Dr. Zhijian He is a Professor at School of Mathematics of South China University of Technology (SCUT). Before joining SCUT, he obtained a Ph.D. in Statistics from Department of Mathematical Science of Tsinghua University, advised by Prof. Xiaoqun Wang. His research interests are quasi-Monte Carlo methods and their applications in quantitative finance and statistics. He was a silver prize recipient of the New World Mathematics Awards (NWMA). He has published in top journals in the fields of statistics and computational mathematics, such as Journal of the Royal Statistical Society: Series B, SIAM Journal on Numerical Analysis, SIAM Journal on Scientific Computing, Mathematics of Computation. Part of his research is supported by National Science Foundation of China (NSFC).
According to https://scimeter.org/clouds/, my research interests are somewhere in that cloud:
◆ Ph.D., Statistics, Tsinghua University, 2015
◆ B.S., Mathematics and Applied Mathematics, South China University of Technology, 2010
◆ Professor, School of Mathematics, South China University of Technology, 2021/08 - current
◆ Associate Professor, School of Mathematics, South China University of Technology, 2018/01 - 2021/08
◆ Research Associate, Lingnan (University) College, Sun Yat-Sen University, 2016/01-2017/11
◆ Lecturer, School of Economics and Commerce, South China University of Technology, 2015/09-12
◆ Visiting Student Researcher, Department of Statistics, Stanford University, 2014/01-07
◆ New World Mathematics Awards (Doctor Thesis), Silver Prize, 2016 [Link]
◆Best Paper Award (with G. Liu and Y. Liu), 14th International Symposium on Financial System Engineering and Risk Management, 2016
◆ Probability and Statistics, Spring 2018
◆ Mathematical Statistics, Fall 2018, Spring 2019
◆ Bayesian Data Analysis, Fall 2018
◆ Quasi-Monte Carlo Simulation in VaR and CVaR Computation, National Science Foundation of China, No. 71601189, 2017 - 2019, PI
Z. He and X. Wang. Convergence analysis of quasi-Monte Carlo sampling for quantile and expected shortfall. Mathematics of Computation, 2020. Accepted. [abstract] [arXiv]
In financial risk measurement, value-at-risk (VaR) and conditional VaR (CVaR)are two important risk measures. They are often estimated by Monte Carlo method combined with importance sampling (IS). Randomized Quasi-Monte Carlo (RQMC)method is an alternative technique, whose convergence rate is asymptotically better than the Monte Carlo method. This paper investigates the combination ofIS and RQMC method. The main contribution is two-fold. First, we prove the consistency of the combined method for both VaR and CVaR estimations. Second,we establish error bounds for the CVaR estimate. Particularly, we show that under some mild conditions, the root mean square error of the CVaR estimate isO(N−1/2−1/(4d−2)+ϵ) for arbitrarily small ϵ>0, where dis the dimension of the problem. As a special case, these results also hold forplain RQMC estimates without using IS.
Z. He and X. Wang. An integrated quasi-Monte Carlo method for handling high dimensional problems with discontinuities in financial engineering. Computational Economics, 2020. Accepted. [abstract] [arXiv]
Quasi-Monte Carlo (QMC) method is playing an increasing role in the problems of pricing and hedging of complex financial derivatives. These problems are usually of high dimensionality and discontinuities. The two factors may significantly deteriorate the performance of the QMC method. This paper develops a method that overcomes the challenges of the high dimensionality and discontinuities concurrently. For this purpose, a smoothing method is proposed to remove the discontinuities for some typical functions arising from financial engineering. To make the smoothing method applicable for more general functions, a new path generation method is designed for simulating the paths of the underlying assets such that the resulting function has the required form. The new path generation method has an additional power to reduce the effective dimension of the target function. Our proposed method caters for a large variety of model specifications, including the Black-Scholes, exponential generalized hyperbolic L\'evy, and Heston models. Numerical experiments dealing with these models show that in the QMC setting the proposed smoothing method in combination with the new path generation method can lead to a dramatic variance reduction for pricing exotic options with discontinuous payoffs and for calculating options' Greeks. The investigation on the effective dimension and the related characteristics explains the significant enhancement of the combined procedure.
X. Fei, M. Giles, Z. He. QMC Sampling from Empirical Datasets. Proceedings of the MCQMC 2018 conference, 2019 (accepted). [abstract] [link]
This paper presents a simple idea for the use of quasi-Monte Carlo samplingwith empirical datasets, such as those generated by MCMC methods. It alsopresents and analyses a related idea of taking advantage of the Hilbert space-filling curve. Theoretical and numerical analyses are provided for both. We find that when applying the proposed QMC sampling methods to datasets coming from a known distribution, they give similar performance as the standard QMC method directly sampling from this known distribution.
Z. He and L. Zhu. Asymptotic Normality of Extensible Grid Sampling. Statistics and Computing, 29 (1), 53-65, 2019. [abstract] [link]
Recently, He and Owen (2016) proposed the use of Hilbert's space filling curve (HSFC) in numerical integration as a way of reducing the dimension from d>1 to d=1. This paper studies the asymptotic normality of the HSFC-based estimate when using scrambled van der Corput sequence as input. We show that the estimate has an asymptotic normal distribution for functions in C1([0,1]d), excluding the trivial case of constant functions. The asymptotic normality also holds for discontinuous functions under mild conditions. It was previously known only that scrambled (0,m,d)-net quadratures enjoy the asymptotic normality for smooth enough functions, whose mixed partial gradients satisfy a H\'older condition. As a by-product, we find lower bounds for the variance of the HSFC-based estimate. Particularly, for nontrivial functions in C1([0,1]d), the lower bound is of order n−1−2/d, which matches the rate of the upper bound established in He and Owen (2016).
F. Xie, Z. He, and X. Wang. An Importance Sampling-Based Smoothing Approach for Quasi-Monte Carlo Simulation of Barrier Options. European Journal of Operational Research, 274 (2), 759-772, 2019. [abstract] [link]
Handling discontinuities in financial engineering is a challenging task when using quasi-Monte Carlo (QMC) method.This paper develops a smoothing approach based on importance sampling (IS) to remove multiple discontinuity structures sequentially arising from pricing barrier options and other simulation problems with similar structures. The IS-based smoothing method yields an unbiased estimate with reduced variance. We find that under the Black-Scholes model and the Variance Gamma model, the order in which the asset prices are simulated may have a significant impact on the efficiency of the IS method. It is optimal to simulate the asset price at the expiration time firstly in both models. Combining with a proper simulation order, the IS method transforms the payoff to make it not only smoother but also having smaller effective dimension. Numerical experiments show that the combined procedure performs consistently much better than commonly used methods for pricing barrier options. Specifically, the IS method combined with the Brownian bridge construction performs best under the Black-Scholes model, while the reverse order construction performs best for the Variance Gamma model. Furthermore, we show that the effective dimension is greatly reduced by applying the proposed procedure, which explains the superiority of the method from another perspective.
Z. He. On the Error Rate of Conditional Quasi-Monte Carlo for Discontinuous Functions. SIAM Journal on Numerical Analysis, 57(2), 854-874, 2019. [abstract] [link]
This paper studies the rate of convergence for conditional quasi-Monte Carlo (QMC), which is a counterpart of conditional Monte Carlo. We focus on discontinuous integrands defined on the whole of Rd, which can be unbounded. Under suitable conditions, we show that conditional QMC not only has the smoothing effect (up to infinitely times differentiable), but also can bring orders of magnitude reduction in integration error compared to plain QMC. Particularly, for some typical problems in options pricing and Greeks estimation, conditional randomized QMC that uses n samples yields a mean error of O(n−1+ϵ) for arbitrarily small ϵ>0. As a by-product, we find that this rate also applies to randomized QMC integration with all terms of the ANOVA decomposition of the discontinuous integrand, except the one of highest order.
Z. He. Quasi-Monte Carlo for Discontinuous Integrands with Singularities along the Boundary of the Unit Cube. Mathematics of Computation, 87 (314), 2857-2870, 2018. [abstract] [link]
This paper studies randomized quasi-Monte Carlo (QMC) sampling for discontinuous integrands having singularities along the boundary of the unit cube [0,1]d. Both discontinuities and singularities are extremely common in the pricing and hedging of financial derivatives and have a tremendous impact on the accuracy of QMC. It was previously known that the root mean square error of randomized QMC is only o(n?1/2) for discontinuous functions with singularities. We find that under some mild conditions, randomized QMC yields an expected error of O(n?1/2?1/(4d?2)+ϵ) for arbitrarily small ϵ>0. Moreover, one can get a better rate if the boundary of discontinuities is parallel to some coordinate axes. As a by-product, we find that the expected error rate attains O(n?1+ϵ) if the discontinuities are QMC-friendly, in the sense that all the discontinuity boundaries are parallel to coordinate axes. The results can be used to assess the QMC accuracy for some typical problems from financial engineering.
C. Weng, X. Wang, and Z. He. Efficient Computation of Option Prices and Greeks by Quasi-Monte Carlo Method with Smoothing and Dimension reduction, SIAM Journal on Scientific Computing, 39 (2), B298-B322, 2017. [abstract] [link]
Discontinuities and high dimensionality are common in the problems of pricing and hedging of derivative securities. Both factors have a tremendous impact on the accuracy of the quasi--Monte Carlo (QMC) method. An ideal approach to improve the QMC method is to transform the functions to make them smoother and having smaller effective dimension. This paper develops a two-step procedure to tackle the challenging problems of both the discontinuity and the high dimensionality concurrently. In the first step, we adopt the smoothing method to remove the discontinuities of the payoff function, improving the smoothness. In the second step, we propose a general dimension reduction method (called the CQR method) to reduce the effective dimension such that the better quality of QMC points in their initial dimensions can be fully exploited. The CQR method is based on the combination of the k-means clustering algorithm and the QR decomposition. 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 matching function of the smoothed function. The matching function serves as an approximation of the smoothed function but has a simpler form, and it is used to find the required transformation. Extensive numerical experiments on option pricing and Greeks estimation demonstrate that the combination of the smoothing method and dimension reduction in QMC achieves substantially larger variance reduction even in high dimension than dealing with either discontinuities or high dimensionality single sidedly.
Z. He and A. B. Owen. Extensible Grids: Uniform Sampling on a Space-Filling Curve, Journal of the Royal Statistical Society: Series B, 78 (4), 917-931, 2016. [] [link] [C++ Code]
C. Weng, X. Wang, and Z. He. An Auto-Realignment Method in Quasi-Monte Carlo for Pricing Financial Derivatives with Jump Structures, European Journal of Operational Research, 254 (1), 304-311, 2016. [abstract] [link]
Discontinuities are common in the pricing of financial derivatives and have a tremendous impact on the accuracy of quasi-Monte Carlo (QMC) method. While if the discontinuities are parallel to the axes, good efficiency of the QMC method can still be expected. By realigning the discontinuities to be axes-parallel, [Wang & Tan, 2013] succeeded in recovering the high efficiency of the QMC method for a special class of functions. Motivated by this work, we propose an auto-realignment method to deal with more general discontinuous functions. The k-means clustering algorithm, a classical algorithm of machine learning, is used to select the most representative normal vectors of the discontinuity surface. By applying this new method, the discontinuities of the resulting function are realigned to be friendly for the QMC method. Numerical experiments demonstrate that the proposed method significantly improves the performance of the QMC method.
C. Schretter, Z. He, M. Gerber, N. Chopin, and H. Niederreiter. Van der Corput and Golden Ratio Sequences Along the Hilbert Space-Filling Curve,Proceedings of the MCQMC 2014 conference, R. Cools and D. Nuyens (Eds.), 531-544, 2016. [abstract] [link]
This work investigates the star discrepancies and squared integration errors of two quasi-random points constructions using a generator one-dimensional sequence and the Hilbert space-filling curve. This recursive fractal is proven to maximize locality and passes uniquely through all points of the d-dimensional space. The van der Corput and the golden ratio generator sequences are compared for randomized integro-approximations of both Lipschitz continuous and piecewise constant functions. We found that the star discrepancy of the construction using the van der Corput sequence reaches the theoretical optimal rate when the number of samples is a power of two while using the golden ratio sequence performs optimally for Fibonacci numbers. Since the Fibonacci sequence increases at a slower rate than the exponential in base 2, the golden ratio sequence is preferable when the budget of samples is not known beforehand. Numerical experiments confirm this observation.
Z. He and X. Wang. On the Convergence Rate of Randomized Quasi-Monte Carlo for Discontinuous Functions, SIAM Journal on Numerical Analysis, 53 (5), 2488-2503, 2015. [abstract] [link]
This paper studies the convergence rate of randomized quasi--Monte Carlo (RQMC) for discontinuous functions, which are often of infinite variation in the sense of Hardy and Krause. It was previously known that the root mean square error (RMSE) of RQMC is only o(n−1/2) for discontinuous functions. For certain discontinuous functions in d dimensions, we prove that the RMSE of RQMC is O(n−1/2−1/(4d−2)+ϵ) for any ϵ>0 and arbitrary n. If some discontinuity boundaries are parallel to some coordinate axes, the rate can be improved to O(n−1/2−1/(4du−2)+ϵ), where du denotes the so-called irregular dimension, that is, the number of axes which are not parallel to the discontinuity boundaries. Moreover, this paper shows that the RMSE is O(n−1/2−1/(2d)) for certain indicator functions.
Z. He and A. B. Owen. Discussion of: 'Sequential Quasi-Monte Carlo' by M. Gerber and N. Chopin, Journal of the Royal Statistical Society: Series B, 77 (3), 563-564, 2015. [] [link]
Z. He and X. Wang. Good Path Generation Methods in Quasi-Monte Carlo for Pricing Financial Derivatives, SIAM Journal on Scientific Computing, 36 (2), B171-B197, 2014. [abstract] [link]
The quasi-Monte Carlo (QMC) method is a powerful numerical tool for pricing complex derivative securities, whose accuracy is affected by the smoothness of the integrands. The payoff functions of many financial derivatives involve two types of nonsmooth factors: an indicator function (called jump structure) and a positive part of a smooth function (called kink structure). This paper develops a good path generation method (PGM) for recovering the superiority of the QMC method on problems involving multiple such structures. This is achieved by realigning these structures such that the associated nonsmooth surfaces are parallel to as many coordinate axes as possible. The proposed method has the advantage of addressing different structures according to their importance. We also offer a systematic measurement of different structures for quantifying and then ranking their importance. Numerical experiments demonstrate that the proposed method is more efficient than traditional PGMs for pricing exotic options, such as straddle Asian options, digital options, and barrier options. The numerical results confirm that both the jumps and kinks have tremendous impacts on the performance of the QMC method.
Z. He. Sensitivity estimation of conditional value at risk using randomized quasi-Monte Carlo. Preprint, arXiv:1908.07232, 2019. [abstract] [arXiv]
Conditional value at risk (CVaR) is a popular measure for quantifying portfolio risk. Sensitivity analysis of CVaR is very useful in risk management and gradient-based optimization algorithms.In this paper, we study the infinitesimal perturbation analysis estimator for CVaR sensitivity using randomized quasi-Monte Carlo (RQMC) simulation. We first prove that the RQMC-based estimator is strong consistent. Under some technical conditions, RQMC that uses d-dimensional points in CVaR sensitivity estimation yields a mean error rate of O(n−1/2−1/(4d−2)+ϵ) for arbitrarily small ϵ>0. The numerical results show that the RQMC method performs better than the Monte Carlo method for all cases. The gain of plain RQMC declines as the dimension $d$ increases, as predicted by the established theoretical error rate.
School of Mathematics
South China University of Technology
Guangzhou, 510641
P. R. China
Office: 4301, Building #4
Email: hezhijian@scut.edu.cn
何志坚，华南理工大学数学学院教授、博导、副院长，国家级青年人才计划获得者，广东省计算数学学会副理事长，首批教育部哲学社会科学创新团队核心成员。于2015年7月在清华大学获得理学博士学位。研究兴趣为随机计算方法与不确定性量化，特别是拟蒙特卡罗方法的理论和应用研究。目前在计算科学、统计学以及运筹管理国际著名刊物发表SCI论文近20篇，其中12篇发表在Journal of the Royal Statistical Society: Series B, SIAM Journal on Numerical Analysis, SIAM Journal on Scientific Computing, Mathematics of Computation。博士论文获得新世界数学奖（ICCM毕业论文奖）银奖。主持两项国家自然科学基金项目以及四项省部级项目。
◆ 2010/09-2015/07, 清华大学，统计学，博士
◆ 2006/09-2010/07, 华南理工大学，数学与应用数学，本科
◆ 2022/11至今, 华南理工大学数学学院，副院长
◆ 2021/09至今, 华南理工大学数学学院，教授
◆ 2018/01-2021/08, 华南理工大学数学学院，副教授
◆ 2016/01-2017/11, 中山大学岭南学院，特聘副研究员
◆ 2015/09-12, 华南理工大学经济与贸易学院，讲师
◆ 2014/01-07, 斯坦福大学统计系，访问学者
◆ 概率论与数理统计（本科）：2018年春季, 2020年春季
◆ 数理统计（本科）：2018年秋季, 2019年春季, 2019秋季, 2020秋季, 2021春季,
2021秋季, 2022春季，2022秋季，2023春季，2023秋季，2024春季，2024秋季
◆ 高等统计（研究生）：2019秋季, 2020秋季, 2021秋季，2022秋季, 2024秋季
◆ 贝叶斯统计与知识推理（研究生）：2018年秋季，2020年春季
◆ 随机过程（研究生）： 2023春季, 2024春季
◆ 广东省现场统计学会理事
◆ 广东省计算数学学会副理事长
◆ 中国运筹学会金融工程与金融风险管理分会理事
◆ 广东省统计局百名统计专家
◆ 国家自然科学基金面上项目：风险测度的敏感性分析与创新算法（编号：12071154，执行期限：2021-2024，主持，在研）
◆ 国家自然科学基金青年项目：基于拟蒙特卡罗模拟的VaR和CVaR计算问题研究（编号：71601189，执行期限：2017-2019，主持，已结题）
◆ 广东自然科学基金面上项目：基于嵌套模拟的金融风险定量计算（执行期限：2021-2023，主持，已结题）
◆ 广州市应用基础研究计划项目：高维近似贝叶斯计算问题的研究（执行期限：2021/4-2023/3，主持，已结题）
◆ 中央高校面上项目：条件拟蒙特卡罗模拟研究（执行期限：2019-2021，主持，已结题）
[12] J. Tan, Z. He*, X. Wang. Extensible grid sampling for quantile estimation, Mathematics of Computation, 94(352), 763-800, 2025. [链接]
[11] D. Ouyang, X. Wang, Z. He*. Achieving High Convergence Rates by Quasi-Monte Carlo and Importance Sampling for Unbounded Integrands, SIAM Journal on Numerical Analysis, 62(5), 2393-2414, 2024. [链接]
[10] Z. He*, Z. Zheng, X. Wang. On the error rate of importance sampling with randomized quasi-Monte Carlo, SIAM Journal on Numerical Analysis, 61(2), 515-538, 2023. [链接]
[9] Z. He, Z. Xu*, X. Wang. Unbiased MLMC-based variational Bayes for likelihood-free inference, SIAM Journal on Scientific Computing, 44(4), A1884-A1910, 2022. [链接]
[8] C. Zhang, X. Wang, and Z. He*. Efficient importance sampling in quasi-Monte Carlo methods for computational finance, SIAM Journal on Scientific Computing, 43(1), B1-B29, 2021. [摘要] [链接]
We consider integration with respect to a d-dimensional spherical Gaussian measure arising from computational finance.Importance sampling (IS) is one of the most important variance reduction techniques in Monte Carlo (MC) methods. In this paper, two kinds of IS are studied in randomized quasi-Monte Carlo (RQMC) setting, namely, the optimal drift IS (ODIS) and the Laplace IS (LapIS). Traditionally, the LapIS is obtained by mimicking the behavior of the optimal IS density, with ODIS as its special case. We prove that the LapIS can also be obtained by an approximate optimization procedure based on the Laplace approximation. We study the promises and limitations of IS in RQMC methods and develop efficient RQMC-based IS procedures. We focus on how to properly combine IS with conditional MC (CMC) and dimension reduction methods in RQMC. In our procedures, the integrands are firstly smoothed by using CMC. Then the LapIS or the ODIS is performed, where several orthogonal matrices are required to be chosen to reduce the effective dimension. Intuitively, designing methods to determine all these optimal matrices seems infeasible. Fortunately, we prove that as long as the last orthogonal matrix is chosen elaborately, the choices of the other matrices can be arbitrary. This helps to significantly simplify the RQMC-based IS procedure. Due to the robustness and the superiority in efficiency of the gradient principal component analysis (GPCA) method, we use the GPCA method as effective dimension reduction method in our RQMC-based IS procedures. Moreover, we prove the integrands obtained by the GPCA method are statistically equivalent. Numerical experiments illustrate the superiority of our proposed RQMC-based IS procedures.
[7] Z. He* and X. Wang. Convergence analysis of quasi-Monte Carlo sampling for quantile and expected shortfall, Mathematics of Computation, 90(327), 303-319 2021. [摘要] [链接]
In financial risk measurement, value-at-risk (VaR) and conditional VaR (CVaR)are two important risk measures. They are often estimated by Monte Carlo method combined with importance sampling (IS). Randomized Quasi-Monte Carlo (RQMC)method is an alternative technique, whose convergence rate is asymptotically better than the Monte Carlo method. This paper investigates the combination ofIS and RQMC method. The main contribution is two-fold. First, we prove theconsistency of the combined method for both VaR and CVaR estimations. Second,we establish error bounds for the CVaR estimate. Particularly, we show that under some mild conditions, the root mean square error of the CVaR estimate isO(N−1/2−1/(4d−2)+ϵ) for arbitrarily small ϵ>0, where dis the dimension of the problem. As a special case, these results also hold forplain RQMC estimates without using IS.
[6] Z. He*. On the Error Rate of Conditional quasi-Monte Carlo for discontinuous functions, SIAM Journal on Numerical Analysis, 57(2), 854-874, 2019. [摘要] [链接]
This paper studies the rate of convergence for conditional quasi-Monte Carlo (QMC), which is a counterpart of conditional Monte Carlo. We focus on discontinuous integrands defined on the whole of Rd, which can be unbounded. Under suitable conditions, we show that conditional QMC not only has the smoothing effect (up to infinitely times differentiable), but also can bring orders of magnitude reduction in integration error compared to plain QMC. Particularly, for some typical problems in options pricing and Greeks estimation, conditional randomized QMC that uses n samples yields a mean error of O(n−1+ϵ) for arbitrarily small ϵ>0. As a by-product, we find that this rate also applies to randomized QMC integration with all terms of the ANOVA decomposition of the discontinuous integrand, except the one of highest order.
[5] Z. He*. Quasi-Monte Carlo for discontinuous integrands with singularities along the boundary of the unit Cube, Mathematics of Computation, 87(314), 2857-2870, 2018. [摘要] [链接]
This paper studies randomized quasi-Monte Carlo (QMC) sampling for discontinuous integrands having singularities along the boundary of the unit cube [0,1]d. Both discontinuities and singularities are extremely common in the pricing and hedging of financial derivatives and have a tremendous impact on the accuracy of QMC. It was previously known that the root mean square error of randomized QMC is only o(n?1/2) for discontinuous functions with singularities. We find that under some mild conditions, randomized QMC yields an expected error of O(n?1/2?1/(4d?2)+ϵ) for arbitrarily small ϵ>0. Moreover, one can get a better rate if the boundary of discontinuities is parallel to some coordinate axes. As a by-product, we find that the expected error rate attains O(n?1+ϵ) if the discontinuities are QMC-friendly, in the sense that all the discontinuity boundaries are parallel to coordinate axes. The results can be used to assess the QMC accuracy for some typical problems from financial engineering.
[4] C. Weng, X. Wang, and Z. He*. Efficient computation of option prices and Greeks by quasi-Monte Carlo method with smoothing and dimension reduction, SIAM Journal on Scientific Computing, 39(2), B298-B322, 2017. [摘要] [链接]
Discontinuities and high dimensionality are common in the problems of pricing and hedging of derivative securities. Both factors have a tremendous impact on the accuracy of the quasi--Monte Carlo (QMC) method. An ideal approach to improve the QMC method is to transform the functions to make them smoother and having smaller effective dimension. This paper develops a two-step procedure to tackle the challenging problems of both the discontinuity and the high dimensionality concurrently. In the first step, we adopt the smoothing method to remove the discontinuities of the payoff function, improving the smoothness. In the second step, we propose a general dimension reduction method (called the CQR method) to reduce the effective dimension such that the better quality of QMC points in their initial dimensions can be fully exploited. The CQR method is based on the combination of the k-means clustering algorithm and the QR decomposition. 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 matching function of the smoothed function. The matching function serves as an approximation of the smoothed function but has a simpler form, and it is used to find the required transformation. Extensive numerical experiments on option pricing and Greeks estimation demonstrate that the combination of the smoothing method and dimension reduction in QMC achieves substantially larger variance reduction even in high dimension than dealing with either discontinuities or high dimensionality single sidedly.
[3] Z. He and A. B. Owen*. Extensible grids: Uniform sampling on a space-filling curve, Journal of the Royal Statistical Society: Series B, 78(4), 917-931, 2016. [] [C++ Code] [链接]
[2] Z. He* and X. Wang. On the Convergence rate of randomized Quasi-Monte Carlo for discontinuous functions, SIAM Journal on Numerical Analysis, 53(5), 2488-2503, 2015. [摘要] [链接]
This paper studies the convergence rate of randomized quasi--Monte Carlo (RQMC) for discontinuous functions, which are often of infinite variation in the sense of Hardy and Krause. It was previously known that the root mean square error (RMSE) of RQMC is only o(n−1/2) for discontinuous functions. For certain discontinuous functions in d dimensions, we prove that the RMSE of RQMC is O(n−1/2−1/(4d−2)+ϵ) for any ϵ>0 and arbitrary n. If some discontinuity boundaries are parallel to some coordinate axes, the rate can be improved to O(n−1/2−1/(4du−2)+ϵ), where du denotes the so-called irregular dimension, that is, the number of axes which are not parallel to the discontinuity boundaries. Moreover, this paper shows that the RMSE is O(n−1/2−1/(2d)) for certain indicator functions.
[1] Z. He* and X. Wang. Good Path generation methods in quasi-Monte Carlo for pricing financial derivatives, SIAM Journal on Scientific Computing, 36(2), B171-B197, 2014. [摘要] [链接]
The quasi-Monte Carlo (QMC) method is a powerful numerical tool for pricing complex derivative securities, whose accuracy is affected by the smoothness of the integrands. The payoff functions of many financial derivatives involve two types of nonsmooth factors: an indicator function (called jump structure) and a positive part of a smooth function (called kink structure). This paper develops a good path generation method (PGM) for recovering the superiority of the QMC method on problems involving multiple such structures. This is achieved by realigning these structures such that the associated nonsmooth surfaces are parallel to as many coordinate axes as possible. The proposed method has the advantage of addressing different structures according to their importance. We also offer a systematic measurement of different structures for quantifying and then ranking their importance. Numerical experiments demonstrate that the proposed method is more efficient than traditional PGMs for pricing exotic options, such as straddle Asian options, digital options, and barrier options. The numerical results confirm that both the jumps and kinks have tremendous impacts on the performance of the QMC method.
[11] Y. Xiong, X. Yang, S. Zhang, and Z. He*. An efficient likelihood-free Bayesian inference method based on sequential neural posterior estimation, Communications in Statistics - Simulation and Computation, 2025. (已接收) [链接]
[10] Z. Xu, Z. He*, and X. Wang. Efficient risk estimation via nested multilevel quasi-Monte Carlo simulation, Journal of Computational and Applied Mathematics, 443, 115745, 2024. [链接]
[9] J. Zhang and Z. He*. GMM-based procedure for multiple hypotheses testing, Communications in Statistics - Simulation and Computation, 53(6), 2605–2623, 2024. [链接]
[8] Z. He*. Sensitivity estimation of conditional value at risk using randomized quasi-Monte Carlo, European Journal of Operational Research, 298(1), 229-242, 2022. [摘要] [链接]
Conditional value at risk (CVaR) is a popular measure for quantifying portfolio risk. Sensitivity analysis of CVaR is very useful in risk management and gradient-based optimization algorithms.In this paper, we study the infinitesimal perturbation analysis estimator for CVaR sensitivity using randomized quasi-Monte Carlo (RQMC) simulation. We first prove that the RQMC-based estimator is strong consistent. Under some technical conditions, RQMC that uses d-dimensional points in CVaR sensitivity estimation yields a mean error rate of O(n−1/2−1/(4d−2)+ϵ) for arbitrarily small ϵ>0. The numerical results show that the RQMC method performs better than the Monte Carlo method for all cases. The gain of plain RQMC declines as the dimension $d$ increases, as predicted by the established theoretical error rate.
[7] Z. He* and X. Wang. An integrated quasi-Monte Carlo method for handling high dimensional problems with discontinuities in financial engineering, Computational Economics, 2021, 57(2), 693-718, 2021. [摘要] [链接]
Quasi-Monte Carlo (QMC) method is playing an increasing role in the problems of pricing and hedging of complex financial derivatives. These problems are usually of high dimensionality and discontinuities. The two factors may significantly deteriorate the performance of the QMC method. This paper develops a method that overcomes the challenges of the high dimensionality and discontinuities concurrently. For this purpose, a smoothing method is proposed to remove the discontinuities for some typical functions arising from financial engineering. To make the smoothing method applicable for more general functions, a new path generation method is designed for simulating the paths of the underlying assets such that the resulting function has the required form. The new path generation method has an additional power to reduce the effective dimension of the target function. Our proposed method caters for a large variety of model specifications, including the Black-Scholes, exponential generalized hyperbolic L\'evy, and Heston models. Numerical experiments dealing with these models show that in the QMC setting the proposed smoothing method in combination with the new path generation method can lead to a dramatic variance reduction for pricing exotic options with discontinuous payoffs and for calculating options' Greeks. The investigation on the effective dimension and the related characteristics explains the significant enhancement of the combined procedure.
[6] X. Fei*, M. Giles, Z. He. QMC Sampling from Empirical Datasets, Proceedings of the MCQMC 2018 conference, 2018. [摘要] [链接]
This paper presents a simple idea for the use of quasi-Monte Carlo samplingwith empirical datasets, such as those generated by MCMC methods. It alsopresents and analyses a related idea of taking advantage of the Hilbert space-filling curve. Theoretical and numerical analyses are provided for both. We find that when applying the proposed QMC sampling methods to datasets coming from a known distribution, they give similar performance as the standard QMC method directly sampling from this known distribution.
[5] Z. He* and L. Zhu. Asymptotic normality of extensible grid sampling, Statistics and Computing, 29(1), 53-65, 2019. [摘要] [链接]
Recently, He and Owen (2016) proposed the use of Hilbert's space filling curve (HSFC) in numerical integration as a way of reducing the dimension from d>1 to d=1. This paper studies the asymptotic normality of the HSFC-based estimate when using scrambled van der Corput sequence as input. We show that the estimate has an asymptotic normal distribution for functions in C1([0,1]d), excluding the trivial case of constant functions. The asymptotic normality also holds for discontinuous functions under mild conditions. It was previously known only that scrambled (0,m,d)-net quadratures enjoy the asymptotic normality for smooth enough functions, whose mixed partial gradients satisfy a H\'older condition. As a by-product, we find lower bounds for the variance of the HSFC-based estimate. Particularly, for nontrivial functions in C1([0,1]d), the lower bound is of order n−1−2/d, which matches the rate of the upper bound established in He and Owen (2016).
[4] F. Xie, Z. He*, and X. Wang. An importance sampling-based smoothing approach for quasi-Monte Carlo Simulation of barrier options. European Journal of Operational Research, 274(2), 759-772, 2019. [摘要] [链接]
Handling discontinuities in financial engineering is a challenging task when using quasi-Monte Carlo (QMC) method.This paper develops a smoothing approach based on importance sampling (IS) to remove multiple discontinuity structures sequentially arising from pricing barrier options and other simulation problems with similar structures. The IS-based smoothing method yields an unbiased estimate with reduced variance. We find that under the Black-Scholes model and the Variance Gamma model, the order in which the asset prices are simulated may have a significant impact on the efficiency of the IS method. It is optimal to simulate the asset price at the expiration time firstly in both models. Combining with a proper simulation order, the IS method transforms the payoff to make it not only smoother but also having smaller effective dimension. Numerical experiments show that the combined procedure performs consistently much better than commonly used methods for pricing barrier options. Specifically, the IS method combined with the Brownian bridge construction performs best under the Black-Scholes model, while the reverse order construction performs best for the Variance Gamma model. Furthermore, we show that the effective dimension is greatly reduced by applying the proposed procedure, which explains the superiority of the method from another perspective.
[3] C. Schretter*, Z. He, M. Gerber, N. Chopin, and H. Niederreiter. Van der Corput and golden ratio sequences along the Hilbert space-filling curve. Proceedings of the MCQMC 2014 conference, R. Cools and D. Nuyens (Eds.), 531-544, 2016. [摘要] [链接]
This work investigates the star discrepancies and squared integration errors of two quasi-random points constructions using a generator one-dimensional sequence and the Hilbert space-filling curve. This recursive fractal is proven to maximize locality and passes uniquely through all points of the d-dimensional space. The van der Corput and the golden ratio generator sequences are compared for randomized integro-approximations of both Lipschitz continuous and piecewise constant functions. We found that the star discrepancy of the construction using the van der Corput sequence reaches the theoretical optimal rate when the number of samples is a power of two while using the golden ratio sequence performs optimally for Fibonacci numbers. Since the Fibonacci sequence increases at a slower rate than the exponential in base 2, the golden ratio sequence is preferable when the budget of samples is not known beforehand. Numerical experiments confirm this observation.
[2] C. Weng, X. Wang, and Z. He*. An auto-realignment method in quasi-Monte Carlo for pricing financial derivatives with jump structures,European Journal of Operational Research, 254(1), 304-311, 2016. [摘要] [链接]
Discontinuities are common in the pricing of financial derivatives and have a tremendous impact on the accuracy of quasi-Monte Carlo (QMC) method. While if the discontinuities are parallel to the axes, good efficiency of the QMC method can still be expected. By realigning the discontinuities to be axes-parallel, [Wang & Tan, 2013] succeeded in recovering the high efficiency of the QMC method for a special class of functions. Motivated by this work, we propose an auto-realignment method to deal with more general discontinuous functions. The k-means clustering algorithm, a classical algorithm of machine learning, is used to select the most representative normal vectors of the discontinuity surface. By applying this new method, the discontinuities of the resulting function are realigned to be friendly for the QMC method. Numerical experiments demonstrate that the proposed method significantly improves the performance of the QMC method.
[1] Z. He and A. B. Owen*. Discussion of: 'Sequential Quasi-Monte Carlo' by M. Gerber and N. Chopin, Journal of the Royal Statistical Society: Series B, 77 (3), 563-564, 2015. [] [链接]
本课题组现有2名在读博士生，4名在读硕士研究生，已经毕业5名硕士生（毕业去向：阿里巴巴、小鹏汽车、中国建设银行等）。
每年计划招生：
1. 1-3名硕士生（概率统计方向），其中包含1-2名推免生，优先考虑有读博意向的学生
2. 1名博士生（计算数学方向）
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现阶段研究方向有
1. 随机模拟算法研究，包括蒙特卡罗方法、拟蒙特卡罗方法、多层蒙特卡罗方法、马尔科夫蒙特卡罗方法
2. 贝叶斯计算研究，包括近似贝叶斯计算、变分推断
3. 科学计算中的机器学习方法
4. 大规模假设检验
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