Generation of cubic power-law for high frequency intra-day returns: Maximum Tsallis entropy framework

A theoretical framework based on the maximum Tsallis entropy is proposed to explain the tail behavior of the intra-day stock returns, providing a rationale for the cubic law behavior for high frequency data. The specification of first two time-dependent moment constraints yields a q-Gaussian distribution for the intra-day stock returns. The value of the parameter q is estimated by minimizing appropriately modified Jensen-Shannon (JS) divergence in Tsallis entropy framework between q-Gaussian distribution and empirical NASDAQ 100 data. The estimated value of q yields the well-known empirically observed cubic law tail behavior of the intra-day stock returns which has been observed for high frequency data sets. To validate the cubic law stylized fact, five more data sets from high frequency NASDAQ 100, S&P 500 and NYSE index have been examined and it is found that the cubic law operates.