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

## TL;DR

This paper introduces a new calibration method for complex four-factor foreign-exchange models that combines particle methods with variance reduction, enabling efficient calibration on real market data.

## Contribution

It presents a generic calibration technique for hybrid local-stochastic volatility models with stochastic rates, improving convergence speed using control variates and particle methods.

## Key findings

- Effective calibration on EUR-USD data
- Comparable run-time to PDE calibration of simpler models
- Applicable to a broad class of hybrid LSV models

## Abstract

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.