  Motivated by the challenges related to the calibration of financial models,
we consider the problem of numerically solving a singular McKean-Vlasov
equation $$ d X_t= \sigma(t,X_t) X_t \frac{\sqrt v_t}{\sqrt {E[v_t|X_t]}}dW_t,
$$ 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.
