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

### Abstract

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

Topics: Probability theory and stochastic processes
Year: 2007
DOI identifier: 10.1016/j.spl.2007.01.008
OAI identifier: oai:generic.eprints.org:623/core69

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