Multi-scaling of moments in stochastic volatility models

Paolo Dai Pra 1 Paolo Pigato 2, 3
2 TOSCA - TO Simulate and CAlibrate stochastic models
CRISAM - Inria Sophia Antipolis - Méditerranée , IECL - Institut Élie Cartan de Lorraine : UMR7502
Abstract : 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.
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Paolo Dai Pra, Paolo Pigato. Multi-scaling of moments in stochastic volatility models. Stochastic Processes and their Applications, Elsevier, 2015, 125 (10), pp.3725-3747. ⟨10.1016/j.spa.2015.04.007⟩. ⟨hal-01407443⟩

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