Financial Information Theory

ArXiv ID: 2511.16339 “View on arXiv”

Authors: Miquel Noguer i Alonso

Abstract

This paper introduces a comprehensive framework for Financial Information Theory by applying information-theoretic concepts such as entropy, Kullback-Leibler divergence, mutual information, normalized mutual information, and transfer entropy to financial time series. We systematically derive these measures with complete mathematical proofs, establish their theoretical properties, and propose practical algorithms for estimation. Using S&P 500 data from 2000 to 2025, we demonstrate empirical usefulness for regime detection, market efficiency testing, and portfolio construction. We show that normalized mutual information (NMI) behaves as a powerful, bounded, and interpretable measure of temporal dependence, highlighting periods of structural change such as the 2008 financial crisis and the COVID-19 shock. Our entropy-adjusted Value at Risk, information-theoretic diversification criterion, and NMI-based market efficiency test provide actionable tools for risk management and asset allocation. We interpret NMI as a quantitative diagnostic of the Efficient Market Hypothesis and demonstrate that information-theoretic methods offer superior regime detection compared to traditional autocorrelation- or volatility-based approaches. All theoretical results include rigorous proofs, and empirical findings are validated across multiple market regimes spanning 25 years of daily returns.

Keywords: Financial Information Theory, Entropy, Kullback-Leibler divergence, Transfer Entropy, Normalized Mutual Information (NMI), Equities (Indices)

Complexity vs Empirical Score

• Math Complexity: 8.5/10
• Empirical Rigor: 7.0/10
• Quadrant: Holy Grail
• Why: The paper presents advanced information theory with rigorous proofs and theorems (high math complexity), while also demonstrating empirical validation on S&P 500 data over 25 years with practical algorithms for estimation and application (high empirical rigor).

  flowchart TD
    A["Research Goal:<br>Develop Financial Information Theory<br>using information-theoretic measures"] --> B["Methodology: Derive & Prove<br>Entropy, KL Divergence, Mutual Info, NMI, Transfer Entropy"]
    B --> C["Data Input:<br>S&P 500 Daily Returns<br>2000-2025"]
    C --> D["Computational Processes:<br>Algorithmic Estimation of Metrics<br>& Sequential Application"]
    D --> E{"Key Findings & Outcomes"}
    E --> F["1. NMI for Regime Detection<br>Identifies 2008 & 2021 Structural Shifts"]
    E --> G["2. Entropy-Adjusted VaR<br>Superior Risk Management Tool"]
    E --> H["3. NMI Market Efficiency Test<br>Quantifies EMH violations<br>vs. Autocorrelation"]
    E --> I["4. Info-Theoretic Diversification<br>Optimized Portfolio Construction"]