A reproducing kernel Hilbert space approach to singular local stochastic volatility McKean–Vlasov models - Christian Bayer; Denis Belomestny; Oleg Butkovsky; John Schoenmakers (Finance and Stochastics, 2024)
Abstract Motivated by the challenges related to the calibration of financial models, we consider the problem of numerically solving a singular McKean–Vlasov equation dXt=σ(t,Xt)XtvtE[vt|Xt]dWt,where Wis a Brownian motion and vis 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.
Additional information: C63;G17
Keywords: 330; Stochastic volatility models; Singular McKean–Vlasov equations; Reproducing kernel Hilbert space