Exchange options with stochastic volatility

In a recent paper by Antonelli and Scarlatti, the problem of pricing plain vanilla options under quite general stochastic dynamics for the volatility of the underlying asset was approached in a novel way. Namely, they developed the option price in a power series of the correlation coefficient between the asset price and the volatility processes around zero. In the present work we perform a first attempt to extend such a technique to a multidimensional setting. Namely, we consider the case of an exchange or Margrabe option, written on two assets whose processes are assumed to be of log-normal type but with stochastic volatilities. In our model, the two assets are correlated and the volatilities follow each an Ornstein-Uhlenbeck dynamics. We first solve the option pricing problem when the correlation is zero and then develop an expansion in powers of the correlation in order to find an approximation of the pricing formula for the case of general correlation values.