Distributions of historic market data: relaxation and correlations

We investigate relaxation and correlations in a class of mean-reverting models for stochastic variances. We derive closed-form expressions for the correlation functions and leverage for a general form of the stochastic term. We also discuss correlation functions and leverage for three specific models- multiplicative, Heston (Cox-Ingersoll-Ross) and combined multiplicative-Heston-whose steady-state probability density functions are Gamma, Inverse Gamma and Beta Prime respectively, the latter two exhibiting "fat" tails. For the Heston model, we apply the eigenvalue analysis of the Fokker-Planck equation to derive the correlation function-in agreement with the general analysis- and to identify a series of time scales, which are observable in relaxation of cumulants on approach to the steady state. We test our findings on a very large set of historic financial markets data.