We implement 15 simulation schemes for the Cox–Ingersoll–Ross (CIR) square root process and 10 schemes for Heston’s stochastic volatility model to generate draws that we investigate for the quality of their mean, variance, skewness and kurtosis estimates. Simulations of continuous-time processes require discretization, and we therefore investigate the quality of currently known simulation techniques from both an accuracy perspective and a timing perspective. We show that no method fits all situations, and we advise the use of different simulation techniques. We also provide an extension to Andersen’s quadratic exponential method to generate returns with skewness or kurtosis closer to their theoretical values in certain settings. A simulation experiment focusing on the estimation of return skewness and kurtosis demonstrates the relevance of using the correct simulation technique and reveals the limitations on the convergence of those estimates to the true moments when volatility is generated by a CIR process for which the Feller condition is not satisfied and the sample is not of relatively large size.