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Beyond Rough Volatility: Decoupling Memory and Scaling via a Generalized Langevin Equation

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  • Andrey Itkin

Abstract

Borrowed from non-equilibrium statistical mechanics, the generalized Langevin equation (GLE) is imported as a framework for stochastic volatility to address the structural limitations of fractional Brownian motion (fBm), the standard engine of rough volatility. The fBm forces a single parameter to set two logically independent properties at once: how volatility scales and how it remembers. The GLE separates them using a memory kernel $K$, a potential $U$, and a noise covariance $C$. Memory becomes a measurable object, and an asymmetric potential supplies a lever on variance skew that the price-variance correlation cannot reach. Physical-measure tests on public datasets decisively reject two constrained corners of the class, a memoryless leverage effect and time-reversal symmetry, while the central rough scaling constraint is left identification-limited rather than refuted. The paper reports these limits honestly, and validation on industry-grade data remains a valuable direction. The risk-neutral construction and the joint SPX--VIX calibration will be developed in a companion paper.

Suggested Citation

  • Andrey Itkin, 2026. "Beyond Rough Volatility: Decoupling Memory and Scaling via a Generalized Langevin Equation," Papers 2609.20293, arXiv.org.
    • Handle: RePEc:arx:papers:2609.20293
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    References listed on IDEAS

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    2. Igor Halperin & Andrey Itkin, 2025. "Marketron Through the Looking Glass: From Equity Dynamics to Option Pricing in Incomplete Markets," Papers 2508.09863, arXiv.org, revised Aug 2025.
    3. Omar Euch & Masaaki Fukasawa & Mathieu Rosenbaum, 2018. "The microstructural foundations of leverage effect and rough volatility," Finance and Stochastics, Springer, vol. 22(2), pages 241-280, April.
    4. Adamantios Ntakaris & Martin Magris & Juho Kanniainen & Moncef Gabbouj & Alexandros Iosifidis, 2018. "Benchmark dataset for mid‐price forecasting of limit order book data with machine learning methods," Journal of Forecasting, John Wiley & Sons, Ltd., vol. 37(8), pages 852-866, December.
    5. Elisa Alòs & Kenichiro Shiraya, 2019. "Estimating the Hurst parameter from short term volatility swaps: a Malliavin calculus approach," Finance and Stochastics, Springer, vol. 23(2), pages 423-447, April.
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