股票是投资人热衷的投资工具， 股票的市场特征分析及其量化投资组合选择问题也一直是学术界和业界重点关注的领域。 大量研究表明， 现实中的金融市场其本质是非线性的、 复杂的， 而传统的有效市场假说及其衍生的金融理论都是线性的， 不能很好地对股票市场中的非线性特征做出解释。 近年来， 非线性科学理论中的分形理论引起金融领域研究者的广泛关注， 而基于多重分形法（MF） 分析股票市场并将其应用到量化投资组合领域， 不仅能揭示不同标度下股票市场的多重分形特征， 还能帮助投资者提升收益、 同时降低风险。

本文使用了多重分形法中的多重分形去趋势波动分析法（MF-DFA） 和多重分形去趋势交互分析法（MF-DCCA）， 对股票市场价格及个股进行特征分析， 针对 计算过程中可能会出现的过拟合现象， 引入 L1 范数对多项式拟合过程进行正则化。 将多重分形方法与传统的均值-方差模型相结合， 构建基于多重分形的股票投资组合模型以及基于 L1 范数多重分形投资组合选择模型， 获得最优投资组合。

主要研究成果包括： （1） 分析了我国股票市场价格指数及个股的多重分形特征。本文首先选取了 2001 年-2022 年之间的股票市场价格指数和个股的日收益率时间序列， 验证了我国股票市场的非正态分布性， 运用 MF-DFA 和 MF-DCCA 方法， 验证了我国股票市场价格指数和个股的多重分形特征以及收益率序列之间交互关系的多重分形特征。 基于 L1 范数多重分形法（L1-MF）， 计算修正的广义 Hurst 指数、 交互广义 Hurst 指数， 验证了 L1-MF 方法在计算广义 Hurst 指数、 交互广义 Hurst 指数有明显的修正效果， 且在个股上修正效果更加明显。（2） 构建基于 MF、 L1-MF 的股票投资组合模型， 与传统的均值-方差模型进行对比分析。 实验结果表明， 基于 MF、L1-MF 的投资组合的累计收益率均高于均值-方差投资组合的累计收益率， 且基于L1-MF 的投资组合累计收益率高于基于 MF 的投资组合累计收益率， 证明了基于 MF的投资组合模型和基于 L1-MF 的投资组合模型是有效的， 且基于 L1-MF 的投资组合模型效果要优于基于 MF 的投资组合模型。 在降低风险的同时， 能够提高投资组合收益率。 本文的研究结果有望为投资者带来更好的股票市场风险评估方法， 从而更有效地进行股票投资组合选择和风险管理， 建立更加合理的投资组合。
 

Stocks are a popular investment tool for investors, and the problem of marketcharacterization of stocks and their quantitative portfolio selection has been a key area ofinterest for academics and industry. A large number of studies have shown that the natureof the real financial market is nonlinear and complex, while the traditional efficient market hypothesis and its derived financial theories are linear and cannot well explain the nonlinear features in the stock market. In recent years, the fractal theory of nonlinear scientific theories has attracted a lot of attention from researchers in the field of finance, and analyzing stock markets based on the multiple fractal method (MF) and applying it to the field of quantitative portfolios can not only reveal the multiple fractal characteristics of stock markets under different scales, but also help investors to enhance returns and reduce
risks at the same time.

In this paper, we use the multifractal detrended volatility analysis (MF-DFA) and the multifractal detrended interaction analysis (MF-DCCA) to characterize stock market prices and individual stocks, and introduce the L1 parametrization to regularize the polynomial
fitting process for the possible overfitting phenomenon in the calculation process. The multifractal method is combined with the traditional mean-variance model to construct a multifractal-based stock portfolio model and an L1-parametric multifractal portfolio
selection model to obtain the optimal portfolio.
The main research results include: (1) analyzing the multifractal characteristics of China's stock market price index and individual stocks. This paper first selects the daily time series of stock market price indices and individual stocks' returns between 2001 and 2022 to verify the non-normal distribution of China's stock market, and applies MF-DFA and MF-DCCA methods to verify the multifractal characteristics of China's stock market price indices and individual stocks as well as the multifractal characteristics of the interaction
relationship between the return series. Based on the L1-parametric multiple fractal method (L1-MF), we calculate the corrected generalized Hurst index and the interaction generalized Hurst index, and verify that the L1-MF method has a significant correction effect in calculating the generalized Hurst index and the interaction generalized Hurst index, and the correction effect is more obvious on individual stocks. (2) The stock portfolio models based on MF and L1-MF are constructed and compared with the
traditional mean-variance model for analysis. The experimental results show that the cumulative returns of MF- and L1-MF-based portfolios are higher than the cumulative returns of mean-variance portfolios, and the cumulative returns of L1-MF-based portfolios
are higher than the cumulative returns of MF-based portfolios, proving that the MF-based portfolio model and L1-MF-based portfolio model are effective, and the L1-MF-based portfolio model is better than the MF-based portfolio model. It can improve the portfolio return while reducing the risk. The results of this paper are expected to bring investors a better method for stock market risk assessment, so that they can make stock portfolio selection and risk management more effectively and build a more reasonable portfolio.
 

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