Approximate Hedging of Options under Jump-Diffusion Processes

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 in the context of Harrison and Pliska (1981) and that there are several risk-neutral measures one can use to price and hedge options (Cont and Tankov, 2004; Miyahara, 2012). As in Jensen (1999) and Leon et al. (2002), 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 as utilized in Cont et al. (2007) and He et al. (2006). Under suitable conditions, we get a unique risk-neutral measure, which is used to determine the option price partial differential equation, along with the asset positions that will replicate the option payoff. This procedure is then implemented on a particular set of stock and option prices, and its performance is compared with the minimal variance and delta hedging strategies.