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 $W$,$\widetilde{W}$. The Brownian motion $W$ drives the stock price, whose volatility and Sharpe ratio are adapted to the filtration $\mathbb{F} := (\widetilde{\mathcal{F}}_{t})_{0 \le t \le T}$ generate...
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| Format: | Journal article |
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2005
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| Summary: | 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. |
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