Blacksmodel

Black's Model Explained: Options Pricing for Futures Contracts

Key Takeaways

Black's Model, or Black 76, is used to price options on futures contracts, expanding upon the Black-Scholes options pricing model.
Developed by Fischer Black in 1976, this model addresses the pricing of options in futures markets.
Black's Model assumes log-normal distribution of future prices, zero trading costs, and uses forward prices at maturity.
The model applies to interest rate derivatives and capped loans, and is useful for financial institutions like banks and hedge funds.

What Is Black's Model?

Black's Model, also known as Black 76, is a financial model used to price options associated with futures contracts. It extends the concepts of the Black-Scholes model, adapting them for the valuation of options where the underlying assets are futures rather than stocks. In using Black's Model, traders and analysts can evaluate the potential outcomes of trading options on futures, helping them make informed decisions in derivatives markets.

How Black's Model Works

In 1976, American economist Fischer Black, one of the co-developers along with Myron Scholes and Robert Merton of the Black-Scholes model for options pricing (which was introduced in 1973), demonstrated how the Black-Scholes model could be modified in order to value European call or put options on futures contracts. He laid out his theory in an academic paper titled, “The Pricing of Commodity Contracts."1 For this reason, the Black model is also referred to as the Black-76 model.

Black’s goals in writing the paper were to improve the current understanding of commodity options and their pricing and introduce a model that could be used to model pricing. Existing models at that time, including Black-Scholes and Merton models, had been unable to address this problem. In his 1976 model, Black describes the futures price of a commodity as, “the price at which we can agree to buy or sell it at a given time in the future without putting up any money now.” He also postulated the total long interest in any commodity contract must equal the total short interest.

Black's model also can apply to other financial instruments typically used by financial institutions such as global banks, mutual funds, and hedge funds: namely interest rate derivatives, caps, and floors (which are designed to offer protection from big swings in interest rates), as well as bond options and swaptions (financial instruments that combine an interest rate swap and an option, they can be used to hedge against interest rate risk and to preserve financing flexibility).

Assumptions Underlying the Black 76 Model

Black’s 76 model makes several assumptions, including that future prices, are log-normally distributed and that the expected change in futures price is zero. One of the key differences between his 1976 model and the Black-Scholes model (which assumes a known risk-free interest rate, options that can only be exercised at maturity, no commissions and that volatility is held constant), is that his revised model uses forward prices to model the value of a futures option at maturity versus the spot prices Black-Scholes used. It also assumes that volatility is dependent on time, rather than being constant.

Journal of Financial Economics. "The Pricing of Commodity Contracts." Accessed Aug. 3, 2021.

Trading

Options and Derivatives

Advanced Concepts