We introduce a class of stochastic volatility models (X_t)_{t≥0} for which the absolute moments of the increments exhibit anomalous scaling: E (|X_{t+h} − X_ t | ^q) scales as h^q/2 for q < q * , but as h^A(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.