Rate of convergence for discrete approximation of option prices

In this article, we study the rate of convergence of prices when a model is approximated by some simplified model. We also provide a method how explicit error formula for more general options can be obtained if such formula is available for digital option prices. We illustrate our results by considering convergence of binomial prices to Black-Scholes prices. We also consider smooth convergence in which the approximation does not oscillate for general class of payoff functions.