Analytically pricing foreign exchange options under a three-factor stochastic volatility and interest rate model: A full correlation structure

Abstract

The current literature usually ignores the dependence between the rate of exchange and two interest rates, i.e., domestic and foreign ones, for the availability of analytical solutions to foreign exchange option prices, while such settings potentially limit the performance of these models. To address this, we introduce two new factors into the Heston–Hull–White model when foreign exchange options are evaluated. This newly formulated three-factor Heston–Hull–White model still admits a closed-form solution to foreign exchange option prices, which is successfully figured out based on the numeraire change technique and the obtained analytical expression of the characteristic function. Based on options on foreign exchange rate between US and Australian Dollar, both the three-factor and one-factor Heston–Hull–White models are calibrated using the Adaptive Simulated Annealing. When evaluating the performance of both models with in- and out-of-sample errors, the greater forecasting ability of the new model equipped with the two new factors over the one-factor Heston–Hull–White model is identified.