This paper presents a generalisation of McKean's free boundary value problem for American options by considering an American strangle position, where the early exercise of one side of the payoff will knock-out the out-of-the-money side. When attempting to evaluate the price of this American strangle, it is not correct to simply price the component American call and put options which make up the strangle, and take the sum of their values. The Fourier transform technique is used to derive the integral equation for the price of our American strangle. From this expression, a coupled integral equation system for the strangle's call- and put-side free boundaries is found. While the equation for the price of the strangle is simply the sum of its component American call and put option equations, the free boundary for each side is shown to have a more complex nature. Anumerical algorithm for solving the coupled integral equation system for the free boundaries is provided, and the resulting approximations are used to determine the price of the American strangle position. Numerical comparisons between the strangle price and the price of a portfolio formed from a long position in both an American call an American put option are presented.
