Theoretical value of a callIn 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:
