I am having a slight brain meltdown because I do not seem to be able to understand the following basic thing.
Consider a BS economy, and two assets $X$ and $Y$ $$ dX = \sigma X dW $$ $$ dY = \nu Y dZ $$ $$ dWdZ = \rho dt $$
I would like to price a Margrabe option $(X_T - Y_T)_+$.
The first and most straightforward method is a change of numeraire approach. In other words $$ E_t(X_T - Y_T)_+ = Y_t E^{Q_Y}( X_T/Y_T -1 )_+ $$ where $Q_Y$ is the measure with $Y$ as numeraire. Now if you evaluate the above expression under this measure you get a relatively simple option price expression, and where the correlation $\rho$ will appear in the formula. Agree?
The second approach is to use conditioning. Does everyone agree that I can also price the options as follows: $$ E_t(X_T - Y_T)_+ = \int_0^\infty C(X_t, y) q(y) dy $$ where $C(X_t, y)$ is the single asset BS option price with strike $y$, and $q(y)$ is the lognormal distribution of $Y$.
I can always calculate using the numerical integration above right? If so, here is where I am confused: how does the correlation parameter $\rho$ appear in the numerical integration? I cannot see it, but it must somehow play a role.
Help!