A mixed Monte Carlo and partial differential equation variance reduction method for foreign exchange options under the Heston–Cox–Ingersoll–Ross model

Abstract

In this paper, we consider the valuation of European and path-dependent options in foreign exchange markets when the currency exchange rate evolves according to the Heston model combined with the Cox-Ingersoll-Ross (CIR) dynamics for the stochastic domestic and foreign short interest rates. The mixed Monte Carlo/partial differential equation method requires that we simulate only the paths of the squared volatility and the two interest rates, while an inner Black-Scholes-type expectation is evaluated by means of a partial differential equation. This can lead to a substantial variance reduction and complexity improvements under certain circumstances depending on the contract and the model parameters. In this work, we establish the uniform boundedness of moments of the exchange rate process and its approximation, and prove strong convergence of the latter in Lρ (ρ ⩾ 1). Then, we carry out a variance reduction analysis and obtain accurate approximations for quantities of interest. All theoretical contributions can be extended to multi-factor short rates in a straightforward manner. Finally, we illustrate the efficiency of the method for the four-factor Heston-CIR model through a detailed quantitative assessment.