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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 ...

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