Esscher transforms and martingale measures in incomplete diffusion models

The minimal entropy and minimal martingale measures are shown to be related by an Esscher transform, involving the mean-variance trade-off, in an incomplete diffusion model containing a traded stock and a correlated non-traded stochastic factor. The coefficients of the diffusions are measurable with respect to the Brownian motion driving the non-traded factor, as is typical in stochastic volatility models. The result is motivated by an analysis of exponential indifference prices, and made rigorous by appealing to a representation equation for the q-optimal measure due to Hobson [14]. The result yields a new representation for the marginal price of a claim on the non-traded factor.