Identifying Extreme Events in the Stock Market: A Topological Data Analysis

Paper: arXiv 2405.16052

Abstract

This paper employs Topological Data Analysis (TDA) to detect extreme events (EEs) in the stock market at a continental level. Previous approaches, which analyzed stock indices separately, could not detect EEs for multiple time series in one go. TDA provides a robust framework for such analysis and identifies the EEs during the crashes for different indices. The TDA analysis shows that $L^1$, $L^2$ norms and Wasserstein distance ($W_D$) of the world leading indices rise abruptly during the crashes, surpassing a threshold of $μ+4σ$ where $μ$ and $σ$ are the mean and the standard deviation of norm or $W_D$, respectively. Our study identified the stock index crashes of the 2008 financial crisis and the COVID-19 pandemic across continents as EEs. Given that different sectors in an index behave differently, a sector-wise analysis was conducted during the COVID-19 pandemic for the Indian stock market. The sector-wise results show that after the occurrence of EE, we have observed strong crashes surpassing $μ+2σ$ for an extended period for the banking sector. While for the pharmaceutical sector, no significant spikes were noted. Hence, TDA also proves successful in identifying the duration of shocks after the occurrence of EEs. This also indicates that the Banking sector continued to face stress and remained volatile even after the crash. This study gives us the applicability of TDA as a powerful analytical tool to study EEs in various fields.

Complexity vs Empirical Score

• Math Complexity: 8.5/10
• Empirical Rigor: 3.0/10
• Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper employs advanced mathematical concepts from topological data analysis, including persistent homology, Vietoris–Rips complexes, and norms of persistence landscapes, indicating high math complexity. However, it lacks detailed backtesting, specific implementation code, or out-of-sample performance metrics, focusing instead on theoretical framework and descriptive results, resulting in low empirical rigor.

Research Flowchart

  flowchart TD
  A[""Research Goal:
  Detect Extreme Events in
  Multivariate Stock Indices""] --> B[""Data Input:
  Continental Stock Indices
  (e.g., 2008 Crisis, COVID-19)""]
  B --> C[""Methodology:
  Topological Data Analysis
  (TDA) via Persistence Diagrams""]
  C --> D[""Computational Process:
  Calculate L1/L2 Norms & Wasserstein Distance (W_D)""]
  D --> E{"Threshold Check:
  Value > μ + 4σ?"}
  E -- Yes --> F[""Outcome 1: Identifies Extreme Events
  (e.g., Global Market Crashes)""]
  E -- No --> G
  F --> H[""Outcome 2: Sector Analysis
  (e.g., Banking vs. Pharma)""]
  H --> G[""Final Outcome:
  TDA successfully detects shock duration;
  Banking sector shows prolonged stress""]
  style A fill:#f9f,stroke:#333,stroke-width:2px
  style G fill:#bbf,stroke:#333,stroke-width:2px

Cited References (63)

1. Statistical Topological Data Analysis using Persistence Landscapes (2015) — link
2. Computational Topology for Data Analysis (2022)
3. Topology and data (2009) — link
4. Critical market crashes (2003)
5. The crash of October 1987 seen as a phase transition: amplitude and universality (1998)
6. Financial crisis dynamics: attempt to define a market instability indicator (2012)
7. COVID-19 and the march 2020 stock market crash. Evidence from S&P1500 (2021)
8. Statistical properties of the aftershocks of stock market crashes revisited: Analysis based on the 1987 crash, financial-crisis-2008 and COVID-19 pandemic (2022)

0 of 63 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.