New Progress Of QMC In High Dimensional And Non-linear Financial Problems

Many financial problems cannot be solved analytically,and one has to resort to Monte Carlo(MC)method or quasi-Monte Carlo(QMC)method.The QMC method is an important numerical tool due to its potential superior convergence rate over the MC method.However,the effectiveness of the QMC method crucially depends on the discontinuities and the dimension of the problem.There are a lot of literatures handling the high-dimensionality and discontinuities in the nearly linear financial models,but few are attributed to the nonlinear models.This thesis shows how these two limitations can be overcome in nonlinear financial problems.Firstly,we propose a general dimension reduction method,called the clustering algorithm based QR(CQR)method,to reduce the effective dimension such that the better quality of QMC points in their initial dimensions can be fully exploited.The general dimension reduction method is based on the method of machine learning.The k-means clustering algorithm,a classical algorithm of machine learning,is used to find some representative linear structures inherent in the function,which are used to construct a match function of the complex payoff.The match function serves as an approximation of the payoff but has a simpler form,and it is used to find the required transform.The pricing of mortgage-backed securities is used as a test problem to study the performance of the CQR method.We also calculate the corresponding effective dimension.The results demonstrate that the CQR method handles the high dimensionality in an effective and robust way.We calculate the corresponding effective dimension-related characteristics and find out that the effective dimensions are incredibly reduced by the CQR method.Motivated by the work of Wang and Tan [1],this thesis proposes an auto-realignment method to deal with more general discontinuous functions.Discontinuities are common in the pricing of financial derivatives and have a tremendous impact on the accuracy of QMC method.The clustering algorithm is used to select the most representative normal vectors of the discontinuity surface.By applying this method,the discontinuities of the resulting function are realigned to be friendly for the QMC method.Numerical experiments demonstrate that the auto-realignment method significantly improves the efficiency of the QMC method.This thesis develops a two-step procedure to tackle the challenging problems of both the discontinuities and the high-dimensionality concurrently.At the first step,we adopt an appropriate smoothing technique to remove the discontinuities of the payoff function,improving the smoothness.At the second step,we use the CQR method to reduce the effective dimension of the new target function,which is obtained from the smoothing step and usually has a rather complex form.This new procedure is called the smoothed QMCCQR method.Extensive numerical experiments on option pricing and Greeks estimation demonstrate that the combination of the smoothing method and dimension reduction in QMC achieves substantial variance reduction than dealing with either discontinuities or high dimensionality single sidedly.Note that the proposed CQR,auto-realignment and smoothed QMC-CQR methods are not only effective when the underlying asset prices follow the classical geometric Brownian motion,but they are equally applicable and effective when the asset prices follow more complicated diffusion process such as the L?evy process.