Black's formula
Quantitative Finance glossary · Derivatives Pricing & Volatility Surfaces
Definition
Closed-form pricing of European options on forwards/futures: C = e^-rT,[F,N(d_1) - K,N(d_2)], with d_1 = (ln(F/K) + 1/2σ^2 T)/(σ√(T)), d_2 = d_1 - σ√(T). The market standard for caplets (replacing F with the forward rate L and multiplying by accrual τ), swaptions (under the annuity measure), and commodity futures options. Black-76 is the workhorse rates-and-commodities cousin of BSM.
Black's formula in plain English
An option is the right (not obligation) to buy something later at a locked-in price, and the deep question is: what's that right worth *today*? Black's formula answers this when the underlying thing is a forward or futures contract — an agreement whose price F is already "delivery later" priced in. The formula, C = e^-rT[F,N(d_1) - K,N(d_2)], blends the forward price F, your locked strike price K, how much time T is left, and how jumpy the price is (σ), where the N(·) terms are probabilities that the option ends up worth exercising. It's the industry workhorse for options on interest rates, commodities like oil, and more — basically the futures-market cousin of the famous Black-Scholes stock-option formula.
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