arXiv2026
Realised kurtosis measures intraday tail risk but compresses the shape of the return distribution into a single scalar, so trading days with similar kurtosis may differ in how probability mass is spread across the peak, the shoulders and the tails. We propose a geometric description of intraday return shape based on the modified Brownian motion model (MBMM), which deforms a Gaussian benchmark through a nonlinear feedback governed by a shape parameter and a distortion intensity. Our main contribution is a peak-shoulder-tail decomposition that measures where the fitted return density gains or loses probability mass relative to the Gaussian, complemented by two feedback landmarks that explain how the shape parameter generates these deformations. We estimate the model at the stock-day level using a bin-free simulation-based minimum-distance estimator that matches empirical and model-implied intraday quantiles, and we document its finite-sample reliability by Monte Carlo. Using regular session one-minute returns for ten liquid U.S. stocks from May 2021 to April 2026, we find that the MBMM provides a substantially better in-sample fit than geometric Brownian motion on every estimator-ready stock-day, and that days with closely matched realised kurtosis can display materially different peak-shoulder-tail geometries. Finally, we show that the estimated shape parameters behave as persistent daily state variables: they are strongly forecastable relative to a random-walk benchmark, though their predictability reflects persistence around a slowly moving level rather than rich autoregressive dynamics. The proposed decomposition offers a risk-shape measurement layer that complements realised kurtosis by describing not only how large intraday tail risk is, but how it is formed, giving investors and risk managers additional distributional information for risk monitoring and, potentially, prediction.