Black-76 Pricing for Forwards and Futures
Black-76 prices options on forwards or futures using forward price, discount factor, and implied volatility as primary inputs.
Black-76 treats the forward or futures price F(0,T) as the underlying and assumes lognormal dynamics with constant volatility under the risk-neutral measure.
European call and put prices take the same closed-form shape as Black–Scholes, with spot S₀ replaced by F(0,T) and cashflows discounted by DF(0,T).
In commodities and rates, markets quote implied volatilities in Black-76 terms because forwards and futures are the natural traded underlyings.
Limitations mirror Black–Scholes: lognormality, constant vol, and European exercise; real power and gas markets exhibit spikes, seasonality, and path-dependent exercise that require richer models.
Black-76 prices options on forwards: the state variable is F(0,T), and discounting is explicit via DF(0,T). Use “implied vol mode” to invert a market option price into σ.
Black-76 uses the forward as the underlying. Discounting is clean: prices scale with DF, and the “moneyness” is set by F/K. The curves show how option value changes with strike holding (F, DF, σ, T) fixed.
In implied-vol mode, you invert one observed option price into σ. That is the market workflow: quotes are often expressed as Black implied vol rather than raw price.
Put-call parity is a fast sanity check. If it fails materially, your inputs (DF, F, price) are inconsistent or you are mixing conventions.