Gain–Loss Asymmetry and Bimodal Structures in Multiscale Financial Returns: Disentangling Market Dynamics from Random Walks

This study presents a multi-scale statistical analysis of financial market dynamics by examining "Trend Returns"(TReturns)—logarithmic returns derived from uninterrupted directional price movements—across four majorglobal indices (DJIA, Nasdaq, Nikkei, and IPC). By contrasting empirical data against a Monte Carlo GeometricBrownian Motion (GBM) null model, we rigorously distinguish between mechanical artifacts and genuinestructural anomalies. We analyze Bimodality in TReturns: while daily returns (d = 1) exhibit unimodal noise,trends persisting for d ≥ 2 structurally separate into distinct positive and negative modes. While the GBMconfirms that this separation is largely a natural topology of trend persistence, empirical data reveal a distinctdivergence at longer durations (d > 6), where the negative mode extinguishes, transitioning back to a positivelyshifted unimodality—a behavior absent in the persistent bimodality of the random walk. Furthermore, weuncovered a fundamental Dual Asymmetry. While the Asymmetry of Frequency (preference for long uptrends)is partially consistent with positive drift, the Asymmetry of Character—quantified via Volatility Ratios (< 1)and the Wasserstein Distance—reveals that downtrends are structurally more volatile ("violent") and possessheavier tails than uptrends, a feature that the Gaussian control fundamentally fails to reproduce. At the macrolevel, this manifests as a universal negative skewness. These findings suggest that standard Gaussian modelsare structurally blind to the specific mechanics of market corrections, leading to systematic underestimation oftail risk.