An Application of Permutation Entropy and Multiscale Complexity in the Gold Market

This study compares the ordinal complexity of daily gold futures log returns rt , overlapping 20-day rolling variance RV20, and log-prices lnPt using permutation entropy (PE) and multiscale permutation entropy (MPE). The sample comprises 4,104 unadjusted closing prices for gold futures (GC=F) from Yahoo Finance, covering 4 January 2010 to 30 April 2026. Full-sample normalized return PE is 0.9963 at embedding order m = 5, close to the theoretical maximum and compatible with weak-form informational efficiency at the daily frequency. High PE describes ordinal randomness, not the absence of forecastability or profitable trading opportunities. At m = 5, PE is lower for log-prices (0.9219) and RV20 (0.9082), indicating stronger ordinal structure. Returns contain all 120 possible patterns for m = 5, and the chi-squared test does not reject uniformity (χ2 = 142.1, d f = 119, p = 0.073). Across scales τ = 1, . . . ,10, the Complexity Index (CI) is 9.826 for returns, 8.695 for RV20, and 8.476 for log-prices. The return CI lies within its 95% reference interval from seeded i.i.d. surrogates; the RV20 and log-price CIs fall below their corresponding intervals. The lowest scale-5 return PE occurs in 2018–2021 (0.9711), but remains within its 95% i.i.d.-surrogate interval and does not establish a statistically distinct crisis regime. The comparison distinguishes near-random daily return orderings from multiscale structure in volatility-related and price-level transformations.