  In this paper we study perpetual American call and put options in an
exponential L\'evy model. We consider a negative effective discount rate which
arises in a number of financial applications including stock loans and real
options, where the strike price can potentially grow at a higher rate than the
original discount factor. We show that in this case a double continuation
region arises and we identify the two critical prices. We also generalize this
result to multiple stopping problems of Swing type, that is, when successive
exercise opportunities are separated by i.i.d. random refraction times. We
conduct an extensive numerical analysis for the Black-Scholes model and the
jump-diffusion model with exponentially distributed jumps.
