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Calibration of local‐stochastic volatility models by optimal transport


  • Ivan Guo
  • Grégoire Loeper
  • Shiyi Wang


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

    1. 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.
    2. 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.
    3. Hadrien de March & Pierre Henry-Labordere, 2019. "Building Arbitrage-Free Implied Volatility: Sinkhorn'S Algorithm And Variants," Working Papers hal-02011533, HAL.
    4. Ivan Guo & Gregoire Loeper & Jan Obloj & Shiyi Wang, 2021. "Optimal transport for model calibration," Papers 2107.01978,
    5. 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.
    6. Christa Cuchiero & Wahid Khosrawi & Josef Teichmann, 2020. "A generative adversarial network approach to calibration of local stochastic volatility models," Papers 2005.02505,, revised Sep 2020.
    7. Frédéric Abergel & Rémi Tachet, 2010. "A nonlinear partial integro-differential equation from mathematical finance," Post-Print hal-00611962, HAL.
    8. Bruno Bouchard & G. Loeper & Y. Zou, 2017. "Hedging of covered options with linear market impact and gamma constraint," Post-Print hal-01611790, HAL.
    9. Bruno Bouchard & G Loeper & Y Zou, 2017. "Hedging of covered options with linear market impact and gamma constraint," Post-Print hal-01247523, HAL.
    10. 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.
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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. Benjamin Joseph & Gregoire Loeper & Jan Obloj, 2023. "The Measure Preserving Martingale Sinkhorn Algorithm," Papers 2310.13797,, revised May 2024.
    3. Julio Backhoff-Veraguas & Gregoire Loeper & Jan Obloj, 2024. "Geometric Martingale Benamou-Brenier transport and geometric Bass martingales," Papers 2406.04016,
    4. 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,, revised Aug 2023.

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