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| Auteurs principaux: | , , , |
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| Format: | Preprint |
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2022
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| Accès en ligne: | https://arxiv.org/abs/2203.01160 |
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| _version_ | 1866913192611938304 |
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| author | Bayer, Christian Belomestny, Denis Butkovsky, Oleg Schoenmakers, John |
| author_facet | Bayer, Christian Belomestny, Denis Butkovsky, Oleg Schoenmakers, John |
| contents | Motivated by the challenges related to the calibration of financial models, we consider the problem of numerically solving a singular McKean-Vlasov equation $$ d X_t= σ(t,X_t) X_t \frac{\sqrt v_t}{\sqrt {E[v_t|X_t]}}dW_t, $$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2203_01160 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | A Reproducing Kernel Hilbert Space approach to singular local stochastic volatility McKean-Vlasov models Bayer, Christian Belomestny, Denis Butkovsky, Oleg Schoenmakers, John Computational Finance Probability 91G20, 65C30, 46E22 Motivated by the challenges related to the calibration of financial models, we consider the problem of numerically solving a singular McKean-Vlasov equation $$ d X_t= σ(t,X_t) X_t \frac{\sqrt v_t}{\sqrt {E[v_t|X_t]}}dW_t, $$ 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. |
| title | A Reproducing Kernel Hilbert Space approach to singular local stochastic volatility McKean-Vlasov models |
| topic | Computational Finance Probability 91G20, 65C30, 46E22 |
| url | https://arxiv.org/abs/2203.01160 |