A large number of alternative formulations to the Black Scholes analysis has been proposed. Very few of them have seen any widespread use, but we will look at some of these alternatives.
Merton has proposed a model where in addition to a Brownian Motion term, the price process of the underlying is allowed to have jumps. The risk of these jumps is assumed to not be priced.
In the following we look at an implementation of a special case of Merton's model, described in (Hull, 1993, pg
454), where the size of the jump has a normal distribution. and are
parameters of the jump distribution. The price of an European call option is