Beating the Omega Clock: An Optimal Stopping Problem with Random Time-horizon under Spectrally Negative L\'evy Models

TL;DR

This paper investigates the optimal stopping problem for American call options under spectrally negative Lévy models with a random time-horizon driven by an Omega default clock, revealing complex optimal strategies depending on model parameters.

Contribution

It introduces a novel analysis of optimal stopping under a random time-horizon modeled by an Omega clock in spectrally negative Lévy processes, including explicit characterization of optimal thresholds.

Findings

Optimal strategies vary with parameters q and y.

Existence of disconnected continuation regions for certain parameters.

Complete characterization of optimal exercise thresholds.

Abstract

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.