Volatility Modeling with Rough Paths: A Signature-Based Alternative to Classical Expansions

Abstract

We study two complementary methodologies for calibrating implied volatility surfaces: analytical approximations and data-driven models based on rough path theory. On the analytical side, we revisit a second-order asymptotic expansion for the Heston model, and we propose a new, VIX-based calibration scheme for the rough Bergomi model. Both methods yield highly accurate and computationally efficient calibration formulas when the underlying dynamics are well specified. In parallel, we develop a signature-based approach in which volatility is represented as a linear functional of the truncated signature of a primary stochastic process, providing a flexible and model-agnostic alternative. Our numerical experiments compare the two approaches across both Markovian and non-Markovian settings. In the Heston case, signature-based models achieve a level of accuracy comparable to analytical expansions. In the rough Bergomi setting, using a fractional Brownian motion as the primary process, the signature approach continues to perform strongly and in some cases improves upon the Markovian specification, reflecting its ability to capture more complex temporal dependencies. Overall, the results illustrate that analytical methods are highly effective when the model is correctly specified, while signature-based methods offer a robust and flexible framework for calibration across a wider range of volatility dynamics.