mixed dynamics: from time series to superstatistics
• Random walker in a temporally deforming higher-order potential forces observed in financial crisis
• Improved estimation of Fokker-Planck equations through optimisation
• Superstatistics in random matrix theory
• Tsallis distributions and 1/f noise from nonlinear stochastic differential equations
• Level Crossing Analysis of the Stock Markets
• Transition from lognormal to chi-square superstatistics for financial time series
• Additive-multiplicative stochastic models of financial mean-reverting processes
• Extended q-Gaussian and q-exponential distributions from Gamma random variables
• Non-extensive Behavior of a Stock Market Index at Microscopic Time Scales
• Scaling Analysis and Evolution Equation of the North Atlantic Oscillation Index Fluctuations
• On superstatistical multiplicative-noise processes
• A (reactive) lattice-gas approach to economic cycles
• Nonextensive and superstatistical generalizations of random-matrix theory
• Random matrix theory within superstatistics
• Dynamical analogy between economical crisis and earthquake dynamics within the nonextensive statistical mechanics framework
• Generalization of the Beck-Cohen superstatistics
• Increment definitions for scale dependent analysis of stochastic data
• Asymmetric Tsallis distributions for modelling financial market dynamics
• Spurious memory in non-equilibrium stochastic models of imitative behavior
• Statistical mixing and aggregation in Feller diffusion
• Superstatistics as the statistics of quasi-equilibrium states: Application to fully developed turbulence
• An Intrinsic Entropy Model for Exchange-Traded Securities
• Searching for self-similarity in switching time and turbulent cascades in ion transport through a biochannel. A time delay asymmetry
• Tsallis entropy for cross-shareholding network configurations
• Analysis of symmetry breaking in quartz blocks using superstatistical random matrix theory
• Distribution of local density of states in superstatistical random matrix theory
• A Comprehensive Analysis of Time Series Segmentation on the Japanese Stock Prices
• Bayesian inference and superstatistics to describe long memory processes of financial time series