Strong existence and uniqueness of a calibrated local stochastic volatility model

TL;DR

This paper proves the strong existence and uniqueness of a two-factor local stochastic volatility model calibrated to market data, with a focus on theoretical well-posedness and particle system propagation.

Contribution

It establishes the well-posedness of a two-factor LSV model with a finite-state volatility driver and proves propagation of chaos for the associated particle system.

Findings

Strong existence and uniqueness of the model solution.

Propagation of chaos for the particle system.

Theoretical validation of calibration algorithms.

Abstract

We study a two-dimensional McKean-Vlasov stochastic differential equation, whose volatility coefficient depends on the conditional distribution of the second component with respect to the first component. We prove the strong existence and uniqueness of the solution, establishing the well-posedness of a two-factor local stochastic volatility (LSV) model calibrated to the market prices of European call options. In the spirit of [Jourdain and Zhou, 2020, Existence of a calibrated regime switching local volatility model.], we assume that the factor driving the volatility of the log-price takes finitely many values. Additionally, the propagation of chaos of the particle system is established, giving theoretical justification for the algorithm [Julien Guyon and Henry-Labord\`ere, 2012, Being particular about calibration.].