# Dimension reduction for worst of basket on $min(S_1, S_2)$

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.

• I think the assumption that min(S1,S2) is lognormal is a weakness in your argument. There is plenty of literature on the probability distribution of the maximum or minimum of 2 correlated normal distributions, which you could look at. – dm63 Aug 27 at 13:51