  We propose a novel and generic calibration technique for four-factor
foreign-exchange hybrid local-stochastic volatility models with stochastic
short rates. We build upon the particle method introduced by Guyon and
Labord\`ere [Nonlinear Option Pricing, Chapter 11, Chapman and Hall, 2013] and
combine it with new variance reduction techniques in order to accelerate
convergence. We use control variates derived from 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. The calibration procedure is performed on real-world 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.
