Derivation and Assumptions
The Black formula is easily derived from use of Margrabe's formula, which in turn is a simple, but clever, application of the Black–Scholes formula.
The payoff of the call option on the futures contract is max (0, F(T) - K). We can consider this an exchange (Margrabe) option by considering the first asset to be and the second asset to be the riskless bond paying off $1 at time T. Then the call option is exercised at time T when the first asset is worth more than K riskless bonds. The assumptions of Margrabe's formula are satisfied with these assets.
The only remaining thing to check is that the first asset is indeed an asset. This can be seen by considering a portfolio formed at time 0 by going long a forward contract with delivery date T and short F(0) riskless bonds (note that under the deterministic interest rate, the forward and futures prices are equal so there is no ambiguity here). Then at any time t you can unwind your obligation for the forward contract by shorting another forward with the same delivery date to get the difference in forward prices, but discounted to present value: . Liquidating the F(0) riskless bonds, each of which is worth, results in a net payoff of .
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