Black Formula (Generalized Version)

The generalized version of Black’s formula is given by

Black Formula (Generalized Version)

This formula could be used to derive the European style option formulas for a derivative follows a lognormal distribution having the payoff


Derivation of Black-Scholes formula using Black’s formula
Now, we will derive Black-Scholes option pricing formula using this Black’s formula. As we know that


Now, we need to put these two values in Black’s formula (generalized version)

Black Formula (Generalized Version)

First take

Now consider

For simplicity, we set

So, now


By putting all these values in Black’s formula, we obtain the formula for European-style call and put option formulas as follows