The Black Formula
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 corresponding put price is
where
and N(.) 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' . Thus the discount factor is replaced by since one must take into account the time value of money. The difference in the two cases is clear from the derivation below.
Read more about this topic: Black Model
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