Black 76 Futures & Options Pricing Model

The Black 76 Model developed by Fischer Black of Black-Scholes was first proposed by him in  Journal of The American Finance Association in 1976 publishing known as Taxes and The Pricing of Options. In it, he expounds upon various improvements and adaptations of the original Black Scholes model conferring greater versatility for the pricing of other derivatives, such as interest swaps, floors, caps,  futures, and limited variable rate securities.

where R = the constant risk free rate ,F(t) is the futures price of a particular underlying with that is log-normal with constant volatility σ. The Black 76 Model states the price of  European call option at maturity T corresponding to a future conferring a strike price K and delivery date.

Criticism

Black’s Model makes various assumptions reminiscent of Black-Scholes, such as a log-normal distribution and an expected change of zero for the futures price. A significant distinction among the Black 76 and Black-Scholes excluding the known assumptions of the latter; (i.e. a risk-free interest rate, execution at maturity, no commissions or fees, and volatility as a constant) is that the former models the value of a futures option at maturity using forward prices as opposed to the latter which derives its model from spot prices. Contrary to the Black Scholes implementation of volatility as constant, the Black 76 model assumes volatility is dependent on time; hence, the lower variability of this model.