  This paper studies a class of optimal multiple stopping problems driven by
L\'evy 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 i.i.d. random refraction times. Under a wide class of two-sided
L\'evy 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 up-crossing times. The corresponding optimal
thresholds are determined explicitly in the single stopping case and
recursively in the multiple stopping case.
