Options On Futures

Buy a call on a stock and exercise it, and you own the stock: a real asset, paid for at the strike, that you could hold forever. Buy a call on a futures contract and exercise it, and you own a futures position — a promise, not an asset, that itself has to be margined and marked to market every day. The option looks the same on a screen, but what it turns into on exercise is completely different, and that difference changes both the cash flows and the pricing formula.

A coupon that redeems for another coupon

Think of a stock option as a coupon redeemable for a physical item — hand it in, get the goods. A futures option is a coupon redeemable for a second coupon: exercise it and you receive a futures contract, which you then still have to manage, post margin against, and mark to market daily until you close it out or it expires. The value you're actually pricing isn't "what will the commodity be worth," it's "what will the futures price be worth," and the futures price already embeds the market's view of storage costs, financing, and dividends. That's why futures options are priced off the forward/futures price $F$ directly, using Black-76 Model, instead of off the spot price with a carry adjustment.

$$C = e^{-rT}\left[F\,N(d_1) - K\,N(d_2)\right], \qquad d_1 = \frac{\ln(F/K) + \tfrac12\sigma^2 T}{\sigma\sqrt{T}},\quad d_2 = d_1 - \sigma\sqrt{T}$$

In plain English: this is the ordinary Black-Scholes call formula with the stock price $S$ replaced everywhere by the futures price $F$, and with the whole result discounted by $e^{-rT}$ because, unlike a stock, holding a futures contract earns no separate financing benefit to offset against — the discounting has to be done explicitly.

Worked example 1: pricing a call on a futures contract. A crude-oil futures contract trades at $F=100$ ($100), strike $K=100$ ($100), $\sigma=25\%$, $T=0.25$ years, $r=5\%$. Then $\ln(F/K)=0$, and $d_1 = (0 + 0.5\times0.0625\times0.25)/(0.25\times0.5) = 0.0078/0.125 = 0.0625$, so $d_2=0.0625-0.125=-0.0625$. Using $N(0.0625)\approx0.5249$ and $N(-0.0625)\approx0.4751$: $C = e^{-0.0125}\left[100(0.5249) - 100(0.4751)\right] = 0.9876 \times 4.98 = 4.92$, i.e. $4.92.

Worked example 2: what exercise actually pays you. You paid $4.92 upfront for this call, same as any option. Suppose the futures price rises to $102 and you exercise. You don't receive oil — you receive a long futures position struck at $100, immediately marked to market against the current $102 futures price. On a contract with a 1,000-barrel multiplier, that mark generates $1{,}000 \times (102 - 100) = 2{,}000$, i.e. $2,000, credited to your account as variation margin, paid in cash the same day. Compare a stock call: exercising it just converts your option into shares worth $102 each, and you'd have to sell them to realise any cash — the futures option pays you the gain directly through the daily settlement mechanism instead.

What this means in practice

Because the underlying is already a forward-looking price, futures options avoid re-deriving a dividend or storage-cost adjustment each time — the carry is already baked into $F$. This is why commodity and interest-rate desks price nearly everything off Black-76 rather than spot Black-Scholes. The margin mechanics also matter for risk: a futures option's exercised position adds to your futures margin requirement immediately, whereas a stock option's exercised position adds a fully-paid asset with no separate margin call.