Black model
The Black model (sometimes known as the Black-76 model) is a variant of the Black–Scholes option pricing model. 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. Black's model can be generalized into a class of models known as log-normal forward models, also referred to as LIBOR market model. 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 formula_1) 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 formula_6 is replaced by formula_7 since one must take into account the time value of money. The difference in the two cases is clear from the derivation below. Derivation and assumptions. The Black formula is easily derived from the 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 formula_8. We can consider this an exchange (Margrabe) option by considering the first asset to be formula_9 and the second asset to be the riskless bond paying off $1 at time formula_10. Then the call option is exercised at time formula_10 when the first asset is worth more than formula_12 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 formula_10 and long formula_14 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 formula_15 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: formula_16. Liquidating the formula_14 riskless bonds, each of which is worth formula_18, results in a net payoff of formula_9. External links. Discussion Online tools