Black Model

In 1976, Fisher Black developed a modification to the Black-Scholes model designed to price options on futures more precisely. The model assumes that futures can be treated the same way as securities, providing a continuous dividend yield equal to the risk-free interest rate.

The model provides a good correction to the original model concerning options on futures. However, it still carries the restrictions of the Black-Scholes evaluation.

Notation

       Theoretical value of a call

       Theoretical value of a put

       Underlying price

       Strike price

        Interest rate

        Time to expiration in years

       Volatility

  Cumulative normal density function

The theoretical values for calls and puts are:

Where:

Note: Although similar, this definition of  is different from the one used in the Black-Scholes model.

An alternative form for  is: