  We consider the L\'evy model of the perpetual American call and put options
with a negative discount rate under Poisson observations. Similar to the
continuous observation case as in De Donno et al. [24], the stopping region
that characterizes the optimal stopping time is either a half-line or an
interval. The objective of this paper is to obtain explicit expressions of the
stopping and continuation regions and the value function, focusing on
spectrally positive and negative cases. To this end, we compute the identities
related to the first Poisson arrival time to an interval via the scale function
and then apply those identities to the computation of the optimal strategies.
We also discuss the convergence of the optimal solutions to those in the
continuous observation case as the rate of observation increases to infinity.
Numerical experiments are also provided.
