IDEAS home Printed from https://ideas.repec.org/a/eee/stapro/v167y2020ics0167715220302029.html
   My bibliography  Save this article

Optimal stopping problems for running minima with positive discounting rates

Author

Listed:
  • Gapeev, Pavel V.

Abstract

We present analytic solutions to some optimal stopping problems for the running minimum of a geometric Brownian motion with exponential positive discounting rates. The proof is based on the reduction of the original problems to the associated free-boundary problems and the solution of the latter problems by means of the smooth-fit and normal-reflection conditions. We show that the optimal stopping boundaries are determined as the minimal solutions of certain first-order nonlinear ordinary differential equations. The obtained results are related to the valuation of perpetual dual American lookback options with fixed and floating strikes in the Black–Merton–Scholes model from the point of view of short sellers.

Suggested Citation

  • Gapeev, Pavel V., 2020. "Optimal stopping problems for running minima with positive discounting rates," Statistics & Probability Letters, Elsevier, vol. 167(C).
    • Handle: RePEc:eee:stapro:v:167:y:2020:i:c:s0167715220302029
    DOI: 10.1016/j.spl.2020.108899
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0167715220302029
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.spl.2020.108899?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Anna Battauz & Marzia De Donno & Alessandro Sbuelz, 2012. "Real options with a double continuation region," Quantitative Finance, Taylor & Francis Journals, vol. 12(3), pages 465-475, April.
    2. Peskir, Goran, 2012. "Optimal detection of a hidden target: The median rule," Stochastic Processes and their Applications, Elsevier, vol. 122(5), pages 2249-2263.
    3. Gapeev, Pavel V. & Rodosthenous, Neofytos, 2016. "Perpetual American options in diffusion-type models with running maxima and drawdowns," Stochastic Processes and their Applications, Elsevier, vol. 126(7), pages 2038-2061.
    4. Kristoffer Glover & Hardy Hulley & Goran Peskir, 2011. "Three-Dimensional Brownian Motion and the Golden Ratio Rule," Research Paper Series 295, Quantitative Finance Research Centre, University of Technology, Sydney.
    5. Gapeev, Pavel V. & Reiß, Markus, 2006. "An optimal stopping problem in a diffusion-type model with delay," Statistics & Probability Letters, Elsevier, vol. 76(6), pages 601-608, March.
    6. Gapeev, Pavel V., 2008. "The integral option in a model with jumps," Statistics & Probability Letters, Elsevier, vol. 78(16), pages 2623-2631, November.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Pavel V. Gapeev, 2022. "Perpetual American Double Lookback Options on Drawdowns and Drawups with Floating Strikes," Methodology and Computing in Applied Probability, Springer, vol. 24(2), pages 749-788, June.
    2. Pavel V. Gapeev & Peter M. Kort & Maria N. Lavrutich & Jacco J. J. Thijssen, 2022. "Optimal Double Stopping Problems for Maxima and Minima of Geometric Brownian Motions," Methodology and Computing in Applied Probability, Springer, vol. 24(2), pages 789-813, June.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Gapeev, Pavel V., 2020. "Optimal stopping problems for running minima with positive discounting rates," LSE Research Online Documents on Economics 105849, London School of Economics and Political Science, LSE Library.
    2. Pavel V. Gapeev & Neofytos Rodosthenous & V. L. Raju Chinthalapati, 2019. "On the Laplace Transforms of the First Hitting Times for Drawdowns and Drawups of Diffusion-Type Processes," Risks, MDPI, vol. 7(3), pages 1-15, August.
    3. Gapeev, Pavel V. & Rodosthenous, Neofytos & Chinthalapati, V.L Raju, 2019. "On the Laplace transforms of the first hitting times for drawdowns and drawups of diffusion-type processes," LSE Research Online Documents on Economics 101272, London School of Economics and Political Science, LSE Library.
    4. T. De Angelis & G. Peskir, 2016. "Optimal prediction of resistance and support levels," Applied Mathematical Finance, Taylor & Francis Journals, vol. 23(6), pages 465-483, November.
    5. Gapeev, Pavel V. & Li, Libo, 2022. "Perpetual American standard and lookback options with event risk and asymmetric information," LSE Research Online Documents on Economics 114940, London School of Economics and Political Science, LSE Library.
    6. Gapeev, Pavel V. & Rodosthenous, Neofytos, 2016. "Perpetual American options in diffusion-type models with running maxima and drawdowns," Stochastic Processes and their Applications, Elsevier, vol. 126(7), pages 2038-2061.
    7. Hongzhong Zhang, 2018. "Stochastic Drawdowns," World Scientific Books, World Scientific Publishing Co. Pte. Ltd., number 10078.
    8. Zbigniew Palmowski & Jos'e Luis P'erez & Kazutoshi Yamazaki, 2020. "Double continuation regions for American options under Poisson exercise opportunities," Papers 2004.03330, arXiv.org.
    9. Pavel V. Gapeev & Peter M. Kort & Maria N. Lavrutich & Jacco J. J. Thijssen, 2022. "Optimal Double Stopping Problems for Maxima and Minima of Geometric Brownian Motions," Methodology and Computing in Applied Probability, Springer, vol. 24(2), pages 789-813, June.
    10. Fry, John & Griguta, Vlad-Marius & Gerber, Luciano & Slater-Petty, Helen & Crockett, Keeley, 2021. "Modelling corporate bank accounts," Economics Letters, Elsevier, vol. 205(C).
    11. Pavel V. Gapeev, 2022. "Perpetual American Double Lookback Options on Drawdowns and Drawups with Floating Strikes," Methodology and Computing in Applied Probability, Springer, vol. 24(2), pages 749-788, June.
    12. Detemple, Jérôme & Laminou Abdou, Souleymane & Moraux, Franck, 2020. "American step options," European Journal of Operational Research, Elsevier, vol. 282(1), pages 363-385.
    13. Anna Battauz & Marzia De Donno & Alessandro Sbuelz, 2015. "Real Options and American Derivatives: The Double Continuation Region," Management Science, INFORMS, vol. 61(5), pages 1094-1107, May.
    14. Marzia De Donno & Zbigniew Palmowski & Joanna Tumilewicz, 2020. "Double continuation regions for American and Swing options with negative discount rate in Lévy models," Mathematical Finance, Wiley Blackwell, vol. 30(1), pages 196-227, January.
    15. Marie Bernhart & Peter Tankov & Xavier Warin, 2010. "A finite dimensional approximation for pricing moving average options," Papers 1011.3599, arXiv.org.
    16. Anna Battauz & Marzia De Donno & Janusz Gajda & Alessandro Sbuelz, 2022. "Optimal exercise of American put options near maturity: A new economic perspective," Review of Derivatives Research, Springer, vol. 25(1), pages 23-46, April.
    17. 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.
    18. Gündüz, Güngör & Gündüz, Yalin, 2016. "A thermodynamical view on asset pricing," International Review of Financial Analysis, Elsevier, vol. 47(C), pages 310-327.
    19. Zhang, Xiang & Li, Lingfei & Zhang, Gongqiu, 2021. "Pricing American drawdown options under Markov models," European Journal of Operational Research, Elsevier, vol. 293(3), pages 1188-1205.
    20. Bayraktar, Erhan & Yao, Song, 2024. "Stochastic control/stopping problem with expectation constraints," Stochastic Processes and their Applications, Elsevier, vol. 176(C).

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:stapro:v:167:y:2020:i:c:s0167715220302029. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Catherine Liu (email available below). General contact details of provider: http://www.elsevier.com/wps/find/journaldescription.cws_home/622892/description#description .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.