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.

Keywords:

• Hedging in incomplete markets
• Stochastic volatility models
• Mean–variance optimal measure
• Change of numéraire
• AMS Subject Classification: 60H30; 90A09
• JEL Subject Classification: G10

Acknowledgment

We wish to thank Maurizio Pratelli for proposing this research, and for his precious advice in many discussions. We also thank Paul Embrechts, for suggesting some examples, and Koichiro Takaoka for his comments, and for a careful reading of an earlier version of the paper.

Notes

^†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.