This paper studies a class of optimal multiple stopping problems driven by Levy processes. Our model allows for 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. Moreover, successive exercise opportunities are separated by independently and identically distributed random refraction times. Under a wide class of two-sided Levy models with a general random refraction time, we rigorously show that the optimal strategy to exercise successive call options is uniquely characterized by a sequence of upcrossing times. The corresponding optimal thresholds are determined explicitly in the single stopping case and recursively in the multiple stopping case.