Perpetual American options under Levy processes

We consider perpetual American options assuming, that under a chosen equivalent martingale measure the shock returns follow a Levy process. For put and call options, their analogs for more general payoffs, and a wide class, of Levy processes, which contains Brownian motion, normal inverse Gaussian processes, hyperbolic processes, truncated Levy processes and their mixtures, we obtain formulas for the optimal exercise price and the fair price of the option in terms of the factors in the Wiener-Hopf factorization formula, i.e., in terms of the resolvents of the supremum and infimum processes, and derive explicit formulas for these factors. For calls, puts and some other options, the results are valid for any Levy process.