Shannon Entropy Analysis of Financial Time Series Volatility

Abstract

Volatility clustering in financial markets, in which large price changes tend to follow large changes of either sign, is
well documented but not consistently linked to a single quantitative measure that anticipates regime shifts before
they appear in realized volatility itself. We computed rolling-window Shannon entropy and permutation entropy of
daily log-return series for 40 large-capitalization equities and 6 broad equity indices over a 22-year sample period
(2003-2024, approximately 5,540 trading days per series), using symbolic discretization into 8 amplitude bins and
embedding dimension 5 for the permutation-entropy estimator, computed over rolling 250-day windows. Mean
normalized Shannon entropy across all 46 series was 0.912 (SD = 0.041) during low-volatility regimes (bottom
tercile of realized volatility) and fell to 0.847 (SD = 0.058) during high-volatility regimes (top tercile), a reduction of
7.1% (Welch's t = 18.4, p < 0.001, n = 46 series), reflecting the more predictable, directionally persistent character
of returns during volatility clusters relative to the comparatively uniform distribution of returns in calm periods.
Entropy declines preceded peaks in realized volatility by a median of 6 trading days across the 46 series
(interquartile range 3-11 days), and a threshold-crossing early-warning rule based on a one-standard-deviation
entropy drop achieved 68% sensitivity and 74% specificity for anticipating volatility regime shifts identified
independently via a two-state Markov-switching model, evaluated by 10-fold cross-validation across
non-overlapping historical windows. Permutation entropy showed a weaker but still significant regime discrimination
(5.3% reduction, high- versus low-volatility regimes, p < 0.001) and correlated with Shannon entropy across series
at r = 0.79 (p < 0.001), indicating substantial but incomplete redundancy between the two measures. Entropy-based
regime discrimination was strongest for financial-sector and technology-sector equities (9.4% and 8.6% mean
entropy reduction, respectively) and weakest for consumer-staples equities (4.2%), a sector difference that
persisted after controlling for mean realized volatility level. These results indicate that rolling Shannon entropy
carries exploitable early-warning information about impending volatility regime shifts beyond what realized volatility
alone provides, with practical implications for risk-management systems that monitor entropy alongside
conventional volatility estimators.