# How to price a futures spread option?

Let's say I have two futures contract $F_1(0,T)$ and $F_2(0,T)$ on two different correlated underlyings.

If I assume that both underlying follow a GBM with volatility $\sigma_1$ and $\sigma_2$ respectively, and their return are correlated by $\rho$, is there a closed-form formula I can use to price an European option on the spread $X_{1,2} = \max( F_1(S,T) - F_2(S,T), 0 )$ at some expiry date $S \leq T$?

Otherwise, how do we usually proceedto price these things? Monte-Carlo using risk-neutral dynamics and discounting payoffs?

• Might be simpler to model the spread directly as a normal process, then price with a Bachelier model? Otherwise you could check Haug's book, I don't have it handy... – experquisite Dec 22 '14 at 18:53
• Do you mean that you want to price payoffs $\max( F_1(S,T) - F_2(S,T), 0 )$ at $S$ or European payoffs $g(X_{1,2})$ (for example, if $g(x) = (x-K)_+$, that would be $\max( F_1(S,T) - F_2(S,T) - K, 0)$). In the first case you have an exchange option and this reduces to the BS formula by using the second underlying as a numeraire. In the second case, there is no closed form. – AFK Dec 24 '14 at 13:52