Mean-variance hedging with random volatility jumps. (English) Zbl 1035.91028

Biagini, Francesca; Guasoni, Paolo

Stochastic Anal. Appl. 20, No. 3, 471-494 (2002).

From authors’ abstract: The authors introduce a general framework for stochastic volatility models, with the risk asset dynamics given by: \[ dX_t(\omega,\eta)= \mu_t(\eta) X_t(\omega, \eta)\,dt+ \sigma_t(\eta) X_t(\omega, \eta)\,dW_t(\omega), \] where \((\omega,\eta)\in (\Omega\times H, {\mathcal T}^\Omega\otimes{\mathcal T}^H, P^\Omega\otimes P^H)\).
In particular, the authors allow for random discontinuities in the volatility \(\sigma\) and the drift \(\mu\). First it is characterized the set of equivalent martingale measures, then compute the mean-variance optimal measure \(\widetilde P\), using some results of Schweizer on the existence of an adjustment process \(\beta\). It is shown the examples where the risk premium \(\lambda= (\mu-r)/\sigma\) follows a discontinuous process, and explicit calculations for \(\widetilde P\).

Reviewer: C. L. Parihar (Indore)

MSC:

91B28	Finance etc. (MSC2000)
60H30	Applications of stochastic analysis (to PDEs, etc.)

Keywords:

stochastic volatility models; mean-variance optimal measure; change of numéraire