  One of the standardized features of financial data is that log-returns are
uncorrelated, but absolute log-returns or their squares namely the fluctuating
volatility are correlated and is characterized by heavy tailed in the sense
that some moment of the absolute log-returns is infinite and typically
non-Gaussian [20]. And this last characteristic change accordantly to different
timescales. We propose to model this long-memory phenomenon by superstatistical
dynamics and provide a Bayesian Inference methodology drawing on
Metropolis-Hasting random walk sampling to determine which superstatistics
among inverse-Gamma and log-Normal describe the best log-returns complexity on
different timescales, from high to low frequency. We show that on smaller
timescales (minutes) even though the Inverse-Gamma superstatistics works the
best, the log-Normal model remains very reliable and suitable to fit the
absolute log-returns probability density distribution with strong capacity of
describing heavy tails and power law decays. On larger timescales (daily), we
show in terms of Bayes factor that the inverse-Gamma superstatistics is
preferred to the log-Normal model. We also show evidence of a transition of
statistics from power law decay on small timescales to exponential decay on
large scale with less heavy tails meaning that on larger time scales the
fluctuating volatility tend to be memoryless, consequently superstatistics
becomes less relevant.
