Crossover behavior of stock returns and mean square displacements of particles governed by the Langevin equation

Abstract

It is found that the mean square log-returns calculated from the high-frequency oneday moving average of US and Taiwan stocks with the time internal τ show ballistic behavior θτα1 with the exponent α1 ≈ 2 for small τ and show diffusion-like behavior Dτα2 with the exponent α2 ≈ 1 for large τ . Such a crossover behavior can be well described by the mean square displacements of particles governed by the Langevin equation of motion. Thus, θ and D can be considered, respectively, as the temperature-like and diffusivity-like kinetic parameters of the market, and they can be used to characterize the behavior of the market.