Simulation III: Numerical Approximation of SDE Models

We cover the numerical approximation of SDE models for which exact sampling is impossible. After introducing the Euler-Maruyama class of numerical schemes for Itô-type SDEs and applying them to SDE models of assets and interest rates, we examine stochastic notions of stability and convergence, providing both theoretical and numerical demonstrations of weak and strong convergence of the explicit Euler-Maruyama scheme applied to a scalar diffusion-only SDE motivated by the SDE representation of a forward price process. We then introduce the Milstein scheme, examining its stability and convergence properties. Finally we investigate Multilevel Monte Carlo sampling, which provides a means of controlling the additional error introduced by the use of a numerical method for sampling from the distribution of an SDE model.