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Null-Validated Topological Signatures of Financial Market Dynamics

Samuel W. Akingbade

TL;DR

The paper tackles the challenge of capturing complex temporal organization in financial markets beyond volatility and linear correlations by introducing a null-validated topological framework. It computes the $L^1$ norm of persistence landscapes from sliding-window delay embeddings of Bitcoin daily log returns and compares it to latent stochastic volatility estimates, revealing non-stationary coupling and bursts of topological structure even during calm periods. Surrogate null models (shuffle and FFT phase-randomized) demonstrate that the observed topological signal arises from temporal ordering and nonlinear temporal dependencies, not merely marginal distributions or linear correlations. The approach provides a principled, statistically validated descriptor of market dynamics that complements volatility models and has potential applicability to other assets and cross-asset systems.

Abstract

Financial markets exhibit temporal organization that is not fully captured by volatility measures or linear correlation structure. We study a null validated topological approach for quantifying market complexity and apply it to Bitcoin daily log returns. The analysis uses the $L^1$ norm of persistence landscapes computed from sliding-window delay embeddings. This quantity shows strong co-movement with stochastic volatility during periods of market stress, but remains intermittently elevated during low volatility regimes, indicating dynamical structure beyond fluctuation scale. Rolling correlation analysis reveals that the dependence between geometry and volatility is not stationary. Surrogate based null models provide statistical validation of these observations. Rejection of shuffle surrogates rules out explanations based on marginal distributions alone, while departures from phase randomized surrogates indicate sensitivity to nonlinear and phase dependent temporal organization beyond linear correlations. These results demonstrate that persistence landscape norms provide complementary information about market dynamics across market conditions.

Null-Validated Topological Signatures of Financial Market Dynamics

TL;DR

The paper tackles the challenge of capturing complex temporal organization in financial markets beyond volatility and linear correlations by introducing a null-validated topological framework. It computes the norm of persistence landscapes from sliding-window delay embeddings of Bitcoin daily log returns and compares it to latent stochastic volatility estimates, revealing non-stationary coupling and bursts of topological structure even during calm periods. Surrogate null models (shuffle and FFT phase-randomized) demonstrate that the observed topological signal arises from temporal ordering and nonlinear temporal dependencies, not merely marginal distributions or linear correlations. The approach provides a principled, statistically validated descriptor of market dynamics that complements volatility models and has potential applicability to other assets and cross-asset systems.

Abstract

Financial markets exhibit temporal organization that is not fully captured by volatility measures or linear correlation structure. We study a null validated topological approach for quantifying market complexity and apply it to Bitcoin daily log returns. The analysis uses the norm of persistence landscapes computed from sliding-window delay embeddings. This quantity shows strong co-movement with stochastic volatility during periods of market stress, but remains intermittently elevated during low volatility regimes, indicating dynamical structure beyond fluctuation scale. Rolling correlation analysis reveals that the dependence between geometry and volatility is not stationary. Surrogate based null models provide statistical validation of these observations. Rejection of shuffle surrogates rules out explanations based on marginal distributions alone, while departures from phase randomized surrogates indicate sensitivity to nonlinear and phase dependent temporal organization beyond linear correlations. These results demonstrate that persistence landscape norms provide complementary information about market dynamics across market conditions.
Paper Structure (27 sections, 26 equations, 9 figures)

This paper contains 27 sections, 26 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Persistent homology
  4. Persistent homology
  5. Persistence diagrams
  6. Persistence landscapes
  7. Application to time series
  8. Choice of embedding dimension and delay.
  9. Topological summaries of Bitcoin log return dynamics
  10. Distribution of L1 norms across sentiment regimes
  11. Stochastic volatility modeling of Bitcoin log returns
  12. Stochastic volatility model
  13. State (process) equation
  14. Observation (measurement) equation
  15. Inference and parameter estimation
  16. ...and 12 more sections

Figures (9)

  • Figure 1: Bitcoin daily log returns.
  • Figure 2: L1 norm of the persistence landscape of Bitcoin log returns.
  • Figure 3: Distribution of $L^1$ norms of the persistence landscape for Bitcoin log returns across Fear & Greed sentiment regimes.
  • Figure 4: Filtered conditional volatility estimate $\hat{\sigma}_t$ obtained from the stochastic volatility model of Bitcoin log returns.
  • Figure 5: Standardized comparison of the $L^1$ norm of persistence landscape (blue) and filtered stochastic volatility (red).
  • ...and 4 more figures