High-Dimensional Financial Computation:the Methods Of Dimension Reduction And Smoothing

The pricing and sensitivity analysis of financial derivatives are the essential problems in computational finance.The prices or sensitivities can be expressed as the expectations under the risk-neutral measure.For path-dependent or multi-assets derivatives,their pricing and hedging involve high-dimensional integrals.Monte Carlo(MC)methods can overcome "the curse of dimensionality".Due to its slow convergence rate,to achieve a desirable accuracy,the required amount of simulation is massive.Quasi-Monte Carlo(QMC)methods can provide a faster convergence rate.However,the efficiency of QMC is critically impacted by the smoothness and dimension of financial problems.In this thesis,to improve QMC,a series of methods are proposed to handle the problems of high-dimesionality and discontinuities in computational finance.The efficiency of QMC is greatly enhanced by using these methods.Since the low discrepancy points used for QMC simulation have perfect projections at the first few dimensions,reducing the effective dimension of the integrands can improve the efficiency of QMC.Based on the theory of effective dimension and ANOVA decomposition of functions,we propose a new adaptive dimension reduction method,which maximizes the truncation variance of the target function step by step.This minimizes the effective dimension of the target functions.By some efficient linear approximations,the procedure is modified to be a manageable dimension reduction technique based on the principle component analysis of the gradients(GPCA),which is able to extract fully the structure feature of the target function.Rather than aiming at decomposing the covariance matrix of the Brownian motions as in the traditional path generation constructions,the new method implements PC A on the gradients of the functions at some sample points.An orthogonal transformation is found by GPCA to reduce the effective dimensions.In pricing exotic options and mortgage-backed securities,the GPCA method prominently improves QMC and shows a remarkable superiority over other path generation constructions or dimension reduction methods.Besides high-dimensionality,the problem of discontinuities,which also occurs com-monly in computational finance,may cause an adverse impact on the performance of QMC,and severely restricts the applications of QMC in computational finance.Inspired by conditional MC,we develop a smoothing method based on conditional expectation,which known as conditional QMC.This method can not only smooth the discontinuous functions but also reduce the variance of the functions.For a class of discontinuous functions commonly arising in the pricing or hedging of Asian and barrier options,we show how the discontinuities are removed completely and how the required conditional expectations can be calculated analytically.Furthermore,a two-step procedure is proposed to handle the discontinuities and to reduce the effective dimensions successively.The first step is to smooth the discontinuous functions by using conditional QMC methods.The second step is to reduce the effective dimensions of the resulting functions by GPCA.Numerical experiments show that the two-step procedure enormously enhances the efficiency of QMC in the pricing and hedging of options with discontinuous functions.Finally,we extend this two-step method to solve problems from the risk management of portfolios.We study the straddle option portfolios and a portfolio model based on t-copula.Since the estimations of VaR(Value-at-Risk)depend on the tail probability of the portfolio loss,the two-step method is used to improve the QMC simulation of rare events.By choosing a proper construction approach to generate correlated underlying assets in the portfolio models,conditional QMC is able to smooth the indicator functions.In the stochastic optimization procedures to estimate VaR,the application of two-step method results in a significant efficient improvement of QMC as well.