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Non--regular McKean--Vlasov equations and calibration problem in local stochastic volatility models

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  • Mao Fabrice Djete

Abstract

In order to deal with the question of the existence of a calibrated local stochastic volatility model in finance, we investigate a class of McKean--Vlasov equations where a minimal continuity assumption is imposed on the coefficients. Namely, the drift coefficient and, in particular, the volatility coefficient are not necessarily continuous in the measure variable for the Wasserstein topology. In this paper, we provide an existence result and show an approximation by $N$--particle system or propagation of chaos for this type of McKean--Vlasov equations. As a direct result, we are able to deduce the existence of a calibrated local stochastic volatility model for an appropriate choice of stochastic volatility parameters. The associated propagation of chaos result is also proved.

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  • Mao Fabrice Djete, 2022. "Non--regular McKean--Vlasov equations and calibration problem in local stochastic volatility models," Papers 2208.09986, arXiv.org.
    • Handle: RePEc:arx:papers:2208.09986
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    References listed on IDEAS

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    1. Frédéric Abergel & Rémi Tachet, 2010. "A nonlinear partial integro-differential equation from mathematical finance," Post-Print hal-00611962, HAL.
    2. 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.
    3. Kusuoka, Seiichiro, 2017. "Continuity and Gaussian two-sided bounds of the density functions of the solutions to path-dependent stochastic differential equations via perturbation," Stochastic Processes and their Applications, Elsevier, vol. 127(2), pages 359-384.
    4. Benjamin Jourdain & Alexandre Zhou, 2016. "Existence of a calibrated regime switching local volatility model and new fake Brownian motions," Papers 1607.00077, arXiv.org, revised Jan 2017.
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