Abstract: An important aspect of the stock price process, which has of ten been ignored in the nancial literature, is that prices on organized exchanges are restricted to a grid. We consider continuous-time models for the stock price process with random waiting times of jumps and discrete jump size. We consider a class of jump processes that are close “to the Black-Scholes model in the sense that as the jump size goes to zero, the jump model converges to geometric Brownianmotion. We study the changes in pricing and hedging caused by discretization. The convergence, estimation, discrete time approximation, and uniform integrability conditions for this model are studied. Upper and lower bounds on option prices are developed. We study the performance of the model with real data. In general, jumpmodels do not admit self-nancing strategies for derivatives. Birth -death processes have the virtue that they allow perfect hedging of derivative securities.