基于拟蒙特卡罗模拟的VaR和CVaR计算问题研究

In recent decades, financial institutions usually have to deal with various financial risks due to the prosperous development of the financial market. Financial risk management is playing an important role in finance, and financial risk measurement is the most important ingredient. This project focuses on investigating value-at-risk (VaR) and conditional value-at-risk (CVaR) calculation in financial risk management. Specifically, the research agenda mainly includes: (1) Quasi-Monte Carlo (QMC) methods in combination with importance sampling technique are used to estimate VaR and CVaR of assets or portfolios; (2) The convergence rate of the proposed method is established in theory; (3) Importance sampling technique is studied in the context of QMC; (4) To handle the “high-dimensionality” challenge resulting from the diversity of risk factors and the complexity of financial models, dimension reduction strategies are investigated, which can break the curse of dimensionality successfully. The efficiency of QMC thus can be improved substantially. The project will overcome some difficulties including high-dimensionality and the discontinuities appeared in modern financial risk management. Additionally, it will develop high performance computational methods in risk management by taking account into practical issues, and establish some basic theories in error estimation. The project connects with practice closely, which is expected to provide novel theory and numerical simulation techniques for financial risk management, and to enhance the level of risk management of financial institutions.

近几十年来金融市场迅猛发展，金融机构常常面临各式各样的金融风险。金融风险管理逐渐成为金融体系中不可或缺的一部分，而金融风险度量则是风险管理的重中之重。本项目将致力于金融风险度量中的VaR和CVaR计算问题的研究，主要包括：（1）用拟蒙特卡罗方法结合重要性抽样技术来估计某资产或者投资组合的VaR和CVaR；（2）理论上证明该方法的收敛速度；（3）研究拟蒙特卡罗框架下的重要性抽样技术；（4）针对风险因子的多样性以及金融模型的复杂性带来的“高维”挑战，研究有效的降维策略以克服“维数灾难”，从本质上提高拟蒙特卡罗方法的效率。本项目的成果预期将有效地处理现代金融风险管理中的“高维”和“间断”等难点，发展现代实际金融问题驱动的高性能风险度量计算方法，并建立相应的误差估计基础理论。本项目紧密联系实际，有望为金融风险管理提供新的理论和数值模拟技术，提升金融机构的风险管理水平。

近几十年来金融市场迅猛发展，金融机构常常面临各式各样的金融风险。金融风险管理逐渐成为金融体系中不可或缺的一部分，而金融风险度量则是风险管理的重中之重。本课题建立基于模拟仿真的金融风险度量计算的理论基础，发展高性能的确定性的拟蒙特卡罗算法和随机化的算法，研究风险度量计算方法的复杂度，为国民经济和金融安全提供重要的数值模拟技术支持。主要结果是：（1）提出一系列降维方法和函数光滑化方法，克服维数灾难和函数的间断性困难，极大提高VaR和CVaR计算效率；（2）理论证明风险度量计算方法的收敛速度，发现所提的随机化算法具有比蒙特卡罗更高阶的收敛速度；（3）证明了一类随机化算法的渐近正态性，构建有效的置信区间。..在本课题资助下，项目主持人以第一作者或通讯作者身份发表SCI论文5篇，其中1篇发表在运筹管理权威期刊Eur. J. Oper. Res., 3篇发表在计算数学顶级期刊SIAM J. Numer. Anal., SIAM J. Sci. Comp.和Math. Comp., 1篇发表在统计学权威期刊Stat. Comput. 培养3名研究生；参加2次国外相关会议和若干次国内会议。