IDEAS home Printed from https://ideas.repec.org/p/arx/papers/2604.22528.html

Malliavin calculus for signatures with applications to finance

Author

Listed:
  • Eduardo Abi Jaber
  • Cl'ement Rey
  • Dimitri Sotnikov

Abstract

Malliavin calculus is a powerful and general framework for the analysis of square-integrable random variables, but it often suffers from a lack of tractability and explicit representations. To address this limitation, we focus on a subclass of random variables given by finite linear combinations of time-extended Brownian motion signatures. The class remains rich due to the universal approximation properties of signatures. Leveraging the algebraic structure of signatures, we first derive explicit formulas for the Malliavin derivative of signatures of continuous It\^o processes. As a consequence, we obtain closed-form expressions for the Clark--Ocone representation, the Ornstein--Uhlenbeck semigroup and its generator, as well as the integration-by-parts formula within the class of Brownian signature variables. These results provide purely algebraic formulations of the classical operators of Malliavin calculus. As an application, we compute Greeks for general path-dependent options under signature volatility models, and numerically compare different choices of Malliavin weights.

Suggested Citation

  • Eduardo Abi Jaber & Cl'ement Rey & Dimitri Sotnikov, 2026. "Malliavin calculus for signatures with applications to finance," Papers 2604.22528, arXiv.org.
    • Handle: RePEc:arx:papers:2604.22528
    as

    Download full text from publisher

    File URL: https://arxiv.org/pdf/2604.22528
    File Function: Latest version
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Daniel Levin & Terry Lyons & Hao Ni, 2013. "Learning from the past, predicting the statistics for the future, learning an evolving system," Papers 1309.0260, arXiv.org, revised Mar 2016.
    2. Davis, Mark H.A. & Johansson, Martin P., 2006. "Malliavin Monte Carlo Greeks for jump diffusions," Stochastic Processes and their Applications, Elsevier, vol. 116(1), pages 101-129, January.
    3. Bruno Dupire, 2019. "Functional Itô calculus," Quantitative Finance, Taylor & Francis Journals, vol. 19(5), pages 721-729, May.
    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. Zhou Fang, 2023. "Continuous-Time Path-Dependent Exploratory Mean-Variance Portfolio Construction," Papers 2303.02298, arXiv.org.
    2. Hans Bühler & Blanka Horvath & Terry Lyons & Imanol Perez Arribas & Ben Wood, 2026. "A Data-Driven Market Simulator for Small Data Environments," Springer Books, in: Dan Crisan & Ilya Chevyrev & Thomas Cass & James Foster & Christian Litterer & Cristopher Salvi (ed.), Stochastic Analysis and Applications 2025, pages 273-310, Springer.
    3. N. Hilber & N. Reich & C. Schwab & C. Winter, 2009. "Numerical methods for Lévy processes," Finance and Stochastics, Springer, vol. 13(4), pages 471-500, September.
    4. El-Khatib, Youssef & Abdulnasser, Hatemi-J, 2011. "On the calculation of price sensitivities with jump-diffusion structure," MPRA Paper 30596, University Library of Munich, Germany.
    5. Anselm Hudde & Ludger Rüschendorf, 2023. "European and Asian Greeks for Exponential Lévy Processes," Methodology and Computing in Applied Probability, Springer, vol. 25(1), pages 1-24, March.
    6. Xin Guo & Binnan Wang & Ruixun Zhang & Chaoyi Zhao, 2025. "On Consistency of Signature Using Lasso," Operations Research, INFORMS, vol. 73(5), pages 2530-2549, September.
    7. Fard, Farzad Alavi & Siu, Tak Kuen, 2013. "Pricing participating products with Markov-modulated jump–diffusion process: An efficient numerical PIDE approach," Insurance: Mathematics and Economics, Elsevier, vol. 53(3), pages 712-721.
    8. Hyungbin Park, 2018. "Sensitivity analysis of long-term cash flows," Finance and Stochastics, Springer, vol. 22(4), pages 773-825, October.
    9. Masafumi Hayashi, 2010. "Coefficients of Asymptotic Expansions of SDE with Jumps," Asia-Pacific Financial Markets, Springer;Japanese Association of Financial Economics and Engineering, vol. 17(4), pages 373-389, December.
    10. Cont, Rama & Lu, Yi, 2016. "Weak approximation of martingale representations," Stochastic Processes and their Applications, Elsevier, vol. 126(3), pages 857-882.
    11. Guy P. Nason & Ben Powell & Duncan Elliott & Paul A. Smith, 2017. "Should we sample a time series more frequently?: decision support via multirate spectrum estimation," Journal of the Royal Statistical Society Series A, Royal Statistical Society, vol. 180(2), pages 353-407, February.
    12. El-Khatib, Youssef & Goutte, Stephane & Makumbe, Zororo S. & Vives, Josep, 2023. "A hybrid stochastic volatility model in a Lévy market," International Review of Economics & Finance, Elsevier, vol. 85(C), pages 220-235.
    13. Blanka Horvath & Josef Teichmann & Zan Zuric, 2021. "Deep Hedging under Rough Volatility," Papers 2102.01962, arXiv.org.
    14. 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.
    15. Eduardo Abi Jaber & Louis-Amand Gérard, 2025. "Hedging with memory: shallow and deep learning with signatures," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) hal-05197836, HAL.
    16. 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.
    17. Chen, Nan & Glasserman, Paul, 2007. "Malliavin Greeks without Malliavin calculus," Stochastic Processes and their Applications, Elsevier, vol. 117(11), pages 1689-1723, November.
    18. Brian Huge & Antoine Savine, 2020. "Differential Machine Learning," Papers 2005.02347, arXiv.org, revised Sep 2020.
    19. Eduardo Abi Jaber & Louis-Amand G'erard, 2025. "Hedging with memory: shallow and deep learning with signatures," Papers 2508.02759, arXiv.org.
    20. Atsushi Takeuchi, 2010. "Bismut–Elworthy–Li-Type Formulae for Stochastic Differential Equations with Jumps," Journal of Theoretical Probability, Springer, vol. 23(2), pages 576-604, June.

    More about this item

    NEP fields

    This paper has been announced in the following NEP Reports:

    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:2604.22528. 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.