# Volatility surface for futures options

When looking at futures options such as CME's Gold options or many equity index futures options, the underlying is not the index but to be precise the closest to delivery futures contract. That means, across a hypothetical surface, the underlying changes in theory.

That's not assumed in BS. I know there is the Black-76 model but I'm suddenly a bit uncertain about how to build or define a volatility surface for options which are in that way not "directly" related? I know this is being pedantic but I would like to understand the potential convexity effects involved.

• Depends on how you want to model the spot / futures basis. If it's deterministic then it's a simple strike shift pricing problem. If you are treating that as stochastic then it's a mess. Commented Apr 17 at 12:26
• @river_rat, can you elaborate? We can focus on the deterministic case for now although for curiosity the stochastic case might be interesting as well. Is there some literature on the topic? Commented Apr 17 at 13:53
• Added the workings for the deterministic case Commented Apr 17 at 15:17

Let $$F(t,T)$$ be the futures contract at time t for delivery at time T. Deterministic rates/basis imply that $$F(T_e, T_d)=\frac{F(t,T_d)}{F(t,T_e)}F(T_e,T_e)$$ Then $$v_t=DF(t,T_e)\mathop{\mathbb{E}}((F(T_e, T_d)-K)^+|\mathcal{F}_t)=DF(t,T_e)\frac{F(t,T_d)}{F(t,T_e)}\mathop{\mathbb{E}}((F(T_e,T_e)-\frac{F(t,T_e)}{F(t,T_d)}K)^+|\mathcal{F}_t)$$ Denote by $$K^* = \frac{F(t,T_e)}{F(t,T_d)}K$$ then we see that $$v_t=DF(t,T_e)\frac{F(t,T_d)}{F(t,T_e)}\mathop{Black}(F(t,T_e), \frac{F(t,T_e)}{F(t,T_d)}K, \mathop{\sigma}(K^*,T_e), T_e)$$ or $$v_t=DF(t,T_e)\mathop{Black}(F(t,T_d), K, \mathop{\sigma}(K^*,T_e), T_e)$$ So we can use Black-76 with the given futures price and forward if we use the volatility for the shifted strike $$K^*$$