Understanding Short-Term Implied Volatility Dynamics: A Model-Independent Approach Beyond Stochastic Volatility

Abstract

This paper examines the short-term asymptotic behavior of the implied volatility surface, focusing on the at-the-money (ATM) skew and curvature. Rather than committing to a specific stochastic differential equation, we adopt a distribution-based approach by imposing cumulant conditions on the log-return distribution. Under these weak assumptions, we derive a quadratic expansion of implied volatility as a function of moneyness for near-the-money options and asymptotic expressions for ATM skew and curvature as time to maturity approaches zero, treating the decay rates of the third and fourth cumulants as independent parameters and introducing a marginal-type classifier. These results highlight differences in ATM asymptotic properties across different types of log return distributions and yield a unified, model-independent characterization of short-term smile dynamics in terms of the scaling laws of the marginal cumulants, covering regular/rough stochastic volatility and distribution-based ones like scalable gamma/CGMY martingale models alike from simple moment information. We subsequently present a distribution-based calibration method that not only effectively validates the analytical approximations, but also exhibits strong interpretability and consistent performance, and discuss potential connections to model-independent path-dependent applications via martingale optimal transport. Overall, our findings provide model-independent analytical tools for evaluating model performance against market stylized features, accurately approximating short-term option prices, and performing robust calibration.