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

TL;DR

This paper establishes the existence of solutions for a class of McKean--Vlasov equations with minimal regularity assumptions, enabling the calibration of local stochastic volatility models in finance.

Contribution

It introduces a novel existence and approximation framework for non-regular McKean--Vlasov equations, facilitating model calibration in local stochastic volatility models.

Findings

Existence of solutions under minimal continuity assumptions.

Approximation via N-particle systems and propagation of chaos.

Application to calibrated local stochastic volatility models.

Abstract

In order to deal with the question of the existence of a calibrated local stochastic volatility model in finance, we investigate a class of McKean--Vlasov equations where a minimal continuity assumption is imposed on the coefficients. Namely, the drift coefficient and, in particular, the volatility coefficient are not necessarily continuous in the measure variable for the Wasserstein topology. In this paper, we provide an existence result and show an approximation by $N$--particle system or propagation of chaos for this type of McKean--Vlasov equations. As a direct result, we are able to deduce the existence of a calibrated local stochastic volatility model for an appropriate choice of stochastic volatility parameters. The associated propagation of chaos result is also proved.