Explain how the Black model values European options on futures contracts
The idea
The Black model prices European options on futures or forwards. Because a futures contract costs nothing to enter, its price replaces the spot and there is no carry term: call = e^(−rT) × [F × N(d1) − X × N(d2)] and put = e^(−rT) × [X × N(−d2) − F × N(−d1)], with d1 = [ln(F ÷ X) + σ²T/2] ÷ (σ√T) and d2 = d1 − σ√T. Both F and X are discounted, so put–call parity for options on futures becomes c = p + e^(−rT)(F − X). The trap is plugging the futures price into the spot BSM formula, which discounts the strike but not F and overstates the call.
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