Forward Start Options Under Stochastic Volatility and Stochastic Interest Rates

Forward start options are examined in Heston's (Review of Financial Studies,Vol. 6, pp. 327–343, 1993) stochastic volatility model with the CIR (Econometrica, Vol. 53, pp. 385–408, 1985) stochastic interest rates. The instantaneous volatility and the instantaneous short rate are assumed to be correlated with the dynamics of stock return. The main result is an analytic formula for the price of a forward start European call option. It is derived using the probabilistic approach combined with the Fourier inversion technique, as developed in Carr and Madan (Journal of Computational Finance, Vol. 2, pp. 61–73, 1999)