Black-76 Model

Black-Scholes needs to know the stock price today, and separately, how it grows before expiry — dividends, financing, storage costs, all bundled into a drift adjustment. For commodities, bonds, and swaps, that drift is often awkward to pin down directly, but the market already publishes a number that has done the work for you: the forward or futures price. Fischer Black's 1976 paper asked the obvious question — why re-derive the carry when the forward price already contains it? — and the resulting formula is simpler for exactly that reason.

Skip the recipe, use the finished dish

Pricing off the spot price is like costing a cake from raw ingredients: flour, sugar, oven time, all separately estimated, with plenty of room for one of them to be wrong. Pricing off the forward price is like costing the same cake from its price on the bakery shelf — someone else already added up the ingredients, and the number is sitting right there, quoted every second in the futures market. Black-76 substitutes the shelf price for the recipe.

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

In plain English: this is Black-Scholes with the stock price $S$ replaced by the forward price $F$, and — because $F$ already carries no further drift under the pricing measure — the whole payoff is simply discounted back at the risk-free rate $e^{-rT}$, with no separate $e^{-qT}$ dividend term needed. $F$ has absorbed all of that already.

Worked example 1: a call on a crude oil future. $F = 75$ ($75), $K = 70$ ($70), $\sigma = 30\%$, $T = 0.5$, $r = 4\%$. Then $\ln(75/70) = 0.0690$, and $\tfrac12\sigma^2 T = 0.5\times0.09\times0.5=0.0225$, so the numerator is $0.0915$. $\sigma\sqrt{T}=0.30\times0.7071=0.2121$, giving $d_1=0.0915/0.2121=0.4314$ and $d_2=0.4314-0.2121=0.2193$. Using $N(0.4314)\approx0.6669$ and $N(0.2193)\approx0.5868$: $C=e^{-0.02}\left[75(0.6669)-70(0.5868)\right]=0.9802\times[50.02-41.08]=0.9802\times8.94=8.76$, i.e. $8.76.

Worked example 2: Black-76 agrees with plain Black-Scholes when it should. Take a non-dividend stock at $S=50$ ($50), $r=5\%$, $T=1$. Its one-year forward price is $F=S e^{rT}=50 e^{0.05}=52.56$, i.e. $52.56. Price a $50-strike call two ways: run ordinary Black-Scholes with $S=50$, $r=5\%$; or run Black-76 with $F=52.56$, then discount by $e^{-rT}$. Both routes must give the same number, because $e^{-rT}F N(d_1)$ is algebraically identical to $S N(d_1)$ once $F=Se^{rT}$ is substituted back in — Black-76 isn't a different model from Black-Scholes, it's the same formula rewritten around the forward price instead of the spot price.

What this means in practice

Black-76 is the default on desks trading interest-rate caps and floors, swaptions, and commodity or bond futures options, precisely because the forward curve is already a market-observed input, sparing the trader from separately estimating storage costs, convenience yield, or a dividend schedule. Options On Futures covers what actually happens mechanically when one of these is exercised.