The minimal entropy measure and an Esscher transform in an incomplete market model

We consider an incomplete market model with one traded stock and two correlated Brownian motions W,W˜. The Brownian motion WW drives the stock price, whose volatility and Sharpe ratio are adapted to the filtration F˜≔(F˜t)0⩽t⩽T generated by W˜. We show that the projections of the minimal entropy and minimal martingale measures onto F˜T are related by an Esscher transform involving the correlation between W,W˜, and the mean-variance trade-off process. The result leads to a new formula for the marginal exponential utility-based price of an F˜T-measurable European claim.