A Modern View on Merton's Jump-Diffusion Model

Merton has provided a formula for the price of a European call option on a single stock where the stock price process contains a continuous Poisson jump component, in addition to a continuous log-normally distributed component. In Merton's analysis, the jump-risk is not priced. Thus the distribution of the jump-arrivals and the jump-sizes do not change under the change of measure. We go onto introduce a Radon-Nikodym derivative process that induces the change of measure from the market measure to an equivalent martingale measure. The choice of parameters in the Radon-Nikodym derivative allows us to price the option under different financial-economic scenarios. We introduce a hedging argument that eliminates the jump-risk in some sort of averaged sense, and derive an integro-partial differential equation of the option price that is related to the one obtained by Merton.