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Suppose we have $X,Y$ independent normally distributed r.v. $X \sim \mathcal N(a,\sigma^2_1)$, $Y \sim \mathcal N(a,\sigma^2_2)$, and $Z=\rho X+\sqrt{1-\rho^2}Y$ where $\rho$ is some constant.

How can I calculate the $\mathbb{E}[\max(0,e^Z-e^Y)]$?

Thanks.

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    I mean $\rho$ to be not the correlation, but just some constant2019-01-17

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Hint: $\mathbb E[ \max(0, \mathrm e^Z - \mathrm e^Y) ] = \mathbb E \left[ \mathrm e^Z - \mathrm e^Y | Z>Y \right] \mathrm P(Z>Y) $