Multiple time scales and the empirical models for stochastic volatility

The most common stochastic volatility models such as the Ornstein–Uhlenbeck (OU), the Heston, the exponential OU (ExpOU) and Hull–White models define volatility as a Markovian process. In this work we check the applicability of the Markovian approximation at separate times scales and will try to answer the question which of the stochastic volatility models indicated above is the most realistic. To this end we consider the volatility at both short (a few days) and long (a few months) time scales as a Markovian process and estimate for it the coefficients of the Kramers–Moyal expansion using the data for Dow-Jones Index. It has been found that the empirical data allow to take only the first two coefficients of expansion to be non-zero that define form of the volatility stochastic differential equation of Itô. It proved to be that for the long time scale the empirical data support the ExpOU model. At the short time scale the empirical model coincides with ExpOU model for the small volatility quantities only.