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 caps / 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 ) is

The corresponding put price is

where

and N(.) is the cumulative normal distribution function.