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

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  • Neofytos Rodosthenous
  • Hongzhong Zhang

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.

Suggested Citation

  • Neofytos Rodosthenous & Hongzhong Zhang, 2017. "Beating the Omega Clock: An Optimal Stopping Problem with Random Time-horizon under Spectrally Negative L\'evy Models," Papers 1706.03724, arXiv.org.
  • Handle: RePEc:arx:papers:1706.03724
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    References listed on IDEAS

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    Cited by:

    1. Mingsi Long & Hongzhong Zhang, 2017. "On the optimality of threshold type strategies in single and recursive optimal stopping under L\'evy models," Papers 1707.07797, arXiv.org, revised Aug 2018.
    2. Hongzhong Zhang, 2018. "Stochastic Drawdowns," World Scientific Books, World Scientific Publishing Co. Pte. Ltd., number 10078, August.

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