Multi-scaling of moments in stochastic volatility models

We introduce a class of stochastic volatility models (Xt)t≥0 for which the absolute moments of the increments exhibit anomalous scaling: double-struck E (| Xt+h - Xt |q) scales as hq/2 for q < q∗, but as hA(q) with A(q) < q/2 for q > q∗, for some threshold 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.