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Calibration of local‐stochastic volatility models by optimal transport

Author

Listed:
  • Ivan Guo
  • Grégoire Loeper
  • Shiyi Wang

Abstract

In this paper, we study a semi‐martingale optimal transport problem and its application to the calibration of local‐stochastic volatility (LSV) models. Rather than considering the classical constraints on marginal distributions at initial and final time, we optimize our cost function given the prices of a finite number of European options. We formulate the problem as a convex optimization problem, for which we provide a PDE formulation along with its dual counterpart. Then we solve numerically the dual problem, which involves a fully non‐linear Hamilton–Jacobi–Bellman equation. The method is tested by calibrating a Heston‐like LSV model with simulated data and foreign exchange market data.

Suggested Citation

  • Ivan Guo & Grégoire Loeper & Shiyi Wang, 2022. "Calibration of local‐stochastic volatility models by optimal transport," Mathematical Finance, Wiley Blackwell, vol. 32(1), pages 46-77, January.
    • Handle: RePEc:bla:mathfi:v:32:y:2022:i:1:p:46-77
    DOI: 10.1111/mafi.12335
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    References listed on IDEAS

    as
    1. Ivan Guo & Gregoire Loeper & Jan Obloj & Shiyi Wang, 2021. "Optimal transport for model calibration," Papers 2107.01978, arXiv.org.
    2. Christa Cuchiero & Wahid Khosrawi & Josef Teichmann, 2020. "A Generative Adversarial Network Approach to Calibration of Local Stochastic Volatility Models," Risks, MDPI, vol. 8(4), pages 1-31, September.
    3. Bruno Bouchard & G Loeper & Y Zou, 2017. "Hedging of covered options with linear market impact and gamma constraint," Post-Print hal-01247523, HAL.
    4. Heston, Steven L, 1993. "A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options," The Review of Financial Studies, Society for Financial Studies, vol. 6(2), pages 327-343.
    5. Christa Cuchiero & Wahid Khosrawi & Josef Teichmann, 2020. "A generative adversarial network approach to calibration of local stochastic volatility models," Papers 2005.02505, arXiv.org, revised Sep 2020.
    6. Frédéric Abergel & Rémi Tachet, 2010. "A nonlinear partial integro-differential equation from mathematical finance," Post-Print hal-00611962, HAL.
    7. Bernd Engelmann & Frank Koster & Daniel Oeltz, 2021. "Calibration of the Heston stochastic local volatility model: A finite volume scheme," International Journal of Financial Engineering (IJFE), World Scientific Publishing Co. Pte. Ltd., vol. 8(01), pages 1-22, March.
    8. repec:hal:journl:hal-01611790 is not listed on IDEAS
    9. Benjamin Jourdain & Alexandre Zhou, 2020. "Existence of a calibrated regime switching local volatility model," Mathematical Finance, Wiley Blackwell, vol. 30(2), pages 501-546, April.
    10. Hadrien de March & Pierre Henry-Labordere, 2019. "Building Arbitrage-Free Implied Volatility: Sinkhorn'S Algorithm And Variants," Working Papers hal-02011533, HAL.
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    Citations

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    Cited by:

    1. 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.
    2. Christoph Reisinger & Maria Olympia Tsianni, 2023. "Convergence of the Euler--Maruyama particle scheme for a regularised McKean--Vlasov equation arising from the calibration of local-stochastic volatility models," Papers 2302.00434, arXiv.org, revised Aug 2023.
    3. Manuel Hasenbichler & Benjamin Joseph & Gregoire Loeper & Jan Obloj & Gudmund Pammer, 2023. "The Martingale Sinkhorn Algorithm," Papers 2310.13797, arXiv.org, revised Mar 2026.
    4. Benjamin Joseph & Grégoire Loeper & Jan Obłój, 2026. "Calibration of local volatility models with stochastic interest rates using optimal transport," Finance and Stochastics, Springer, vol. 30(2), pages 397-439, April.
    5. Julio Backhoff & Gregoire Loeper & Jan Obloj, 2024. "Geometric Martingale Benamou-Brenier transport and geometric Bass martingales," Papers 2406.04016, arXiv.org, revised Feb 2025.

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