This dissertation consists of two parts. In the first part we present a quasi-Monte Carlo implementation of the de-biased Monte Carlo estimator in the context of stochastic differential equations. We combine the quasi-Monte Carlo implementation with path generation techniques, and compare the accuracy of the resulting methods with the original de-biased Monte Carlo estimator when they are applied to option pricing problems under the geometric Brownian motion and Cox-Ingersoll-Ross models. In the second part we consider the application of the multilevel Monte Carlo methods to the LIBOR market model framework. The LIBOR market model is a popular interest rate model used for pricing interest rate derivatives like caplets, caps, and swaptions. Recently, long-dated interest rate derivatives have been popular in the interest rate derivative market, and the practitioners typically price them using the standard Monte Carlo method. To achieve real time pricing, practitioners often use very few Monte Carlo samples, typically in the low hundreds. We use multilevel Monte Carlo, low-discrepancy sequences, and path generation techniques to develop fast and accurate algorithms that achieve significant error reduction for small sample sizes, for pricing long-dated interest rate derivatives in the LIBOR market model framework.