I am looking to find/estimate the "greeks"/option price sensitivities/derivatives for a basket option situation. In specific the change in price of a put option associated with a change in weight of a given asset in a portfolio. I think a finite difference method (FDM) with MCMC simulations would work for a two asset situation but wasn't sure how to carry it out for more than two assets.
Say you have three assets/stocks (with no dividends): A,B,C in a basket with their respective estimated mean returns, estimated standard deviations, and estimated correlations/covariance matrix. Instead of three strikes for each asset, you have a single portfolio strike that is 10% below the 'current' portfolio. The 'current' portfolio weights are presented/given by the 'client'.
You can relatively simply MCMC simulate the three stocks, and assign the given weights from initiation to calculate portfolio terminal value (assuming no rebalancing), and estimate various 'standard' greeks. And for a two asset portfolio, the
weight of asset 2 is just
1 - weight of asset 1, so in terms of finding the change in option price due to a change in weight of each asset, which is what originally looked like a two variable problem, (i.e. to find the change in option price by changing the weight of asset 1 and asset 2), becomes a one variable problem, as any change in the weight of asset 1 will implicitly change the weight of asset 2, making simulation and calculation of the sensitivities easier. But this simplification/dimension reduction does not exist for a three or more asset situation, so I was wondering how would one do it for a three or more asset situation?