Novel Invariant Features of the US Stock Market



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Price fluctuations in financial markets are influenced by a multitude of economic, societal, and other factors. Rather than attempt to understand the cause and effect of such myriad complexities, we use traditional tools of physics and mathematics in addressing large systems. Specifically, we model price fluctuations as a random motion that cannot be predicted nor replicated. Scientific measurements rely on invariance, the reproducibility of conclusions in nominally identical experiments. Only they can be used to characterize the underlying process. This thesis will outline some novel invariant features related to the fluctuations observed in the US stock market. First-passage-time distribution, which presents the likelihood of a stock reaching a pre-specified price at a given time, is useful in risk assessment and in valuation of certain option contracts. This is one invariant feature of a random process. Price fluctuations depend on both time and on the current price. Recently, variable-diffusion models, which account for these dependencies, have been proposed. Here we present a model for the US stock market and investigate its first-passage-time distribution. For a Wiener process it has a single peak, while that for stocks exhibits a notable second peak within a trading day. We show here that the first-passage-time distribution of our two-stage variable-diffusion model does exhibit this feature. Periodicity in financial markets has long been speculated on. Seasonal cycles have been conjectured and proposed longer-term cycles include major recessions and depressions. Koopman decomposition is a generalization of normal mode analysis that identifies oscillating structures within data. Although Fourier analysis decomposes any signal into a sum of periodic functions, the challenge is to separate the invariant (robust) features from the noise and transient features. A recent study using a novel "robust mode analysis" reported four robust modes in the US stock market with periods between 75 and 250 days. Here, we present a more detailed analysis of over 1000 stocks spanning 18 years. We have confirmed the presence of four robust modes and identified a fifth. In addition, we find a high level of phase coherence for some of the modes within specific market sectors.



Econophysics, Quantitative, Finance, Stochastic, Variable diffusion, First-passage, Investment horizon, Volatility, Koopman, Dynamic mode decomposition


Portions of this document appear in: Hua, Jia-Chen, Lijian Chen, Liberty Falcon, Joseph L. McCauley, and Gemunu H. Gunaratne. "Variable diffusion in stock market fluctuations." Physica A: Statistical Mechanics and its Applications 419 (2015): 221-233. And in: Barney, Liberty, and Gemunu H. Gunaratne. "First-Passage-Time Distribution for Variable-Diffusion Processes." Journal of Statistical Physics 167, no. 3-4 (2017): 878-891.