An important aspect of the stock price process, which has often been ignored in the financial literature, is that prices on organized exchanges are restricted to lie on 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 pure 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 Brownian motion. We study the changes in pricing caused by discretization. Upper and lower bounds on option prices are developed. We show that it is possible to hedge options if one restricts to jumps of size one, that is, if one models the stock price process as a birth and death process. One needs the stock and another market traded derivative to hedge an option in this setting. We obtain parameter estimates using Generalized Method of Moments. We use filtering equations for inference in the stochastic intensity setting. We present real data applications to study the performance of our modeling and estimation techniques.