Mapping Futures Options to a Black-76 Volatility Surface

Summary

The document explains how to relate an option on a delivery-date futures contract to a Black-76 valuation using a different futures contract associated with the option's expiry. Under deterministic rates and basis, it expresses the delivery-date futures price at option expiry as a scaled version of the expiry-date futures price. Rewriting the option payoff then transforms the strike: the relevant volatility is the surface volatility at the adjusted strike, rather than simply at the original strike. The resulting price can be written using Black-76 with the delivery futures price and the original strike.

This gives a framework for defining or reading a volatility surface when the futures underlying varies across maturities. The derivation assumes deterministic rates and basis, so it does not resolve the more general convexity effects that can arise with stochastic rates or basis. It explains a pricing transformation, but does not provide market data, calibration steps, or evidence that the assumptions fit a particular futures-options market.

Key ideas

• The option's delivery-date futures underlying can be related to the expiry-date futures under deterministic rates and basis.
• The transformation rescales the strike used to select volatility from the surface.
• The transformed valuation can be expressed with Black-76 using the delivery futures price and original strike.
• Stochastic rates or basis may introduce convexity effects beyond the stated derivation.

Tags

# Volatility surface for futures options


# 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.

## Answer by river_rat (score 1)

https://quant.stackexchange.com/a/79023

The deterministic basis/rates case is simpler, as we can use the fact the forwards and futures are effectively the same product.

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 $K^*$

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.