This master thesis will demonstrate how to price perpetual American options with linear programming. American options are used both for hedging and speculation, and being able to price derivatives, without creating arbitrage opportunities, are of importance. First we introduce a deterministic security market model and exploit the mathematical structure. Then European and American put and call options are presented. With dynamic programming we show how to price American options. Dynamic programming is based on an idea that an investor would act optimally at all trading dates and the objective is yielding the maximum profit, despite the risk of not knowing the true future value of the option. With this technique, we investigate perpetual American options on a ternary Markov chain model. Perpetual options are without an expiration date. Markov chain models are only dependent of the current state when determining the future value, thus simplifying the computations. The solution, based on dynamic programming, is the smallest payoff that is greater than the discounted expected value of the option at the next trading date. The value and the payoff must not be confused, as an investor may be willing to pay more than the payoff today, if the value of the option might rise in the future. The solution is obtained by formulating the problem as an optimization problem and then using linear programming theory.