Monte Carlo methods in derivative modelling

This thesis addresses issues in discretization and variance reduction methods for Monte Carlo simulation. For the discretization methods, we investigate the convergence properties of various Itˆo-Taylor schemes and the strong Taylor expansion (Siopacha and Teichmann [77]) for the LIBOR market model. We also provide an improvement on the strong Taylor expansion method which produces lower pricing bias. For the variance reduction methods, we have four contributions. Firstly, we formulate a general stochastic volatility model nesting many existing models in the literature. Secondly, we construct a correlation control variate for this model. Thirdly, we apply the model as well as the new control variate to pricing average rate and barrier options. Numerical results demonstrate the improvement over using old control variates alone. Last but not least, with the help of our model and control variate, we explore the variations in barrier option pricing consistent with the implied volatility surface