Perpetual dual American barrier options for short sellers

We obtain closed-form solutions to the problems of pricing of perpetual American put and call barrier options in the one-dimensional Black-Merton-Scholes model from the point of view of short sellers. The proof is based on the reduction of the original optimal stopping problems for a one-dimensional geometric Brownian motion with positive exponential discounting rates to the equivalent free-boundary problems and the solution of the latter problems by means of the smooth-fit conditions.