股票的可观测矩阵动力学
Observable Matrix Dynamics of Stocks
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中文总结 AI 辅助
研究将可观测矩阵动力学方法应用于标准普尔500指数在三个危机十年的横截面数据，通过距离矩阵等指标监测市场，揭示行业轮动等情况，还将收益和波动率排名建模为马尔可夫链，发现各矩阵在市场崩溃时有连贯变化，能预测部分危机。
中文摘要 AI 辅助
可观测矩阵动力学（OMD）方法通过固定大小距离矩阵及其频谱的轨迹来监测复杂非线性系统的时间发展。我们将其应用于标准普尔500指数在三个危机十年（2001年互联网泡沫破裂、2007 - 2008年金融危机和2020年新冠疫情引发的股市暴跌）期间的横截面数据，在固定的股票池中使用三个固定大小的可观测指标。滚动收益相关性的反余弦距离矩阵反映了相关几何结构：其有效维度在2008年和2020年危机时崩溃，而2001年泡沫破裂是分散式 unwind。与机器学习距离矩阵对比，其频谱处于未放松、未学习状态，没有低维流形，市场从未学习其相关结构或放松到稳定几何结构。减去市场因素后揭示出连贯的行业轮动，通过名称级归因可确定哪些股票驱动每次危机及顺序。短期回顾时这些信号能解决先兆并预测2008年内生危机，但无法预测2020年外生冲击。另外两个可观测指标将每日收益和波动率排名建模为其排名空间上的马尔可夫链。收益链有持续的、由防御性股票主导的领头羊且动态近乎可逆。波动率链更具持续性，由金融行业主导，是唯一带有微弱、间歇性时间箭头的，在市场压力下爆发，与波动率聚类和祖姆巴赫效应相匹配。所有三个矩阵在市场崩溃期间都显示出连贯变化。
英文摘要
The Observable Matrix Dynamics (OMD) approach monitors the time development of complex non-linear systems through the trajectory of a fixed-size distance matrix and its spectrum. We apply it to the S\&P 500 cross section over three crisis decades, the 2001 dot-com bust, the 2007--2008 financial crisis, and the 2020 Covid crash, with three fixed-size observables on a fixed universe. The arccos distance matrix of the rolling return correlations reads the correlation geometry: its effective dimension collapses at the 2008 and 2020 crises, while the 2001 bust is a dispersed unwind. Read against machine-learning distance matrices, its spectrum stays in the un-relaxed, pre-learning regime with no low-dimensional manifold, so the market never learns its correlation structure or relaxes to a stationary geometry. Subtracting the market factor exposes a coherent sector rotation, whose name-level attribution identifies which stocks drive each crisis and in what order. At a short lookback these signals resolve precursors and forecast the endogenous 2008 crisis, though not the exogenous 2020 shock. The other two observables model the daily return and volatility rankings as Markov chains on their ranking spaces. The return chain has persistent, defensive-led bellwethers and near-reversible dynamics. The volatility chain is far more persistent, led by the financial sector, and is the only one to carry a weak, episodic arrow of time, flaring at market stress and matching volatility clustering and the Zumbach effect. All three matrices show coherent changes during market crashes.
