Suppose we want to price an exotic equity which is a function of $min(S_1, S_2)$. To do this, I'm trying to compute an implied volatility surface for $min(S_1, S_2)$ and then price the option using that surface. It's essentially a reduction in dimension. In particular, we have the implied vol surfaces for $S_1$ and $S_2$ and we need the implied volatility surface for $min(S_1, S_2)$.
Here's what I've tried to do: For each $S_1$ and $S_2$, we have a forward and an implied volatility space which can be used to price options. For each strike and tenor, I am computing an option price (put for low strike, call for higher strike) on the $min(S_1, S_2)$ by matching moments, i.e. assume that $min(S_1, S_2)$ is lognormal and then analytically compute the moments of $min(S_1, S_2)$, imply the GBM params and price the option using the B-S equation.
Now I have option prices for $min(S_1, S_2)$ for each strike and tenor. Finally, I can imply the implied volatility surface using B-S. My question is which forward should I use in implying back out the implied volatility? I think I should use the zero-strike implied option price using the aforementioned dimension reduction ; however, I'm also thinking it might be correct to use the actual basket forward, i.e. literally compute the $min(F_1, F_2)$ at each tenor. Please let me know your thoughts. Really struggling with this.