Fractional market dynamics

A new extension of a fractality concept in financial mathematics has been developed. We have introduced a new fractional Langevin-type stochastic differential equation that differs from the standard Langevin equation: (i) by replacing the first-order derivative with respect to time by the fractional derivative of order μ; and (ii) by replacing “white noise” Gaussian stochastic force by the generalized “shot noise”, each pulse of which has a random amplitude with the α-stable Lévy distribution. As an application of the developed fractional non-Gaussian dynamical approach the expression for the probability distribution function (pdf) of the returns has been established. It is shown that the obtained fractional pdf fits well the central part and the tails of the empirical distribution of S&P 500 returns.