Hey emme1.

There are three main approaches to prove the Black Scholes formula. They include:

1) The PDE Approach (you transform the problem to a heat equation PDE and solve)

2) The Stochastic DE approach (You use Ito's result and solve)

3) The Probability Approach (You find the random variable that corresponds to a Price at time t and then take the expectation to get the fair price)

All approaches will give the same answer and each requires a different kind of thinking and reasoning.