Forward Start Foreign Exchange Options Under Heston’s Volatility and the CIR Interest Rates

We examine the valuation of forward start foreign exchange options in the Heston (Rev. Financ. Stud. 6:327–343, 1993) stochastic volatility model for the exchange rate combined with the CIR (see Cox et al. in Econometrica 53:385–408, 1985) dynamics for the domestic and foreign interest rates. The instantaneous volatility is correlated with the dynamics of the exchange rate, whereas the domestic and foreign short-term rates are assumed to be independent of the dynamics of the exchange rate volatility. The main results are derived using the probabilistic approach combined with the Fourier inversion technique developed in Carr and Madan (J. Comput. Finance 2:61–73, 1999). They furnish two alternative semi-analytical formulae for the price of the forward start foreign exchange European call option. As was argued in Ahlip and Rutkowski (Quant. Finance 13:955–966, 2013), the setup examined here is the only analytically tractable version of the foreign exchange market model that combines the Heston stochastic volatility model for the exchange rate with the CIR dynamics for interest rates.