My apologies if this question might be better suited elsewhere, however it regards probability and mathematical finance, so I thought I would post it here.
The question is:
Assume a universe where Black-Scholes is valid and Alice wants to sell a basket of $X$ call options to Bob on $Y$ different stocks with weights given by the vector $W$, subject to $X>Y$ and that the sum of $W=1$. She is given a vector of strike prices $K$ which she is unable to change, additionally she is given a co-variance matrix $\Sigma$.
If she wants to minimize the expected amount she has to pay at expiration to Bob by only changing the weights in the vector $W$ then how would you go about calculating this?
My own take is that I would just use the Markowitz minimum variance portfolio, but I am actually unsure whether this would yield a valid result.