Expansion method for pricing foreign exchange options under stochastic volatility and interest rates

Some expansion methods have been proposed for pricing options approximately in analytical form. One of these is the smart expansion method based on the Malliavin calculus, which is used to price options in the Heston stochastic volatility model with deterministic interest rates. In this paper, we apply the method to the Heston–Hull–White model, which admits stochastic interest rates to enhance the model, and we obtain the expansion formula for pricing options in the model up to second order. Then numerical studies are performed to compare our approximation formula with the Monte Carlo simulation. Our formula shows numerically comparable results with another method using the approximation of the characteristic function, and can also be applied for parameter configurations where the latter method is not useful. The control variate is also used to improve the accuracy for high volatility-of-volatility cases.