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I don´t know how to derivate the Expected Value for the following problem:

Suppose that the random vector (S_1, S_2) has a bivariate lognormal distribution with parameter vector (u_1, u_2, v_1, v_2, p) such that vector (U_1, U_2)=[(ln{S_1}-u_1)/v_1, (ln{S_2}-u_2)/v_2] has a standard bivariate normal distribution with correlation p

... Now, how would you derivate expected positive difference of the bivariate lognormal spread, t.i. : E[max(S_1 - S_2, 0)]?

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if they are stocks, this problem is called pricing a Margrabe option and it is generally solved by change of numeraire. Take $S_2$ to be the numeraire. Then the value of the option is $$ S_2(0) \mathbb{E}_{S_2}( (S_1(T)/S_2(T)-1)_+) $$ where the expectation is taken in the measure that has $S_1/S_2$ as a martingale. Since it's a martingale and log-normal at time T the expectation is easy to compute and you are done.

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