高维度、非线性模型下的金融资产定价和风险定量计算

Financial asset pricing and risk control are the core of financial engineering, their solutions require accurate modelling and a large amount of complicated analysis and computation. Because of the complexity of the financial products, and the high dimensionality and non-linearlity of the financial econometric models, analytical tools and traditional numerical methods face challenges. More advanced analytical tools and computational methods are needed. This project aims to establish accurate mathematical models for financial assets that are consistent with the nature of the financial markets and are consistent with the stylized facts in financial data, and to develop statistical and computational methods for financial models. The project focuses on the following aspects: (1) Asset pricing and volatility modelling, including the asset returns, the modelling for volatility, especially the multi-factor and nonlinear models; (2) High performance computational methods in asset pricing, including the computational methods for high-dimensional integrals and the constructions of low discrepancy point sets, simulation methods for stochastic differential equations, dimension reduction methods for high-domensional financial problems, pricing methods for complex exotic derivatives and American options. (3) Risk measures and their computation, including new coherent risk measures and their effective computation, the computation of sensitivities (Greeks) of financial derivatives. Our aim is to develop effective, robust and real-time methods for financial data analysis and computation, to establish general theories of convergence and error estimation, and to propose methods for solving high-dimensional nonlinear problems in pricing and hedging. We hope that our research can provide key methods for financial modelling, data analysis and computation in financial practice.

金融资产定价和风险控制是金融工程的核心课题，需要准确的计量建模和大量复杂的分析和数值计算。由于金融产品的复杂性和模型的高维度和非线性等特征，解析方法和传统计算方法均面临挑战。本项目致力于建立符合金融市场规律、并与数据典型特征相匹配的金融资产的数学模型，发展处理复杂海量金融数据的统计方法和数值方法。重点研究：（1）资产定价与波动率建模，包括金融资产的收益规律、波动率发生机制与建模，特别是多因子和非线性模型。（2）资产定价中的高性能算法，包括高维积分的计算方法和低偏差点列的构造、随机微分方程的模拟方法、高维问题的降维方法、复杂衍生产品和美式期权的定价。（3）风险度量与计算，包括新的一致性风险度量与计算以及风险敏感性参数的计算。目标是发展高效、稳健、实时的新型分析和计算方法，建立一般性的收敛性和误差估计理论，探索解决高维非线性问题的新途径，为金融实践提供关键的金融建模、数据分析和数值模拟方法。

金融资产定价和风险控制是金融工程的核心课题，需要准确的计量建模和大量复杂的分析和数值计算。由于金融产品的复杂性和模型的高维度和非线性等特征，解析方法和传统计算方法均面临挑战。本项目致力于建立符合金融市场规律、并与数据典型特征相匹配的金融资产的数学模型，发展处理复杂海量金融数据的统计方法和数值方法。重点研究以下问题：（1）资产定价与波动率建模。（2）资产定价中的高性能算法，包括高维积分的计算方法和低偏差点列的构造、随机微分方程的模拟方法、高维问题的降维方法、复杂衍生产品和美式期权的定价。（3）风险度量的分析与计算，包括一致性风险度量和风险敏感性参数的计算。项目圆满完成预定计划，取得一系列创新性研究成果。提出了一系列高效、稳健、实时的新型分析和计算方法，包括高维问题的降维方法和间断函数的光滑化方法，极大地提高了计算效率，拓展了方法的应用领域和范围。建立了一般性的收敛性和误差估计理论，为解决高维复杂金融问题提供了新途径。可为金融实践提供关键的金融建模、数据分析和数值模拟方法。在金融和其它学科领域中有明显的潜在应用价值。