Game options are American-type options with the additional property that the seller of the option has the right the cancel the option at any time prior to the buyer exercise or the expiration date of the option. The cancelation by the seller can be achieved through a payment of an additional penalty to the exercise payoff or using a payoff process greater than or equal to the exercise value.

The main contribution of this thesis is a numerical framework for computing the value of such options with finite maturity time as well as in the perpetual setting. This framework employs the theory of weak solutions of parabolic and elliptic variational inequalities. These solutions will be computed using finite element methods.

The computational advantage of this framework is that it allows the user to go from one type of process to another by changing the stiffness matrix in the algorithm. Several types of Levy processes will be used to show the functionality of this method. The processes considered are of pure diffusion type (Black-Scholes model), the CGMY process as a pure jump model and a combination of the two for the case of jump diffusion.

Computational results of the option prices as well as exercise, hold and cancelation regions are shown together with numerical estimates of the error convergence rates with respect to the L₂ norm and the energy norm.