Multi-scaling of moments in stochastic volatility models

Abstract

We introduce a class of stochastic volatility models $(X_t{)}_{t\ge 0}$ for which the absolute moments of the increments exhibit anomalous scaling: $\E\left(|X_{t+h} - X_t|^q \right)$ scales as $h^{q/2}$ for $q<q^{\ast }$, but as $h^{A(q)}$ with $A(q)<q/2$ for $q>q^{\ast }$, for some threshold $q^{\ast }$. 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.