IDEAS home Printed from https://ideas.repec.org/p/arx/papers/1812.03526.html
   My bibliography  Save this paper

Path Dependent Optimal Transport and Model Calibration on Exotic Derivatives

Author

Listed:
  • Ivan Guo
  • Gregoire Loeper

Abstract

In this paper, we introduce and develop the theory of semimartingale optimal transport in a path dependent setting. Instead of the classical constraints on marginal distributions, we consider a general framework of path dependent constraints. Duality results are established, representing the solution in terms of path dependent partial differential equations (PPDEs). Moreover, we provide a dimension reduction result based on the new notion of "semifiltrations", which identifies appropriate Markovian state variables based on the constraints and the cost function. Our technique is then applied to the exact calibration of volatility models to the prices of general path dependent derivatives.

Suggested Citation

  • Ivan Guo & Gregoire Loeper, 2018. "Path Dependent Optimal Transport and Model Calibration on Exotic Derivatives," Papers 1812.03526, arXiv.org, revised Sep 2020.
    • Handle: RePEc:arx:papers:1812.03526
    as

    Download full text from publisher

    File URL: http://arxiv.org/pdf/1812.03526
    File Function: Latest version
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Julio Backhoff-Veraguas & Mathias Beiglbock & Martin Huesmann & Sigrid Kallblad, 2017. "Martingale Benamou--Brenier: a probabilistic perspective," Papers 1708.04869, arXiv.org, revised Jan 2019.
    2. Mikami, Toshio & Thieullen, Michèle, 2006. "Duality theorem for the stochastic optimal control problem," Stochastic Processes and their Applications, Elsevier, vol. 116(12), pages 1815-1835, December.
    3. Frédéric Abergel & Rémi Tachet, 2010. "A nonlinear partial integro-differential equation from mathematical finance," Post-Print hal-00611962, HAL.
    4. Buckdahn, Rainer & Ma, Jin & Zhang, Jianfeng, 2015. "Pathwise Taylor expansions for random fields on multiple dimensional paths," Stochastic Processes and their Applications, Elsevier, vol. 125(7), pages 2820-2855.
    5. Marco Avellaneda & Craig Friedman & Richard Holmes & Dominick Samperi, 1997. "Calibrating volatility surfaces via relative-entropy minimization," Applied Mathematical Finance, Taylor & Francis Journals, vol. 4(1), pages 37-64.
    6. Pierre Henry-Labordère & Nizar Touzi, 2016. "An explicit martingale version of the one-dimensional Brenier theorem," Finance and Stochastics, Springer, vol. 20(3), pages 635-668, July.
    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. Sergey Badikov & Antoine Jacquier & Daphne Qing Liu & Patrick Roome, 2017. "No-arbitrage bounds for the forward smile given marginals," Quantitative Finance, Taylor & Francis Journals, vol. 17(8), pages 1243-1256, August.
    2. Samuel Daudin, 2022. "Optimal Control of Diffusion Processes with Terminal Constraint in Law," Journal of Optimization Theory and Applications, Springer, vol. 195(1), pages 1-41, October.
    3. Lassalle, Rémi & Cruzeiro, Ana Bela, 2019. "An intrinsic calculus of variations for functionals of laws of semi-martingales," Stochastic Processes and their Applications, Elsevier, vol. 129(10), pages 3585-3618.
    4. A. Monteiro & R. Tütüncü & L. Vicente, 2011. "Estimation of risk-neutral density surfaces," Computational Management Science, Springer, vol. 8(4), pages 387-414, November.
    5. Sergey Badikov & Mark H. A. Davis & Antoine Jacquier, 2018. "Perturbation analysis of sub/super hedging problems," Papers 1806.03543, arXiv.org, revised May 2021.
    6. Carr, Peter & Geman, Helyette & Madan, Dilip B., 2001. "Pricing and hedging in incomplete markets," Journal of Financial Economics, Elsevier, vol. 62(1), pages 131-167, October.
    7. Martin Larsson & Shukun Long, 2024. "Markovian projections for It\^o semimartingales with jumps," Papers 2403.15980, arXiv.org.
    8. Jiangze Du & Shaojie Lai & Kin Keung Lai & Shifei Zhou, 2021. "A novel term structure stochastic model with adaptive correlation for trend analysis," International Journal of Finance & Economics, John Wiley & Sons, Ltd., vol. 26(4), pages 5485-5498, October.
    9. Tapiero, Oren J., 2013. "A maximum (non-extensive) entropy approach to equity options bid–ask spread," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 392(14), pages 3051-3060.
    10. David Hobson & Dominykas Norgilas, 2019. "Robust bounds for the American put," Finance and Stochastics, Springer, vol. 23(2), pages 359-395, April.
    11. Thomas Breuer & Martin Summer, 2013. "Stress Test Robustness: Recent Advances and Open Problems," Financial Stability Report, Oesterreichische Nationalbank (Austrian Central Bank), issue 25, pages 74-86.
    12. Ivan Guo & Gregoire Loeper & Jan Obloj & Shiyi Wang, 2021. "Optimal transport for model calibration," Papers 2107.01978, arXiv.org.
    13. Ariel Neufeld & Julian Sester, 2021. "On the stability of the martingale optimal transport problem: A set-valued map approach," Papers 2102.02718, arXiv.org, revised Apr 2021.
    14. Fairouz Tchier & Ioannis Dassios & Ferdous Tawfiq & Lakhdar Ragoub, 2021. "On the Approximate Solution of Partial Integro-Differential Equations Using the Pseudospectral Method Based on Chebyshev Cardinal Functions," Mathematics, MDPI, vol. 9(3), pages 1-14, February.
    15. Erhan Bayraktar & Shuoqing Deng & Dominykas Norgilas, 2023. "Supermartingale Brenier’s Theorem with Full-Marginal Constraint," World Scientific Book Chapters, in: Robert A Jarrow & Dilip B Madan (ed.), Peter Carr Gedenkschrift Research Advances in Mathematical Finance, chapter 17, pages 569-636, World Scientific Publishing Co. Pte. Ltd..
    16. Neufeld, Ariel & Sester, Julian, 2021. "On the stability of the martingale optimal transport problem: A set-valued map approach," Statistics & Probability Letters, Elsevier, vol. 176(C).
    17. Marcel Nutz & Florian Stebegg, 2016. "Canonical Supermartingale Couplings," Papers 1609.02867, arXiv.org, revised Nov 2017.
    18. Ilia Bouchouev & Victor Isakov & Nicolas Valdivia, 2002. "Recovery of volatility coefficient by linearization," Quantitative Finance, Taylor & Francis Journals, vol. 2(4), pages 257-263.
    19. Pierre Henry-Labordère, 2019. "(Martingale) Optimal Transport And Anomaly Detection With Neural Networks: A Primal-Dual Algorithm," Working Papers hal-02095222, HAL.
    20. Anton Kolotilin & Roberto Corrao & Alexander Wolitzky, 2022. "Persuasion with Non-Linear Preferences," Papers 2206.09164, arXiv.org, revised Aug 2022.

    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:1812.03526. 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: http://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.