BEATING THE OMEGA CLOCK: AN OPTIMAL STOPPING PROBLEM WITH RANDOM TIME-HORIZON UNDER SPECTRALLY NEGATIVE LEVY MODELS

• 成果类型: Article
• 署名作者: Rodosthenous, Neofytos; Zhang, Hongzhong
• 署名单位: University of London; Queen Mary University London; Columbia University
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/17-AAP1322
• 发表日期: 2018
• 页码: 2105-2140
• 关键词: occupation times; 1st passage; drawdowns

摘要

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