On the Derivation of the Black-Scholes Formula

Methods of proving the Black-Scholes formula for the price of an European call option fall into two categories: the bond replication method (the original one by Black and Scholes), and the call replication method (originated by Merton). These two methods are not equivalent. While the call replication argument is simple and requires only continuity of the call price C, the bond replication method puts more restrictions on C in order for the argument to work. Moreover, we show that the typical bond replication method fails if the call option delta is equal to one. That implies that at each point either C satisfies the Black-Scholes PDE, or it satisfies the linear PDE given by delta equal to one. We then show that these two PDEs cannot coexist if C is assumed continuously differentiable, which proves that the Black-Scholes PDE holds everywhere