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A reproducing kernel Hilbert space approach to singular local stochastic volatility McKean–Vlasov models

Author

Listed:
  • Christian Bayer

    (Weierstrass Institute)

  • Denis Belomestny

    (Duisburg-Essen University)

  • Oleg Butkovsky

    (Weierstrass Institute)

  • John Schoenmakers

    (Weierstrass Institute)

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 = σ ( t , X t ) X t v t E [ v t | X t ] d W t , $$ d X_{t}= \sigma (t,X_{t}) X_{t} \frac{\sqrt{v}_{t}}{\sqrt{\mathbb{E}[v_{t}|X_{t}]}}dW_{t}, $$ where W $W$ is a Brownian motion and v $v$ is an adapted diffusion process. This equation can be considered as a singular local stochastic volatility model. While such models are quite popular among practitioners, its well-posedness has unfortunately not yet been fully understood and in general is possibly not guaranteed at all. We develop a novel regularisation approach based on the reproducing kernel Hilbert space (RKHS) technique and show that the regularised model is well posed. Furthermore, we prove propagation of chaos. We demonstrate numerically that a thus regularised model is able to perfectly replicate option prices coming from 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, 2024. "A reproducing kernel Hilbert space approach to singular local stochastic volatility McKean–Vlasov models," Finance and Stochastics, Springer, vol. 28(4), pages 1147-1178, October.
    • Handle: RePEc:spr:finsto:v:28:y:2024:i:4:d:10.1007_s00780-024-00541-5
    DOI: 10.1007/s00780-024-00541-5
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    References listed on IDEAS

    as
    1. Vincent Lemaire & Thibaut Montes & Gilles Pagès, 2022. "Stationary Heston model: calibration and pricing of exotics using product recursive quantization," Quantitative Finance, Taylor & Francis Journals, vol. 22(4), pages 611-629, April.
    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.
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    More about this item

    Keywords

    Stochastic volatility models; Singular McKean–Vlasov equations; Reproducing kernel Hilbert space;
    All these keywords.

    JEL classification:

    • C63 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Computational Techniques
    • G17 - Financial Economics - - General Financial Markets - - - Financial Forecasting and Simulation

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