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EDIT:

Maybe there is no way to get explicit solutions for basket options (maybe the Black-Scholes differential equation can't be solved directly ??).

Q3: How do you price and hedge ( S1(T) + S2(T) - K )+ at time t. S1 evolves in $ S2 evolves in €, and the flows are in \$ ??

An alternative solution (if S1 and S2 were in $) might be Monte-Carlo simulation of S1 and S2 under risk-free hypothesis. The hedging is done using finite differiation method in the simulations.

Thank you already ;D

Guillaume

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well there are approximations for the prices but no exact formula since you have a sum of lognormals.

Take the USD bank account as numeraire. Then the drift of S1 is the drift of the riskless account r. The drift of S2 is $r +C_{f2}$ where $C_{f2}$ is the instantaneous covariance between $S2$ and the FX rate.

Now just compute $$ \mathbb{E} \left( (S_1(T) + S_2(T) - K)_+ \right)e^{-rT}. $$ This is a 2-dim integral so Monte Carlo is overkill. Just use a simple 2d-integral method.

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