Options on Futures

Options, Futures and Derivative Securities

Lorenzo Naranjo

Spring 2026

Options on Futures

  • The underlying asset is a futures contract, usually on a commodity, a precious metal, a currency, an interest rate or a bond.
  • These options usually expire on or a few days before the earliest delivery date of the underlying futures contract.
  • If a call futures option is exercised, the buyer gets a long position in the futures plus a cash amount equal to the excess of the futures price at the time of the most recent settlement over the strike price.
  • If a put futures option is exercised, the buyer acquires a short position in the futures plus a cash amount equal to the excess of the strike price over the futures price at the time of the most recent settlement.

Example 1 A September call option contract on copper futures has a strike of 425 cents per pound. A CME contract is written on 25,000 pounds of copper. The call is exercised when the futures price is 426.95 cents and the most recent settlement is 426.45 cents. The trader receives a long September futures contract on copper and 25{,}000 \times (4.2645 - 4.2500) = \$362.50.

Example 2 A September put option contract on soybean futures has a strike price of 1,380 cents per bushel. A CME contract is written over 5,000 bushels. The put is exercised when the futures price is 1372 cents per bushel and the most recent settlement price is 1365 cents per bushel. The trader receives a short September futures contract on soybean and 5{,}000 \times (13.80 - 13.65) = \$750 in cash.

Why Options on Futures

  • For commodities, the futures contract is what trades: the spot price of a commodity is the price of the nearby futures.
  • Exercising a futures option delivers a futures position, which can be closed or rolled easily, not the physical good.
  • Futures options and futures usually trade side by side on the same exchange, such as the CME, so futures options are liquid and may entail lower transaction costs.

Replicating a Futures Option

  • A desk sells a European option H with maturity T on a futures contract that expires at T_{F} \geq T, and replicates it with N_{F} futures and N_{Z} bonds.
  • Entering a futures contract costs nothing, and gains are marked to market, so all the value is in the bond: V = N_{Z} Z \quad \Rightarrow \quad dV = N_{F} dF + r V dt
  • Itô’s lemma for the option: dV = \left(\frac{1}{2} \sigma^{2} F^{2} \frac{\partial^{2} V}{\partial F^{2}} + \frac{\partial V}{\partial t}\right) dt + \frac{\partial V}{\partial F} dF
  • Matching both: N_{F} = \partial V / \partial F and \frac{1}{2} \sigma^{2} F^{2} \frac{\partial^{2} V}{\partial F^{2}} + \frac{\partial V}{\partial t} - r V = 0
  • This is the dividend-yield PDE with \delta = r: a futures price behaves as if it paid a dividend yield equal to r.

The Risk-Neutral Process for the Futures Price

  • A futures position costs nothing to enter and pays dF, so in a risk-neutral economy its expected gain must be zero.
  • The futures price is a martingale under the risk-neutral measure: dF = \sigma F dB^{*}
  • This holds even with random interest rates: the futures price is the risk-adjusted expected spot price at maturity (Cox et al. 1981; Shreve 2004, Theorem 5.6.5).
  • The call price is then C = e^{-r T} \operatorname{E}^{*}\left((F_{T} - K) 1\kern-0.25em\text{l}_{\{F_{T} > K\}}\right)

Black’s Model for European Futures Options

  • The prices of European call and put futures options are (Black 1976): \begin{aligned} C & = e^{-r T} \left[F \operatorname{Φ}(d_{1}) - K \operatorname{Φ}(d_{2})\right] \\ P & = e^{-r T} \left[K \operatorname{Φ}(-d_{2}) - F \operatorname{Φ}(-d_{1})\right] \end{aligned} where d_{1} = \frac{\ln(F/K) + \frac{1}{2} \sigma^{2} T}{\sigma \sqrt{T}} \quad \text{and} \quad d_{2} = d_{1} - \sigma \sqrt{T}
  • Put-call parity: C - P = e^{-r T} (F - K).
  • The delta of the call is e^{-r T} \operatorname{Φ}(d_{1}) futures contracts, and that of the put is -e^{-r T} \operatorname{Φ}(-d_{1}).
  • \sigma is the volatility of the futures price.

Black’s Model in Practice

  • When the option and the futures mature on the same date, F_{T} = S_{T}, so a European option on the futures is a European option on the underlying.
  • Financial underlyings: F = S e^{(r - \delta) T}, and Black’s model gives back the dividend-yield formulas.
    • Currency traders prefer the forward form: forward rates are quoted directly as forward points.
  • Commodities: the futures price is not tied to the cash price of the physical good, so it is the input to the formula.

Example 3 Consider a 6-month European call option on gold whose payoff is set by the price of gold in six months, when the 6-month futures contract expires. The 6-month futures price is $1,806, the 6-month risk-free rate is 1% per year continuously-compounded, the strike price is $1,820, and the volatility of the futures price is 20% per year. Then \begin{aligned} d_{1} & = \frac{\ln(1806/1820) + \frac{1}{2} (0.20)^{2} (0.5)}{0.20 \sqrt{0.5}} = 0.0161\Rightarrow \operatorname{Φ}(d_{1}) = 0.5064 \\ d_{2} & = 0.0161 - 0.20 \sqrt{0.5} = -0.1253 \Rightarrow \operatorname{Φ}(d_{2}) = 0.4501 \\ C & = 1806 e^{-0.01 (0.5)} (0.5064) - 1820 e^{-0.01 (0.5)} (0.4501) = 94.88 \end{aligned} To hedge it, a dealer who sold the call buys e^{-0.01 (0.5)} (0.5064) = 0.5039 futures contracts per unit of the option.

Futures Style Options

  • A futures-style option is a futures contract on a futures option.
  • The premium is not paid upfront: the position is margined and marked to market like a futures contract.
  • The futures price of a call futures-style option is: \phi_{C} = C e^{r T} = F \operatorname{Φ}(d_{1}) - K \operatorname{Φ}(d_{2})
  • Similarly, for a put futures-style option: \phi_{P} = P e^{r T} = K \operatorname{Φ}(-d_{2}) - F \operatorname{Φ}(-d_{1})

References

Black, Fischer. 1976. “The Pricing of Commodity Contracts.” Journal of Financial Economics 3 (1–2): 167–79. https://doi.org/10.1016/0304-405X(76)90024-6.
Cox, John C., Jonathan E. Ingersoll, and Stephen A. Ross. 1981. “The Relation Between Forward Prices and Futures Prices.” Journal of Financial Economics 9 (4): 321–46. https://doi.org/10.1016/0304-405X(81)90002-7.
Shreve, Steven E. 2004. Stochastic Calculus for Finance II: Continuous-Time Models. Springer Finance. Springer.