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

Denis Belomestny, O. A. Butkovsky, Christian Bayer, John Schoenmakers

Abstract: Motivated by the challenges related to the calibration of financial models, we consider the problem of solving numerically a singular McKean-Vlasov equation

d S_t= sigma(t,S_t) S_t frac{sqrt v_t}{sqrt {E[v_t|S_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.

Mathematics Subject Classification ID

Derivative securities (option pricing, hedging, etc.) (91G20) Numerical solutions to stochastic differential and integral equations (65C30) Hilbert spaces with reproducing kernels (= (proper) functional Hilbert spaces, including de Branges-Rovnyak and other structured spaces) (46E22)

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