IDEAS home Printed from https://ideas.repec.org/p/arx/papers/2509.14529.html

Unbiased Rough Integrators and No Free Lunch in Rough-Path-Based Market Models

Author

Listed:
  • Tomoyuki Ichiba
  • Qijin Shi

Abstract

Built to generalise classical stochastic calculus, rough path theory provides a natural and pathwise framework to model continuous non-semimartingale assets. This paper investigates the capacity of this framework to support frictionless continuous No-Free-Lunch markets \`a la Kreps-Yan. We establish a "Rough Kreps-Yan" theorem, which links a No Controlled Free Lunch (NCFL) condition to the unbiasedness of the driver of the price process as a rough integrator. The central work of this paper is a classification of these unbiased rough integrators with respect to different classes of controlled paths as integrands, under some assumptions. As the admissible strategies are enlarged from Markovian-type portfolios to signature-type and adaptedly scaled signature-type portfolios, the admissible random rough paths collapse first to Gaussian-Hermite rough paths, and ultimately to the It\^o rough path lift of a standard Brownian motion, up to a time change. Notably, simple strategies do not appear in the theory. This implies that within our framework, continuous frictionless markets based on rough path theory are inevitably constrained to the classical semimartingale paradigm, clarifying the limits of this approach. Our framework covers $\alpha-$H\"older continuous rough paths for $\alpha>0$ arbitrarily small in the tensor algebra setting.

Suggested Citation

  • Tomoyuki Ichiba & Qijin Shi, 2025. "Unbiased Rough Integrators and No Free Lunch in Rough-Path-Based Market Models," Papers 2509.14529, arXiv.org, revised May 2026.
    • Handle: RePEc:arx:papers:2509.14529
    as

    Download full text from publisher

    File URL: https://arxiv.org/pdf/2509.14529
    File Function: Latest version
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Eduardo Abi Jaber & Louis-Amand G'erard, 2024. "Signature volatility models: pricing and hedging with Fourier," Papers 2402.01820, arXiv.org, revised Jun 2025.
    2. Robert J. Elliott & John Van Der Hoek, 2003. "A General Fractional White Noise Theory And Applications To Finance," Mathematical Finance, Wiley Blackwell, vol. 13(2), pages 301-330, April.
    3. Christian Bayer & Peter Friz & Jim Gatheral, 2016. "Pricing under rough volatility," Quantitative Finance, Taylor & Francis Journals, vol. 16(6), pages 887-904, June.
    4. Eduardo Abi Jaber & Louis-Amand Gérard, 2025. "Signature volatility models: pricing and hedging with Fourier," Post-Print hal-04435238, HAL.
    5. Bruno Dupire, 2019. "Functional Itô calculus," Quantitative Finance, Taylor & Francis Journals, vol. 19(5), pages 721-729, May.
    6. Andrew L. Allan & Christa Cuchiero & Chong Liu & David J. Prömel, 2023. "Model‐free portfolio theory: A rough path approach," Mathematical Finance, Wiley Blackwell, vol. 33(3), pages 709-765, July.
    7. Christa Cuchiero & Francesca Primavera & Sara Svaluto-Ferro, 2025. "Universal approximation theorems for continuous functions of càdlàg paths and Lévy-type signature models," Finance and Stochastics, Springer, vol. 29(2), pages 289-342, April.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Jingtang Ma & Xianglin Wu & Wenyuan Li, 2025. "Option pricing under non-Markovian stochastic volatility models: A deep signature approach," Papers 2508.15237, arXiv.org, revised May 2026.
    2. Pere Diaz-Lozano & Thomas K. Kloster, 2026. "A Wiener Chaos Approach to Martingale Modelling and Implied Volatility Calibration," Papers 2602.16232, arXiv.org.
    3. Eduardo Abi Jaber & Donatien Hainaut & Edouard Motte, 2025. "The Volterra Stein-Stein model with stochastic interest rates," Papers 2503.01716, arXiv.org, revised Jul 2025.
    4. Akmal Xodarev, 2026. "On the Structural Foundations of Signature Volatility Models: Existence, Arbitrage, Completeness, and the Hedging-Error Decomposition," Papers 2605.17142, arXiv.org.
    5. Mihriban Ceylan & Anna P. Kwossek & David J. Promel, 2026. "Universal approximation with signatures of non-geometric rough paths," Papers 2602.05898, arXiv.org.
    6. Ofelia Bonesini & Antoine Jacquier & Alexandre Pannier, 2023. "Rough volatility, path-dependent PDEs and weak rates of convergence," Papers 2304.03042, arXiv.org, revised May 2026.
    7. Cuchiero, Christa & Primavera, Francesca & Svaluto-Ferro, Sara, 2026. "Holomorphic jump-diffusions," Stochastic Processes and their Applications, Elsevier, vol. 191(C).
    8. Alexandre Pannier, 2023. "Path-dependent PDEs for volatility derivatives," Papers 2311.08289, arXiv.org, revised Jul 2025.
    9. Christian Bayer & Luca Pelizzari & John Schoenmakers, 2023. "Primal and dual optimal stopping with signatures," Papers 2312.03444, arXiv.org, revised Feb 2025.
    10. Yuecai Han & Xudong Zheng, 2022. "Approximate Pricing of Derivatives Under Fractional Stochastic Volatility Model," Papers 2210.15453, arXiv.org.
    11. Bingyan Han & Hoi Ying Wong, 2019. "Time-inconsistency with rough volatility," Papers 1907.11378, arXiv.org, revised Dec 2021.
    12. Ioannis Gasteratos & Alexandre Pannier, 2025. "Kolmogorov equations for stochastic Volterra processes with singular kernels," Papers 2509.21608, arXiv.org.
    13. Qi Feng & Man Luo & Zhaoyu Zhang, 2021. "Deep Signature FBSDE Algorithm," Papers 2108.10504, arXiv.org, revised Aug 2022.
    14. Zhou Fang, 2023. "Continuous-Time Path-Dependent Exploratory Mean-Variance Portfolio Construction," Papers 2303.02298, arXiv.org.
    15. Christian Bayer & Peter K. Friz & Paul Gassiat & Jorg Martin & Benjamin Stemper, 2020. "A regularity structure for rough volatility," Mathematical Finance, Wiley Blackwell, vol. 30(3), pages 782-832, July.
    16. Eduardo Abi Jaber, 2022. "The characteristic function of Gaussian stochastic volatility models: an analytic expression," Working Papers hal-02946146, HAL.
    17. Eduardo Abi Jaber, 2022. "The characteristic function of Gaussian stochastic volatility models: an analytic expression," Finance and Stochastics, Springer, vol. 26(4), pages 733-769, October.
    18. Stefano De Marco, 2020. "On the harmonic mean representation of the implied volatility," Papers 2007.03585, arXiv.org.
    19. Changqing Teng & Guanglian Li, 2025. "Efficient Simulation and Calibration of the Rough Bergomi Model via Wasserstein Distance," Papers 2512.00448, arXiv.org, revised Apr 2026.
    20. Mikkel Bennedsen & Asger Lunde & Mikko S. Pakkanen, 2017. "Decoupling the short- and long-term behavior of stochastic volatility," CREATES Research Papers 2017-26, Department of Economics and Business Economics, Aarhus University.

    More about this item

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:arx:papers:2509.14529. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: arXiv administrators (email available below). General contact details of provider: https://arxiv.org/ .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.