Based on the seminal work by Aït-Sahalia (2002), we propose a simple recursive algorithm for approximating the transition density of a stochastic differential equation (SDE). We then introduce a new method for simulating SDEs. Unlike Euler's and other schemes, our discretization step is fixed and does not need to approach zero. We apply our methods through simulations to some wellknown financial models, specifically the Vasicek and CIR models.