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. Our numerical experiments show that because of the dramatic variance reduction we are able to calibrate the four-factor model at almost no extra computational cost when the corresponding calibrated two-factor model is at our disposal. 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.
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