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A Reproducing Kernel Hilbert Space approach to singular local stochastic volatility McKean-Vlasov models

Author

Listed:
  • Christian Bayer
  • Denis Belomestny
  • Oleg Butkovsky
  • John Schoenmakers

Abstract

Motivated by the challenges related to the calibration of financial models, we consider the problem of numerically solving a singular McKean-Vlasov equation $$ d X_t= \sigma(t,X_t) X_t \frac{\sqrt v_t}{\sqrt {E[v_t|X_t]}}dW_t, $$ where $W$ is a Brownian motion and $v$ is an adapted diffusion process. This equation can be considered as a singular local stochastic volatility model. Whilst such models are quite popular among practitioners, unfortunately, its well-posedness has not been fully understood yet and, in general, is possibly not guaranteed at all. We develop a novel regularization approach based on the reproducing kernel Hilbert space (RKHS) technique and show that the regularized model is well-posed. Furthermore, we prove propagation of chaos. We demonstrate numerically that a thus regularized model is able to perfectly replicate option prices due to typical local volatility models. Our results are also applicable to more general McKean--Vlasov equations.

Suggested Citation

  • Christian Bayer & Denis Belomestny & Oleg Butkovsky & John Schoenmakers, 2022. "A Reproducing Kernel Hilbert Space approach to singular local stochastic volatility McKean-Vlasov models," Papers 2203.01160, arXiv.org, revised Jan 2024.
    • Handle: RePEc:arx:papers:2203.01160
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    References listed on IDEAS

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    1. 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.
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    Cited by:

    1. 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.
    2. Mao Fabrice Djete, 2022. "Non--regular McKean--Vlasov equations and calibration problem in local stochastic volatility models," Papers 2208.09986, arXiv.org, revised Oct 2024.
    3. Enrico Dall’Acqua & Riccardo Longoni & Andrea Pallavicini, 2023. "Rough-Heston Local-Volatility Model," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 26(06n07), pages 1-18, November.

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