The price of the underlying asset, , is assumed to follow a geometric
Brownian Motion process, conveniently written
in either of the shorthand forms

or

where and are constants, and is Brownian motion.

Using Ito's lemma, the assumption of no arbitrage, and the ability to trade
continuously, Black and Scholes showed that the price of any contingent
claim written on the underlying must solve the following partial
differential equation:

For any particular contingent claim, the terms of the claim will give a number of boundary conditions that determines the form of the pricing formula.

We will start by discussing the original example solved by Black, Scholes, Merton: European call and put options.