An exploration of the mathematical structure and behavioural biases of 21st century financial crises

ArXiv ID: 2307.15402 “View on arXiv”

Authors: Unknown

Abstract

In this paper we contrast the dynamics of the 2022 Ukraine invasion financial crisis with notable financial crises of the 21st century - the dot-com bubble, global financial crisis and COVID-19. We study the similarity in market dynamics and associated implications for equity investors between various financial market crises and we introduce new mathematical techniques to do so. First, we study the strength of collective dynamics during different market crises, and compare suitable portfolio diversification strategies with respect to the unique number of sectors and stocks for optimal systematic risk reduction. Next, we introduce a new linear operator method to quantify distributional distance between equity returns during various crises. Our method allows us to fairly compare underlying stock and sector performance during different time periods, normalising for those collective dynamics driven by the overall market. Finally, we introduce a new combinatorial portfolio optimisation framework driven by random sampling to investigate whether particular equities and equity sectors are more effective in maximising investor risk-adjusted returns during market crises.

Keywords: financial crisis analysis, linear operator method, distributional distance, portfolio optimization, Equities

Complexity vs Empirical Score

• Math Complexity: 8.0/10
• Empirical Rigor: 6.0/10
• Quadrant: Holy Grail
• Why: The paper introduces new mathematical methods like a linear operator for distributional distance and a combinatorial optimization framework, indicating high mathematical density. While it uses real historical market data for multiple crises, the empirical rigor is moderate as it focuses on methodological innovation rather than providing extensive backtesting code or implementation-heavy trading strategies.

  flowchart TD
    A["Research Goal:<br>Analyze mathematical structure & biases<br>in 21st Century Financial Crises"] --> B["Data Inputs"]
    
    B --> B1["Equity Returns: 2022 Ukraine Crisis,<br>Dot-com Bubble, GFC, COVID-19"]
    B --> B2["Sector & Stock Metadata"]
    
    B --> C["Methodology: Phase 1 -<br>Collective Dynamics"]
    C --> C1["Measure market correlation strength"]
    C --> C2["Test diversification strategies<br>by sector/stock count"]
    
    B --> D["Methodology: Phase 2 -<br>Linear Operator Method"]
    D --> D1["Quantify distributional distance<br>between returns"]
    D --> D2["Normalize for market-wide dynamics<br>to isolate specific crisis effects"]
    
    B --> E["Methodology: Phase 3 -<br>Combinatorial Optimization"]
    E --> E1["Random sampling framework"]
    E --> E2["Optimize portfolios for<br>risk-adjusted returns"]
    
    C2 --> F["Key Findings & Outcomes"]
    D2 --> F
    E2 --> F
    
    F --> F1["Identified optimal portfolio<br>configurations per crisis type"]
    F --> F2["Validated distributional<br>similarity between crises"]
    F --> F3["Revealed sector-specific<br>performance patterns"]