Beating the omega clock: An optimal stopping problem with random time-horizon under spectrally negative Lévy models

We study the optimal stopping of an American call option in a random time-horizon under exponential spectrally negative Lévy 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 Lévy 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.