Editing Black model (section)

==The Black formula==
The Black formula is similar to the [[Black–Scholes formula]] for valuing [[stock option]]s 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 (finance)|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 <math>T' \geq T</math>) is

:<math> c = e^{-r T} [FN(d_1) - KN(d_2)]</math>

The corresponding put price is

:<math> p = e^{-r T} [KN(-d_2) - FN(-d_1)]</math>

where

:<math> d_1 = \frac{\ln(F/K) + (\sigma^2/2)T}{\sigma\sqrt{T}} </math>

:<math> d_2 = \frac{\ln(F/K) - (\sigma^2/2)T}{\sigma\sqrt{T}} = d_1 - \sigma\sqrt{T}, </math>

and <math>N(\cdot)</math> is the [[Normal distribution#Cumulative distribution function|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 <math>e^{-rT}</math> is replaced by <math>e^{-rT '}</math> since one must take into account the [[time value of money]]. The difference in the two cases is clear from the derivation below.