Black Option Pricing Model

A variation of the Black-Scholes option pricing model, the Black Option Pricing model is occasionally referred to as the Black-76 model. Its function is mainly to price swaptions, interest rate caps, interest rate floors, and bond options. Fischer Black first introduced the Black Option Pricing model in 1976. The Black Option Pricing model is a LIBRO market model, which is a log-normal forward model.

Formula for Black Option Pricing Model

The difference between the Black Option Pricing model and the Black-Scholes formula is that the forward price F replaces the spot price for stock options valuations in the Black formula.

Black Option Pricing Model Derivation

The pricing formulas in the Black Option Pricing model are almost identical to the Black-Scholes model. The biggest difference is the assumption that the option is log-normally distributed at the forward price upon maturity rather than the assumption that the log-normal process is followed by the spot price as is the case with the Black-Scholes model. Other than this one change, the derivation is exactly the same as the Black-Scholes model and the ultimate formula used is the same other than the forward replaces the spot price. In this case, the undiscounted expected future value is represented by the forward price.

Black Option Pricing Model Assumptions

The only difference in the Black Option Pricing Model and Black-Scholes model is the option is log-normally distributed assumption. With the Black-Scholes model the assumption is that the spot price follows the log-normal price.

Black Option Pricing Challenges

There are challenges when it comes to Black Option Pricing due to the fact that commodity prices evolve and change regularly. Being aware of the changes in price based on historical information is important in order to effectively price options. For example, during the winter months natural gas is generally higher than during the summer. Another example would be that before harvest spot prices will typically increase for agricultural products and then fall after the harvest is complete. These are not random rises and falls in price, but rather anticipated rises and falls. As a result, the Black-Scholes models will not apply.

Application of Black Option Pricing Model

Fischer Black addressed this particular problem in a paper published in 1976. He provided a solution to the problem by modeling forward prices rather than spot prices. Forward prices, futures, and physical commodities are frequently modeled with the Black Option Pricing Model. This particular model also is used to price interest rate floors and interest rate caps.