RKHS regularization of singular local stochastic volatility McKean-Vlasov models

Abstract

Motivated by the challenges related to the calibration of financial models, we consider the problem of solving numerically a singular McKean-Vlasov equationwhere 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 wellposed. 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.

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

“…As mentioned in the introduction, one of the famous application of the SDE like Equation (1.4) or Equation (2.2) is in the calibration of local stochastic volatility models (see Lipton [20], Piterbarg [22], Guyon and Henry-Labordere [11], Tian, Zhu, Lee, Klebaner, and Hamza [25], Saporito, Yang, and Zubelli [23], Bayer, Belomestny, Butkovsky, and Schoenmakers [4] ). Let us here describe this model more precisely.…”