Here is the substitution that makes the rest of the module possible.
Black-Scholes-Merton prices an option on a spot asset, and it carries terms for whatever the asset does between now and expiry: a dividend yield on an index, a foreign interest rate on a currency, a storage cost and convenience yield on a commodity. Each of those terms exists to convert today's spot into what the asset will be worth at expiry in a risk-neutral world.
But for many instruments that conversion has already been done by the market. A futures price is the forward price. A forward rate implied by the discount curve is the market's price for money over that future period. When the underlying of the option is already a forward, carrying the spot around and then adjusting it is redundant.
What is the Black 76 formula?
Substitute F = S·e^{(r-q)T} into the Black-Scholes-Merton call formula and the exponentials collapse. What survives is
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c = e^{-rT} [ F·N(d1) - K·N(d2) ]
p = e^{-rT} [ K·N(-d2) - F·N(-d1) ]
d1 = [ ln(F/K) + (sigma^2 / 2)·T ] / (sigma·sqrt(T))
d2 = d1 - sigma·sqrt(T)There is no q. There is no cost of carry. The discount factor sits outside the bracket and does nothing but move the expected payoff back to today. That is the Black model, published in 1976 for commodity futures options and now the market standard for interest-rate options, and the practice set checks numerically that it agrees with Black-Scholes-Merton to within 4 × 10⁻¹³ when the forward is built from the spot.
Three properties of the formula are worth stating before it gets used.
The sigma is the volatility of the forward, not of the spot. Those differ, sometimes substantially, and a volatility lifted from the wrong series is a silent error.
Put-call parity becomes c - p = e^{-rT}(F - K), which is cleaner than its spot cousin because there is no dividend term to remember. It is also the fastest check on any implementation: price both, subtract, and compare against the discounted difference between forward and strike.
The formula is European. It contains no exercise decision, so it cannot price early exercise, and an option on a futures contract is very often American. That is the caveat the next section makes numerical rather than rhetorical.
Options on futures
An option on a futures contract gives the holder the right to acquire a futures position at the strike. Exercise does not deliver the commodity or the index. It delivers a long futures position at the strike price for a call, a short one for a put, and the exchange immediately marks that position to the settlement price, so the economic effect is a cash payment of the intrinsic value.
Black prices the European version directly, with F the current futures price and T the option's expiry. What Black cannot do is decide whether to exercise early, and for futures options early exercise is a live question in a way that it is not for a non-dividend-paying stock call.
The reason is that exercising an in-the-money futures option converts an asset whose payoff is discounted into a futures position that is marked to market in cash immediately. The holder of a deep in-the-money put on a futures contract at 78.40 with a strike of 86.00 can take the 7.60 today, or hold an option whose discounted intrinsic value is only 7.5131. Whether the option's remaining time value covers that gap is an arithmetic question, and the answer depends on how deep the option is and how far away expiry is.
How do you value early exercise on a futures option?
The tool for the answer is the same binomial tree DV1.02 built, with one change. When the underlying is a futures price, the risk-neutral probability is
p = (1 - d) / (u - d)with no e^{rΔt} in the numerator, because a futures position costs nothing to enter and therefore has zero expected return under the risk-neutral measure. Everything else is identical: build the lattice, apply the payoff at expiry, roll back discounting one step at a time, and at each node compare the continuation value against immediate exercise.
Worked example 4 runs that comparison on a real-shaped position and finds an early-exercise premium of $0.029111 per barrel on a European put worth $9.778335, which is 0.298%. Small. But the same tree shows early exercise firing at three of the five interior nodes of a three-step lattice, so "small in value" and "rare in occurrence" are different claims, and only one of them is true here.