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An Analytic Recursive Method For Optimal Multiple Stopping: Canadization And Phase-Type Fitting

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  • TIM LEUNG

    (IEOR Department, Columbia University, New York, NY 10027, USA)

  • KAZUTOSHI YAMAZAKI

    (Department of Mathematics, Kansai University, Osaka, Japan)

  • HONGZHONG ZHANG

    (Statistics Department, Columbia University, New York, NY 10027, USA)

Abstract

We study an optimal multiple stopping problem for call-type payoff driven by a spectrally negative Lévy process. The stopping times are separated by constant refraction times, and the discount rate can be positive or negative. The computation involves a distribution of the Lévy process at a constant horizon and hence the solutions in general cannot be attained analytically. Motivated by the maturity randomization (Canadization) technique by Carr (1998), we approximate the refraction times by independent, identically distributed Erlang random variables. In addition, fitting random jumps to phase-type distributions, our method involves repeated integrations with respect to the resolvent measure written in terms of the scale function of the underlying Lévy process. We derive a recursive algorithm to compute the value function in closed form, and sequentially determine the optimal exercise thresholds. A series of numerical examples are provided to compare our analytic formula to results from Monte Carlo simulation.

Suggested Citation

  • Tim Leung & Kazutoshi Yamazaki & Hongzhong Zhang, 2015. "An Analytic Recursive Method For Optimal Multiple Stopping: Canadization And Phase-Type Fitting," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 18(05), pages 1-31.
    • Handle: RePEc:wsi:ijtafx:v:18:y:2015:i:05:n:s0219024915500326
    DOI: 10.1142/S0219024915500326
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    Citations

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    Cited by:

    1. Jos'e-Luis P'erez & Kazutoshi Yamazaki, 2023. "L\'evy bandits under Poissonian decision times," Papers 2301.07798, arXiv.org.
    2. Jos'e-Luis P'erez & Kazutoshi Yamazaki & Alain Bensoussan, 2018. "Optimal periodic replenishment policies for spectrally positive L\'evy demand processes," Papers 1806.09216, arXiv.org, revised Sep 2020.
    3. Ankush Agarwal & Sandeep Juneja & Ronnie Sircar, 2016. "American options under stochastic volatility: control variates, maturity randomization & multiscale asymptotics," Quantitative Finance, Taylor & Francis Journals, vol. 16(1), pages 17-30, January.
    4. Kei Noba & Jos'e-Luis P'erez & Kazutoshi Yamazaki & Kouji Yano, 2017. "On optimal periodic dividend strategies for L\'evy risk processes," Papers 1708.01678, arXiv.org, revised Feb 2018.
    5. 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.
    6. Tim Leung & Kazutoshi Yamazaki & Hongzhong Zhang, 2015. "Optimal Multiple Stopping with Negative Discount Rate and Random Refraction Times under Levy Models," Papers 1505.07313, arXiv.org.
    7. Zbigniew Palmowski & José Luis Pérez & Kazutoshi Yamazaki, 2021. "Double continuation regions for American options under Poisson exercise opportunities," Mathematical Finance, Wiley Blackwell, vol. 31(2), pages 722-771, April.
    8. Pérez, José-Luis & Yamazaki, Kazutoshi, 2017. "On the optimality of periodic barrier strategies for a spectrally positive Lévy process," Insurance: Mathematics and Economics, Elsevier, vol. 77(C), pages 1-13.
    9. Czarna, Irmina & Pérez, José-Luis & Yamazaki, Kazutoshi, 2018. "Optimality of multi-refraction control strategies in the dual model," Insurance: Mathematics and Economics, Elsevier, vol. 83(C), pages 148-160.
    10. Long, Mingsi & Zhang, Hongzhong, 2019. "On the optimality of threshold type strategies in single and recursive optimal stopping under Lévy models," Stochastic Processes and their Applications, Elsevier, vol. 129(8), pages 2821-2849.
    11. Jos'e-Luis P'erez & Kazutoshi Yamazaki, 2016. "Hybrid continuous and periodic barrier strategies in the dual model: optimality and fluctuation identities," Papers 1612.02444, arXiv.org, revised Jan 2018.
    12. Noba, Kei & Pérez, José-Luis & Yamazaki, Kazutoshi & Yano, Kouji, 2018. "On optimal periodic dividend strategies for Lévy risk processes," Insurance: Mathematics and Economics, Elsevier, vol. 80(C), pages 29-44.
    13. Zbigniew Palmowski & José Luis Pérez & Budhi Arta Surya & Kazutoshi Yamazaki, 2020. "The Leland–Toft optimal capital structure model under Poisson observations," Finance and Stochastics, Springer, vol. 24(4), pages 1035-1082, October.
    14. Søren Asmussen & Patrick J. Laub & Hailiang Yang, 2019. "Phase-Type Models in Life Insurance: Fitting and Valuation of Equity-Linked Benefits," Risks, MDPI, vol. 7(1), pages 1-22, February.

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