An exchange option is a financial instrument that allows the holder to trade one asset for another upon exercise. Under the Black-Scholes framework, the Margrabe formula gives its price when the correlation between the two underlying assets is assumed to be constant, which is not realistic, as shown by the data. A mean-reverting stochastic process is proposed to model correlation, some properties are examined, and a revised Black-Scholes PDE is derived through dynamic hedging. Numerical solutions are found. Asymptotic prices are found in the limit of small stochastic variation, which can be written in terms of the Margrabe formula with perturbed parameters. Similar results are found for various payoff functions. Numerical results are also found for American-style option prices and the free exercise boundaries.