Euler and Milstein Discretization

"Monte Carlo simulation" in the context of option pricing refers to a set of techniques to generate underlying values–typically stock prices or interest rates–over time. Typically the dynamics of these stock prices and interest rates are assumed to be driven by a continuous-time stochastic process. Simulation, however, is done at discrete time steps. Hence, the …rst step in any simulation scheme is to …nd a way to "discretize" a continuous-time process into a discrete time process. In this Note we present two discretization schemes, Euler and Milstein discretization, and illustrate both with the Black-Scholes and the Heston models. We assume that the stock price St is driven by the stochastic di¤erential equation (SDE) dSt = (St; t) dt+ (St; t) dWt (1)