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Functional Itô calculus

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  • Bruno Dupire

Abstract

We extend some results of the Itô calculus to functionals of the current path of a process to reflect the fact that often the impact of randomness is cumulative and depends on the history of the process, not merely on its current value. We express the differential of the functional in terms of adequately defined partial derivatives to obtain an Itô formula. We develop an extension of the Feynman-Kac formula to the functional case and an explicit expression of the integrand in the Martingale Representation Theorem. We establish that under certain conditions, even path dependent options prices satisfy a partial differential equation in a local sense. We exploit this fact to find an expression of the price difference between two models and compute variational derivatives with respect to the volatility surface.

Suggested Citation

  • Bruno Dupire, 2019. "Functional Itô calculus," Quantitative Finance, Taylor & Francis Journals, vol. 19(5), pages 721-729, May.
    • Handle: RePEc:taf:quantf:v:19:y:2019:i:5:p:721-729
    DOI: 10.1080/14697688.2019.1575974
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    Citations

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    Cited by:

    1. Zhou Fang, 2023. "Continuous-Time Path-Dependent Exploratory Mean-Variance Portfolio Construction," Papers 2303.02298, arXiv.org.
    2. Georgii Riabov & Aleh Tsyvinski, 2021. "Policy with stochastic hysteresis," Papers 2104.10225, arXiv.org.
    3. Andrew L. Allan & Christa Cuchiero & Chong Liu & David J. Promel, 2021. "Model-free Portfolio Theory: A Rough Path Approach," Papers 2109.01843, arXiv.org, revised Oct 2022.
    4. Bingyan Han & Hoi Ying Wong, 2019. "Time-inconsistency with rough volatility," Papers 1907.11378, arXiv.org, revised Dec 2021.
    5. Christian Bayer & Paul Hager & Sebastian Riedel & John Schoenmakers, 2021. "Optimal stopping with signatures," Papers 2105.00778, arXiv.org.
    6. Christa Cuchiero & Janka Moller, 2023. "Signature Methods in Stochastic Portfolio Theory," Papers 2310.02322, arXiv.org, revised Oct 2024.
    7. Andrew L. Allan & Chong Liu & David J. Promel, 2021. "A C\`adl\`ag Rough Path Foundation for Robust Finance," Papers 2109.04225, arXiv.org, revised May 2023.
    8. Blanka Horvath & Josef Teichmann & Zan Zuric, 2021. "Deep Hedging under Rough Volatility," Papers 2102.01962, arXiv.org.
    9. Bruno Bouchard & Xiaolu Tan, 2021. "A quasi-sure optional decomposition and super-hedging result on the Skorokhod space," Finance and Stochastics, Springer, vol. 25(3), pages 505-528, July.
    10. Nam, Kihun, 2021. "Locally Lipschitz BSDE driven by a continuous martingale a path-derivative approach," Stochastic Processes and their Applications, Elsevier, vol. 141(C), pages 376-411.
    11. Christian Bayer & Luca Pelizzari & John Schoenmakers, 2023. "Primal and dual optimal stopping with signatures," Papers 2312.03444, arXiv.org.
    12. 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.
    13. Ofelia Bonesini & Antoine Jacquier & Alexandre Pannier, 2023. "Rough volatility, path-dependent PDEs and weak rates of convergence," Papers 2304.03042, arXiv.org.
    14. Li, Xiaoyue & Mao, Xuerong & Song, Guoting, 2024. "An explicit approximation for super-linear stochastic functional differential equations," Stochastic Processes and their Applications, Elsevier, vol. 169(C).
    15. Alexandre Pannier, 2023. "Path-dependent PDEs for volatility derivatives," Papers 2311.08289, arXiv.org, revised Jan 2024.
    16. Kiseop Lee & Seongje Lim & Hyungbin Park, 2022. "Option pricing under path-dependent stock models," Papers 2211.10953, arXiv.org, revised Aug 2023.
    17. Brian Huge & Antoine Savine, 2020. "Differential Machine Learning," Papers 2005.02347, arXiv.org, revised Sep 2020.
    18. Cont, Rama & Kalinin, Alexander, 2020. "On the support of solutions to stochastic differential equations with path-dependent coefficients," Stochastic Processes and their Applications, Elsevier, vol. 130(5), pages 2639-2674.
    19. Qi Feng & Man Luo & Zhaoyu Zhang, 2021. "Deep Signature FBSDE Algorithm," Papers 2108.10504, arXiv.org, revised Aug 2022.
    20. Shreya Bose & Ibrahim Ekren, 2021. "Multidimensional Kyle-Back model with a risk averse informed trader," Papers 2111.01957, arXiv.org.
    21. Henry Chiu & Rama Cont, 2023. "A model‐free approach to continuous‐time finance," Mathematical Finance, Wiley Blackwell, vol. 33(2), pages 257-273, April.
    22. Anton Plaksin, 2020. "Minimax and Viscosity Solutions of Hamilton–Jacobi–Bellman Equations for Time-Delay Systems," Journal of Optimization Theory and Applications, Springer, vol. 187(1), pages 22-42, October.

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