How does the Black model price options on forward contracts?

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AcadiFi · Accepted Answer

The Black-76 model prices European options on forwards or futures. Unlike Black-Scholes which uses spot price, Black uses the forward price F(0,T) as the underlying and discounts the entire payoff at the risk-free rate. The call price is c = e^(-rT)[F(0,T)N(d1) - KN(d2)] where d1 = [ln(F/K) + (sigma^2/2)T] / (sigma*sqrt(T)) and d2 = d1 - sigma*sqrt(T). For Prairie Grain's wheat forward option with F = $7.20, K = $7.00, T = 0.5, sigma = 25%, r = 4%: d1 = [ln(7.20/7.00) + (0.0625/2)(0.5)] / (0.25*0.707) = 0.249, N(d1) = 0.599. d2 = 0.072, N(d2) = 0.529. c = e^(-0.02)[7.20(0.599) - 7.00(0.529)] = 0.98*(4.313 - 3.703) = $0.598. ```mermaid flowchart LR F[Forward
price F] --> M[Black
model] K[Strike
price K] --> M V[Volatility
sigma] --> M M --> C[Call price
discounted] ``` Black-76 is preferred when the underlying is a forward/futures because carry costs and convenience yields are already embedded in F. It's heavily used for commodities, interest rate caps/floors, and swaptions.

How does the Black model price options on forward contracts?

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