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

Paper: arXiv 2507.23392

Authors: Elisa Alòs, Òscar Burés, Rafael de Santiago, Josep Vives

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.

Complexity vs Empirical Score

• Math Complexity: 8.0/10
• Empirical Rigor: 6.0/10
• Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: The paper presents rigorous mathematical methodology combining path signatures with classical stochastic volatility asymptotic expansions. It offers novel contributions in VIX-based calibration for rough Bergomi dynamics while providing solid numerical benchmarks across Markovian and non-Markovian settings.

Research Flowchart

  flowchart TD
    A[Research Goal: Calibrate Implied Volatility via Analytical vs Signature-Based Models] --> B[Input Data & Processes: Options, Heston, Rough Bergomi & fBM Path Data]
    
    B --> C[Analytical Branch: Revisit Heston 2nd-Order Expansion & New Rough Bergomi VIX Calibration]
    B --> D[Data-Driven Branch: Signature Approach via Linear Functional of Truncated Signatures]
    
    C --> E[Numerical Calibration Experiments & Model Comparison]
    D --> E
    
    E --> F[Outcomes: Analytical holds high accuracy when well-specified; Signature model offers robust, model-agnostic flexibility across Markovian & non-Markovian dynamics]

Cited References (50)

1. Calibration of stochastic volatility models via second-order approximation: The Heston case (2015)
2. On the short-time behavior of the implied volatility for jump-diffusion models with stochastic volatility (2007)
3. On Smile Properties of Volatility Derivatives: Understanding the VIX Skew (2022) — link
4. On the Implied Volatility of European and Asian Call Options under the Stochastic Volatility Bachelier Model — → 2308.15341
5. Long memory in continuous-time stochastic volatility models (1998)
6. Short-time at-the-money skew and rough fractional volatility (2017)
7. Approximation and calibration of short-term implied volatilities under jump-diffusion stochastic volatility (2007)
8. System control and rough paths (2002)

4 of 50 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.