What Is Black's Model?
Black's Model, sometimes called Black-76, is an adjustment of his earlier Black-Scholes options pricing model. Unlike the earlier model, the revised model is useful for valuing options on futures. Black's Model is used in the application of capped variable rate loans, and is also applied to price a variety of derivatives.
These include 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).
Black's Model Explained
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." 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 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.