Quasi-monte Carlo In Asset Pricing And Data Science

Quasi-Monte Carlo(QMC)method is an essential numerical simulation tool which has wide application scope.The usage of QMC method is as simple as the Monte Carlo(MC)method,but QMC has the potential to give higher convergence rate.This advantage results from the low-discrepancy points used in QMC method,which possess better uniformity than the random points used in MC.However,QMC does not exhibit consistent superiority in all applications.The reason is that the efficiency of QMC highly depends on the smoothness and dimensionality of the problem.These motivate us to investigate how to handle these influential factors properly.This thesis aims at schemes for making use of QMC optimally in a variety of QMC applications.The related situations include financial asset pricing and data science.The first problem we are concerned with is pricing barrier options.The biggest challenge tackled by QMC here is the complex discontinuities embedded in their payoff functions.For the purpose of smoothing,we propose a so-called sequential importance sampling(SIS)method.The method removes multiple discontinuity structures sequentially rather than just realigning the discontinuities.We find that the order of path generation influences the variance of the SIS estimator.Inspired by the findings,we give general rules for choosing optimally the first generation step.Specifically,we develop a good path generation method with the smoothed estimator under the Black –Scholes model and models based on subordinated Brownian motion(e.g.,Variance Gamma process).As confirmed by numerical experiments,the SIS method combined with a carefully chosen path generation method can significantly reduce the variance with improved rate of convergence.In addition,we show that the effective dimension is greatly reduced by the combined method,explaining the superiority of the proposed procedure from another perspective.We also carry out a deep research on the efficiency of the multi-level QMC(MLQMC)method.Through substantial numerical experiments,we investigate how the efficiency of MLQMC may be influenced by the construction of low-discrepancy points,the high dimensionality and the discontinuity.We find that the curse of dimensionality limits the efficiency of the standard MLQMC,whereas problem-dependent path generation methods effectively reduce the effective dimension of the estimator on each level and thereby improve the computation efficiency.Discontinuity also affect MLQMC negatively.We modify the SIS method slightly to accommodate the MLQMC framework,and show significant improvement in accuracy.Besides,we recommend Sobol points rather than rank-1 lattice rule to be used in MLQMC,as Sobol points mostly outperform rank-1 lattice rule in MLQMC.The third problem we are interested in is how to apply QMC to data science.We develop a method to sample from high-dimensional empirical datasets using QMC.The method fully exploits good features of the low-discrepancy points,ensuring the drawn samples could well represent the distribution of the original dataset.Numerical results demonstrate that when applied to datasets coming from known distributions,the effectiveness of the proposed QMC sampling method is very similar to the standard QMC applied to those distributions.