# Modelling forward and correlation dynamics for options on the best-of multiple commodity spreads

I am looking at best-of options on futures (commodities), let's take for example the following payoff specification:

$$max(a_{1}-c_{1}, \ a_{2}-c_{1}, \ a_{3}-c_{1}, \ b_{1}-c_{1}, \ b_{2}-c_{1}, \ b_{3}-c_{1}, 0)$$

$$a_{1}, \ a_{2}, \ a_{3}$$ represent forward contracts on asset $$a$$ at delivery period $$1, 2, 3$$.

$$b_{1}, \ b_{2}, \ b_{3} \$$ represent forward contracts on asset $$b$$ at delivery period $$1, 2, 3$$.

$$c_{1}$$ represent a forward contract on asset $$c$$ at delivery period $$1$$.

I was not yet able to find an analytical solution in books\papers for more than two assets (example: https://carmona.princeton.edu/download/fe/spread.pdf). Given the payoff complexity and the number of underlyings I guess Monte Carlo is the optimal solution, how would you model the underlyings and correlation? Should I simulate a forward curve dynamic (for example via PCA) or ignore the fact that there are two distinct underlyings but with three different contract expirations?

Assume there are no options contracts listed, so it is not possible to input implied volatilities and correlations.