## The Annals of Applied Probability

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

#### Abstract

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.

#### Article information

Source
Ann. Appl. Probab., Volume 28, Number 4 (2018), 2105-2140.

Dates
Revised: March 2017
First available in Project Euclid: 9 August 2018

https://projecteuclid.org/euclid.aoap/1533780269

Digital Object Identifier
doi:10.1214/17-AAP1322

Mathematical Reviews number (MathSciNet)
MR3843825

Zentralblatt MATH identifier
06974747

#### Citation

Rodosthenous, Neofytos; Zhang, Hongzhong. Beating the omega clock: An optimal stopping problem with random time-horizon under spectrally negative Lévy models. Ann. Appl. Probab. 28 (2018), no. 4, 2105--2140. doi:10.1214/17-AAP1322. https://projecteuclid.org/euclid.aoap/1533780269

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