Convergence of Discrete-time European Option Problems with Hereditary Price Structures to Continuous-time Versions It has long been recognized that the solution to the European option pricing problem depends on how the bank account and stock account are modeled. Recently, the standard model for these has been under question. This dissertation develops a new discrete-time model for these accounts which overcomes many of the objections to the standard model. It is then shown that the discrete-time model converges to a continuous-time model. The European option pricing problem is then solved for both cases. Then, it is shown in this dissertation that the solutions to the discrete-time problems converge to that of the continuous-time problem. Finally, two applications are shown.