Quasi-monte Carlo Methods In Computational Finance:smoothing And Importance Sampling

Quasi-Monte Carlo(QMC)method,usually used to estimate high-dimensional in-tegrals,is a method with higher convergence rate compared with Monte Carlo(MC)method,and therefore has wide applications in computational finance.However,the computational efficiency of QMC method is deeply affected by the smoothness and di-mensions of the integrands.In this thesis,by using smoothing,importance sampling(IS)and dimension reduction methods,the superiority of QMC methods is demonstrated and the computational efficiency is greatly increased.The calculation of option Greeks is important in financial risk management.However,the traditional pathwise method is not applicable to options with discontinuous payoffs.In this thesis,using the idea of conditional QMC method to smoothing the payoff functions,we generalize the traditional pathwise method to calculate the first-and high-order Greeks.If the payoffs satisfy the variable separation conditions,by taking a conditional expectation,the discontinuous integrand is smoothed.More importantly,the interchange of expectation and differentiation is proved to be possible for the new integrands,and therefore unbiased estimators of Greeks are obtained.We show that the assumptions in this thesis are satisfied and the calculations of the conditional expectations and the derivatives with respect to the parameter of interest analytically are feasible for many common options.The new estimators for Greeks usually have good smoothness,so using QMC method to estimate the expectations improves the efficiency significantly.Actually,we prove that for Asian and binary Asian option Greeks under Black-Scholes model,our methods have a convergence rate of O(N^-1+?)with arbitrarily small?>0,an obvious advantage over MC methods with a convergence rate of O(N^-1/2).Moreover,we also study the relationships of our method with several others in literature,and show that our method is an extension and improved version of these methods.Numerical experiments are performed to demonstrate the high efficiency of the proposed method.This thesis also focuses on the effective combination of QMC methods and impor-tance sampling(IS).IS is one of the most important variance reduction techniques in MC methods,but its promises and limitations in QMC methods is yet to be explored.In this thesis,two kinds of IS are studied,namely,the optimal drift IS(ODIS)and the Laplace IS(Lap IS).Traditionally,the Lap IS is obtained by mimic the behavior of the optimal IS density based on the second order Taylor approximation,with ODIS as its special case.This thesis presents another new viewpoint of Lap IS,namely,it can also be obtained by an optimization procedure based on the Laplace approximation.Moreover,we also provide a variance reduction comparison of the ODIS and Lap IS in MC schemes.Both QMC-based ODIS algorithm and QMC-based Lap IS algorithm are developed.At the first step,the integrands are smoothed if possible,and then the IS and dimension reduction methods are applied,where several orthogonal matrices are required to choose to reduce the effective dimension.We prove that as long as the last orthogonal matrix,dimension reduction methods are designed to find,is chosen elaborately,the choices of others can be arbitrary.This greatly simplify the complexity of applications of IS methods in QMC schemes.Numerical experiments in computing option Greeks illustrate the superiority of our proposed QMC-based IS algorithms.