  We study the optimal stopping of an American call option in a random
time-horizon under exponential spectrally negative L\'evy models. The random
time-horizon is modeled as the so-called Omega default clock in insurance,
which is the first time when the occupation time of the underlying L\'evy
process below a level $y$, exceeds an independent exponential random variable
with mean $1/q>0$. We show that the shape of the value function varies
qualitatively with different values of $q$ and $y$. In particular, we show that
for certain values of $q$ and $y$, some quantitatively different but
traditional up-crossing strategies are still optimal, while for other values we
may have two disconnected continuation regions, resulting in the optimality of
two-sided exit strategies. By deriving the joint distribution of the
discounting factor and the underlying process under a random discount rate, we
give a complete characterization of all optimal exercising thresholds. Finally,
we present an example with a compound Poisson process plus a drifted Brownian
motion.
