# Black-Scholes-Merton Formula

A step-by-step derivation.

Written on September 27, 2018

Given that:

where

we have:

Let

the probability desnity function $f_S(S)$ and cumulative distribution function $F_S(S)$ can be written as:

Consider the value of a vanilla European call option:

The first term:

The second term:

Combing the two terms, we have:

where