Pricing Foreign Currency and Cross-Currency Options Under GARCH

This paper generalizes the GARCH option pricing methodology in Duan (1995, Mathematical Finance) to a two-country setting. Specifically, we assume a bivariate nonlinear GARCH system for the exchange rate and the foreign asset price, and generalize the local risk-neutral valuation relationship in Duan (1995). We derive the equilibrium GARCH processes for the exchange rate and the foreign asset price in the two-country economy. Foreign currency options and cross-currency options can then be valued using the well-known risk-neutral valuation technique. Our setup allows rich empirical regularities such as stochastic volatility, fat tails, and the so-called leverage effect. We also run simulations to price quanto options and find that when the true environment is GARCH a constant variance model is not reliable in most cases. The proposed equilibrium valuation framework to price foreign currency and cross-currency options is the first of its kind in the literature