APPROXIMATE HEDGING OF OPTIONS UNDER JUMP-DIFFUSION PROCESSES

Abstract

We consider the problem of hedging a European-type option in a market where asset prices have jump-diffusion dynamics. It is known that markets with jumps are incomplete and that there are several risk-neutral measures one can use to price and hedge options. In order to address these issues, we approximate such a market by discretizing the jumps in an averaged sense, and complete it by including traded options in the model and hedge portfolio. Under suitable conditions, we get a unique risk-neutral measure, which is used to determine the option price integro-partial differential equation, along with the asset positions that will replicate the option payoff. Upon implementation on a particular set of stock and option prices, our approximate complete market hedge yields easily computable asset positions that equal those of the minimal variance hedge, while at the same time offers protection against upward jumps and higher profit compared to delta hedging.