A mixed Monte Carlo/PDE variance reduction method under the Heston-CIR model

In this paper, the valuation of European and path-dependent options in FX markets is considered when the currency exchange rate evolves according to the Heston model combined with the Cox-Ingersoll-Ross dynamics for the stochastic domestic and foreign interest rates. The mixed Monte Carlo/PDE 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 PDE. 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 in $L^p$ ($p\geq1$) of the latter. Then, we carry out a theoretical variance reduction analysis and illustrate the efficiency of the method through a detailed quantitative assessment based on a European call and an up-and-out put option by comparison with alternative numerical methods.