摘 要：随机矩阵理论（Random Matrix Theory, RMT）作为统计物理学的重要分支，近年来被广泛应用于金融投资组合优化领域。本研究旨在探讨RMT在处理高维金融数据中的潜力，通过分离市场噪声与有效信息，提升投资组合构建的稳健性。基于RMT的核心思想，本文提出了一种改进的协方差矩阵估计方法，能够有效识别并剔除由有限样本引起的虚假相关性。通过对多个金融市场的真实数据进行实证分析，结果表明该方法显著降低了投资组合的风险水平，并提高了夏普比率。此外，本文首次将非线性特征值修正技术引入RMT框架，进一步增强了模型对极端市场波动的适应能力。研究表明，RMT不仅为理解复杂金融系统的内在结构提供了新视角，还为量化投资策略的设计带来了重要启示，其创新性贡献在于实现了噪声过滤与风险控制的有机结合，从而为金融实践提供了更为可靠的理论支持。
关键词：随机矩阵理论；协方差矩阵估计；投资组合优化；噪声过滤；非线性特征值修正
Application of Random Matrix Theory in Financial Portfolio Optimization
英文人名
Directive teacher:×××
Abstract：Random Matrix Theory (RMT), as a significant branch of statistical physics, has been increasingly applied to the field of financial portfolio optimization in recent years. This study investigates the potential of RMT in handling high-dimensional financial data by distinguishing between market noise and effective information, thereby enhancing the robustness of portfolio construction. Based on the core principles of RMT, an improved covariance matrix estimation method is proposed, which effectively identifies and removes spurious correlations caused by finite sample sizes. Empirical analyses conducted on real datasets from multiple financial markets demonstrate that this method significantly reduces the risk level of portfolios while improving the Sharpe ratio. Moreover, this paper introduces nonlinear eigenvalue correction techniques into the RMT fr amework for the first time, further strengthening the model's adaptability to extreme market fluctuations. The findings indicate that RMT not only offers a new perspective for understanding the intrinsic structure of complex financial systems but also provides critical insights for the design of quantitative investment strategies. Its innovative contribution lies in the organic integration of noise filtering and risk control, offering more reliable theoretical support for financial practice.
Keywords: Random Matrix Theory；Covariance Matrix Estimation；Portfolio Optimization；Noise Filtering；Nonlinear Eigenvalue Correction
目 录
引言 1
一、随机矩阵理论基础与金融背景 1
(一)随机矩阵理论概述 1
(二)金融数据的随机特性分析 2
(三)理论在金融中的适用性探讨 2
二、投资组合优化中的噪声过滤 3
(一)噪声对投资组合的影响 3
(二)随机矩阵方法的噪声识别 3
(三)噪声过滤的实际应用案例 4
三、相关性结构的建模与分析 4
(一)金融资产相关性的特征提取 4
(二)随机矩阵在相关性建模中的作用 5
(三)相关性结构对投资组合的影响 5
四、实证研究与策略改进 6
(一)数据选取与实验设计 6
(二)随机矩阵理论的应用效果评估 6
(三)基于理论的投资组合策略优化 7
结论 7
参考文献 9
致谢 9
