Jump diffusion processes: Discrete -time option replication and pricing European call options in the presence of transaction costs

Whereas the discrete-time hedging strategies and hedging error problems have been examined by several researchers under the Black-Scholes assumption of geometric Brownian motion, nothing has been done to the problems when the stocks have discontinuous returns. This dissertation, examines two major issues in options pricing and hedging: the problem of discrete-time hedging and hedging error on the one hand, and pricing European call options in the presence of transaction costs on the other, when the underlying securities follow a jump-diffusion process. Under the assumptions of the continuous time models presented by Bardhan and Chao (1993), we develop discrete-time hedging strategies using a fixed revision interval and constant parameters for European call option, and analyzed the associated hedging errors associated. We proved that the total hedging error converges to zero in probability as the time between rebalancing points goes to zero. For small revision time intervals, we derive an approximate conditional distribution for the individual one period hedging errors. We derive an exact closed form expression for the total expected hedging error, conditional on the information at time when the call option is written. The results obtained can be used in risk management to monitor the performance of the strategies. We also developed an equation for European call options when the underlying asset follows the jump-diffusion process in the presence of non-zero transaction costs: extending an equation of Leland's (1985).