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 LÏ (Ï ; ⩾ 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.