Is Persistent Entropy Simply a Volatility Proxy? Evidence from the WIG20 Index

Aim: This article investigates whether persistent homology and persistence entropy capture structural properties of financial time series beyond variance-based risk measures. Using data from the WIG20 index (2019–2024), the study examines whether topological descriptors reflect intrinsic geometric and temporal organization rather than merely volatility intensity. Methodology: Logarithmic returns are embedded using sliding-window delay coordinates and analysed with Vietoris–Rips persistent homology. Betti numbers, persistence diagrams and rolling 𝐻1 persistence entropy are computed. Relationships with classical risk diagnostics are evaluated using linear correlations, nonlinear dependence measures, regime comparisons and shuffle-based tests. Results: Persistence entropy shows weak association with volatility and second-moment risk. Stronger relationships appear with higher-order distributional characteristics such as skewness and kurtosis. Volatility-based regimes do not significantly separate entropy, whereas a structural split around the 2022 geopolitical shock reveals a significant increase, indicating a shift in return geometry. Shuffle experiments confirm dependence on temporal ordering. Implications: Persistence entropy captures structural and temporal organization of financial returns and may complement classical econometric risk measures. Originality/value: The study shows that persistent homology reflects structural organization of return dynamics rather than acting as a volatility proxy.