In this talk we
analyze 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 numerical analysis for the Black-Scholes model and
the jump-diffusion model with exponentially distributed jumps.