Mean–variance hedging with random volatility jumps

Abstract

We introduce a general framework for stochastic volatility models, with the risky asset dynamics given by: where (ω,η)∈(Ω×H,F Ω⊗F H ,P Ω⊗P H ). In particular, we allow for random discontinuities in the volatility σ and the drift μ. First we characterize the set of equivalent martingale measures, then compute the mean–variance optimal measure P˜, using some results of Schweizer on the existence of an adjustment process β. We show examples where the risk premium λ=(μ−r)/σ follows a discontinuous process, and make explicit calculations for P˜. †The first draft of this paper was completed while the second author was affiliated to Scuola Normale Superiore. The views expressed in this paper are those of the authors and do not involve the responsibility of the Bank of Italy.