Topological Data Analysis of Financial Time Series: Landscapes of Crashes

Abstract

We explore the evolution of daily returns of four major US stock market indices during the technology crash of 2000, and the financial crisis of 2007-2009. Our methodology is based on topological data analysis (TDA). We use persistence homology to detect and quantify topological patterns that appear in multidimensional time series. Using a sliding window, we extract time-dependent point cloud data sets, to which we associate a topological space. We detect transient loops that appear in this space, and we measure their persistence. This is encoded in real-valued functions referred to as a 'persistence landscapes'. We quantify the temporal changes in persistence landscapes via their $L^p$-norms. We test this procedure on multidimensional time series generated by various non-linear and non-equilibrium models. We find that, in the vicinity of financial meltdowns, the $L^p$-norms exhibit strong growth prior to the primary peak, which ascends during a crash. Remarkably, the average spectral density at low frequencies of the time series of $L^p$-norms of the persistence landscapes demonstrates a strong rising trend for 250 trading days prior to either dotcom crash on 03/10/2000, or to the Lehman bankruptcy on 09/15/2008. Our study suggests that TDA provides a new type of econometric analysis, which goes beyond the standard statistical measures. The method can be used to detect early warning signals of imminent market crashes. We believe that this approach can be used beyond the analysis of financial time series presented here.

References in corpus (17)

• Superstatistics
• Dynamical foundations of nonextensive statistical mechanics
• Statistical topological data analysis using persistence landscapes
• A persistence landscapes toolbox for topological statistics
• Nonextensive statistical mechanics and economics
• Dynamical model and nonextensive statistical mechanics of a market index on large time windows
• Volatility in Financial Markets: Stochastic Models and Empirical Results
• Analysis of Kolmogorov Flow and Rayleigh-Bénard Convection using Persistent Homology
• Superstatistical fluctuations in time series: Applications to share-price dynamics and turbulence
• Persistent topological features of dynamical systems
• Topological classifier for detecting the emergence of epileptic seizures
• Model for Non-Gaussian Intraday Stock Returns
• Microscopic Origin of Non-Gaussian Distributions of Financial Returns
• A logistic map approach to economic cycles I. The best adapted companies
• Bridging stylized facts in finance and data non-stationarities
• Non-parametric segmentation of non-stationary time series
• A Non-Gaussian Option Pricing Model with Skew

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