Calibration of a Hybrid Local-Stochastic Volatility Stochastic Rates Model with a Control Variate Particle Method

© 2019 Society for Industrial and Applied Mathematics. We propose a novel and generic calibration technique for four-factor foreign-exchange hybrid local-stochastic volatility (LSV) models with stochastic short rates. We build upon the particle method introduced by Guyon and Henry-Labord`ere [Nonlinear Option Pricing, Chapman and Hall, 2013, Chapter 11] and combine it with new variance reduction techniques in order to accelerate convergence. We use control variates derived from the following: a calibrated pure local volatility model, a two-factor Heston-type LSV model (both with deterministic rates), and the stochastic (CIR) short rates. The method can be applied to a large class of hybrid LSV models and is not restricted to our particular choice of the diffusion. However, we address in the paper some specific difficulties arising from the Heston model, notably by a new PDE formulation and finite element solution to bypass the singularities of the density when zero is attainable by the variance. The calibration procedure is performed on market data for the EUR-USD currency pair and has a comparable run-time to the PDE calibration of a two-factor LSV model alone.