  Financial markets are highly non-linear and non-equilibrium systems. Earlier
works have suggested that the behavior of market returns can be well described
within the framework of non-extensive Tsallis statistics or superstatistics.
For small time scales (delays), a good fit to the distributions of stock
returns is obtained with q-Gaussian 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
non-symmetric. 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 q-Gaussians with different
non-extensivity 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
q-Gaussian 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 q-Gaussian distribution model with q independent of
time lag.
