Its primary applications are for pricing options on future contracts, bond options, interest rate cap and floors, and swaptions.
It was first presented in a paper written by Fischer Black in 1976.
The Black formula is similar to the Black–Scholes formula for valuing stock options except that the spot price of the underlying is replaced by a discounted futures price F. Suppose there is constant risk-free interest rate r and the futures price F(t) of a particular underlying is log-normal with constant volatility σ.
Then the Black formula states the price for a European call option of maturity T on a futures contract with strike price K and delivery date T' (with
is the cumulative normal distribution function.
Note that T' doesn't appear in the formulae even though it could be greater than T. This is because futures contracts are marked to market and so the payoff is realized when the option is exercised.
If we consider an option on a forward contract expiring at time T' > T, the payoff doesn't occur until T' .
since one must take into account the time value of money.
The difference in the two cases is clear from the derivation below.
The payoff of the call option on the futures contract is
We can consider this an exchange (Margrabe) option by considering the first asset to be
riskless bonds paying off $1 at time
Then the call option is exercised at time
The assumptions of Margrabe's formula are satisfied with these assets.
This can be seen by considering a portfolio formed at time 0 by going long a forward contract with delivery date
riskless bonds (note that under the deterministic interest rate, the forward and futures prices are equal so there is no ambiguity here).
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: