Multilevel Monte Carlo for jump processes

This thesis consists of two parts. The first part (Chapters 2-4) considers multilevel Monte Carlo for option pricing in finite activity jump-diffusion models. We use a jump-adapted Milstein discretisation for constant rate cases and with the thinning method for bounded state-dependent rate cases. Multilevel Monte Carlo estimators are constructed for Asian, lookback, barrier and digital options. The computational efficiency is numerically demonstrated and analytically justified. The second part (Chapter 5) deals with option pricing problems in exponential Lévy models where the increments of the underlying process can be directly simulated. We discuss several examples: Variance Gamma, Normal Inverse Gaussian and alpha-stable processes and present numerical experiments of multilevel Monte Carlo for Asian, lookback, barrier options, where the running maximum of the Lévy process involved in lookback and barrier payoffs is approximated using discretely monitored maximum. To analytically verify the computational complexity of multilevel method, we also prove some upper bounds on Lp convergence rate of discretely monitored error for a broad class of Lévy processes.This thesis is not currently available in ORA