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I am working on a project that involves pricing european call options in incomplete markets. Now I need to find a unique measure $Q^*$ such that

$$Q^* = \min_{M_e} E_Q [V(T)-F(w)]^2 = \min_{u} E_Q [V(T)-F(w)]^2$$

where $V(T)$ is the terminal value of a portfolio given

$$V(T) = V(0)e^{rt} + \int_0^T e^{r(t-u)}\beta(u)S(u)\sigma dB(u)$$

and

$$F(w)=(e^{\sigma B(T)+(r-\frac{1}{2}\sigma^2)T}-K)$$

$M_e$ is the set of all equivalent martingales.

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I dont see a question? –  phi Apr 22 '13 at 12:51
2  
The whole first line does not make sense. On the LHS you have a measure $Q^*$ and on the right $\min_{M_e}E_Q[V(T)-F(\omega)]^2$. And what should $u$ be? The expression you want to minimize does not depend on $u$! Furthermore, in the definition of $F(\omega)$, is it meant to be $F(\omega)=(e^{\sigma B_T(\omega)+(r-\frac{1}{2}\sigma^2)T}-K)$? So it should be an $\omega$ instead of $w$? –  user8 Apr 23 '13 at 17:03

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