Multi-scaling of moments in stochastic volatility models

We introduce a class of stochastic volatility models (Xt)t≥0(Xt)t≥0 for which the absolute moments of the increments exhibit anomalous scaling: E(∣Xt+h−Xt∣q)E(∣Xt+h−Xt∣q) scales as hq/2hq/2 for qq∗q>q∗, for some threshold q∗q∗. This multi-scaling phenomenon is observed in time series of financial assets. If the dynamics of the volatility is given by a mean-reverting equation driven by a Levy subordinator and the characteristic measure of the Levy process has power law tails, then multi-scaling occurs if and only if the mean reversion is superlinear.