# Has anyone ever derived an analytical basket option which gives terminal asset prices individually, by asset?

Random thought I had around what would be an ideal analytical basket formula. If the formula gave terminal prices of each asset instead of a single basket price, you could price any number of exotic payoffs. Which would in theory make it (nearly) as useful as MC.

Could you emulate the same with closed-form solutions that are already published? A creative way to back-solve for each underlying price at expiration? Let's for the sake of keeping things simple assume GBM, or ABM if that would be easier.

Note that my question is rather than a closed-form solution for a single basket value, a closed-form solution that provides the terminal values (prices) at expiration OF EACH ASSET IN THE BASKET. If that's possible or not I do not know.

There are people that attempt to arbitrage the mispricing between portfolio vs components. In options, one method is dispersion. There are other markets where this happens as well such as ETF arbitrage. I don’t know if anyone has a closed form solution to gauge this mispricing as much as they use brute force to price all of the underlying components of the portfolio. And then construct the basket with the individual components, taking into account portfolio effects such as correlation etc., to arrive at the basket prices.

• I don't disagree at all. Valuing the items which comprise a basket vs. the Index is a bit of a different question than what I was trying to say. I tried to clarify my question so that it is more clear.
– Matt
Apr 19 at 18:55
• @Matt. In the scenario I describe of brute force, since there is no closed form solution to value the portfolio, there is no invertible function so the reverse is also true. There are an old infinite number of component values that could achieve a portfolio value. Is there a particular product or market where you are attempting to arrive at component values? If so, that might steer others toward a solution for you. Apr 19 at 19:22
• Well the specific usage of this is in commodities for valuation of long term contracts - one I'm valuing has 30 underling assets which become 3 "best of" options of ~10 crude grades. I currently price them via Randomized Quasi-MC simulation w/ GBM assumptions, but that's quite slow, hence the reason for my question.
– Matt
Apr 19 at 19:30
• @Matt I remember there is a book by Peter G. Zhang who uses closed from approximations for exotic derivatives. As far as i recall FinCad also uses these approximations to price various instruments that should be valued using MC. Maybe this could assist you? Not sure whether this book will provide some of the answers you might habe..
– T123
Apr 21 at 6:37
• @T123 thank you, that's exactly what I was looking for
– Matt
Apr 21 at 13:49