Asymmetric Tsallis distributions for modelling financial market dynamics
Abstract
Financial markets are highly nonlinear and nonequilibrium systems. Earlier works have suggested that the behavior of market returns can be well described within the framework of nonextensive Tsallis statistics or superstatistics. For small time scales (delays), a good fit to the distributions of stock returns is obtained with qGaussian distributions, which can be derived either from Tsallis statistics or superstatistics. These distributions are symmetric. However, as the time lag increases, the distributions become increasingly nonsymmetric. In this work, we address this problem by considering the data distribution as a linear combination of two independent normalized distributions  one for negative returns and one for positive returns. Each of these two independent distributions are half qGaussians with different nonextensivity parameter q and temperature parameter beta. Using this model, we investigate the behavior of stock market returns over time scales from 1 to 80 days. The data covers both the .com bubble and the 2008 crash periods. These investigations show that for all the time lags, the fits to the data distributions are better using asymmetric distributions than symmetric qGaussian distributions. The behaviors of the q parameter are quite different for positive and negative returns. For positive returns, q approaches a constant value of 1 after a certain lag, indicating the distributions have reached equilibrium. On the other hand, for negative returns, the q values do not reach a stationary value over the time scales studied. In the present model, the markets show a transition from normal to superdiffusive behavior (a possible phase transition) during the 2008 crash period. Such behavior is not observed with a symmetric qGaussian distribution model with q independent of time lag.
 Publication:

arXiv eprints
 Pub Date:
 February 2021
 arXiv:
 arXiv:2102.04532
 Bibcode:
 2021arXiv210204532D
 Keywords:

 Quantitative Finance  Statistical Finance
 EPrint:
 13 figures, 28 pages