Black76

Black-76 model, also known as the Black model, is a mathematical framework used to price European options on futures and forwards. It was introduced by Fischer Black in 1976 in his work on pricing options and futures contracts. In this model the price of an option on a futures contract is expressed in terms of the current forward price F0 of the futures, the strike price K, the risk-free interest rate r, the time to maturity T, and the volatility sigma of the futures price.

For a European call option on a futures price, the present value C is:

C = e^{-rT} [F0 N(d1) - K N(d2)],

where

d1 = [ln(F0/K) + (sigma^2) T / 2] / (sigma sqrt(T)),

d2 = d1 - sigma sqrt(T),

and N() denotes the cumulative distribution function of the standard normal distribution.

Similarly, the put price is:

P = e^{-rT} [K N(-d2) - F0 N(-d1)].

Usage and scope

The Black-76 model is widely used for pricing options on commodity futures (such as oil and other energy products), as well as interest-rate futures and other forwards where the underlying is a futures price. It assumes the futures price follows a lognormal process under the risk-neutral measure with constant volatility, and that there are no dividends or storage costs explicitly included in the basic form. Pricing is done in present value terms, reflecting the risk-free discount factor.

Limitations

The model relies on constant volatility and a lognormal price process, which can be violated in real markets. It does not handle American-style exercise, stochastic volatility, or jumps. In practice, alternative models (such as stochastic volatility or jump-diffusion variants) are used when these features are significant.