I have a problem which involves optimisation of a portfolio containing one stock and multiple call options written on it, with the same maturity and different strikes. In order to use optimisation technique, I need a covariance matrix between the options and the stock. Usually to calculate covariance matrix one would use a matrix of returns, but what do you do if you use Monte Carlo simulation instead of historic data and portfolio containing derivatives rather than simple stocks?
evolve the stock to the requisite time horizon using some model. Get its value and that of the options on it. Compute the returns implied by these. Store this vector.
Do this many times.
Compute the implied covariance matrix of these vectors of returns.