IDEAS home Printed from https://ideas.repec.org/p/arx/papers/1707.06138.html
   My bibliography  Save this paper

American Options with Discontinuous Two-Level Caps

Author

Listed:
  • Jerome Detemple
  • Yerkin Kitapbayev

Abstract

This paper examines the valuation of American capped call options with two-level caps. The structure of the immediate exercise region is significantly more complex than in the classical case with constant cap. When the cap grows over time, making extensive use of probabilistic arguments and local time, we show that the exercise region can be the union of two disconnected set. Alternatively, it can consist of two sets connected by a line. The problem then reduces to the characterization of the upper boundary of the first set, which is shown to satisfy a recursive integral equation. When the cap decreases over time, the boundary of the exercise region has piecewise constant segments alternating with non-increasing segments. General representation formulas for the option price, involving the exercise boundaries and the local time of the underlying price process, are derived. An efficient algorithm is developed and numerical results are provided.

Suggested Citation

  • Jerome Detemple & Yerkin Kitapbayev, 2017. "American Options with Discontinuous Two-Level Caps," Papers 1707.06138, arXiv.org.
  • Handle: RePEc:arx:papers:1707.06138
    as

    Download full text from publisher

    File URL: http://arxiv.org/pdf/1707.06138
    File Function: Latest version
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Phelim P. Boyle & Stuart M. Turnbull, 1989. "Pricing and hedging capped options," Journal of Futures Markets, John Wiley & Sons, Ltd., vol. 9(1), pages 41-54, February.
    2. Don M. Chance, 1994. "The Pricing And Hedging Of Limited Exercise Caps And Spreads," Journal of Financial Research, Southern Finance Association;Southwestern Finance Association, vol. 17(4), pages 561-584, December.
    3. Broadie, Mark & Detemple, Jerome, 1995. "American Capped Call Options on Dividend-Paying Assets," The Review of Financial Studies, Society for Financial Studies, vol. 8(1), pages 161-191.
    4. Chance, Don M, 1994. "The Pricing and Hedging of Limited Exercise Caps and Spreads," Journal of Financial Research, Southern Finance Association;Southwestern Finance Association, vol. 17(4), pages 561-584, Winter.
    Full references (including those not matched with items on IDEAS)

    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. Lim, Terence & Lo, Andrew W. & Merton, Robert C. & Scholes, Myron S., 2006. "The Derivatives Sourcebook," Foundations and Trends(R) in Finance, now publishers, vol. 1(5–6), pages 365-572, April.
    2. Mark Broadie & Jérôme Detemple, 1996. "Recent Advances in Numerical Methods for Pricing Derivative Securities," CIRANO Working Papers 96s-17, CIRANO.
    3. B. Gao J. Huang, "undated". "The Valuation of American Barrier Options Using the Decomposition Technique," New York University, Leonard N. Stern School Finance Department Working Paper Seires 99-002, New York University, Leonard N. Stern School of Business-.
    4. Lihua Zhang & Weiguo Zhang & Weijun Xu & Xiang Shi, 2014. "A Modified Least-Squares Simulation Approach to Value American Barrier Options," Computational Economics, Springer;Society for Computational Economics, vol. 44(4), pages 489-506, December.
    5. Weihan Li & Jin E. Zhang & Xinfeng Ruan & Pakorn Aschakulporn, 2024. "An empirical study on the early exercise premium of American options: Evidence from OEX and XEO options," Journal of Futures Markets, John Wiley & Sons, Ltd., vol. 44(7), pages 1117-1153, July.
    6. Jérôme Detemple, 1999. "American Options: Symmetry Properties," CIRANO Working Papers 99s-45, CIRANO.
    7. Jérôme Detemple, 2014. "Optimal Exercise for Derivative Securities," Annual Review of Financial Economics, Annual Reviews, vol. 6(1), pages 459-487, December.
    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. Detemple, Jérôme & Laminou Abdou, Souleymane & Moraux, Franck, 2020. "American step options," European Journal of Operational Research, Elsevier, vol. 282(1), pages 363-385.
    10. Pawe{l} Stc{e}pniak & Zbigniew Palmowski, 2025. "Pricing time-capped American options using Least Squares Monte Carlo method," Papers 2503.01040, arXiv.org.
    11. Steinar Ekern & Svein-Arne Persson, 1996. "Exotic Unit-Linked Life Insurance Contracts," The Geneva Risk and Insurance Review, Palgrave Macmillan;International Association for the Study of Insurance Economics (The Geneva Association), vol. 21(1), pages 35-63, June.
    12. Tsvetelin S. Zaevski, 2024. "Quadratic American Strangle Options in Light of Two-Sided Optimal Stopping Problems," Mathematics, MDPI, vol. 12(10), pages 1-27, May.
    13. 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.
    14. Mark Broadie & Jérôme Detemple, 1996. "American Options on Dividend-Paying Assets," CIRANO Working Papers 96s-16, CIRANO.
    15. F. Andre-le Pogamp & F. Moraux, 2004. "Valuing callable convertible bonds: a reduced approach," Applied Financial Economics, Taylor & Francis Journals, vol. 14(10), pages 743-749.
    16. Jérôme Detemple & Weidong Tian, 2002. "The Valuation of American Options for a Class of Diffusion Processes," Management Science, INFORMS, vol. 48(7), pages 917-937, July.
    17. Dayanik, Savas & Karatzas, Ioannis, 2003. "On the optimal stopping problem for one-dimensional diffusions," Stochastic Processes and their Applications, Elsevier, vol. 107(2), pages 173-212, October.
    18. Don M. Chance, 1994. "The Pricing And Hedging Of Limited Exercise Caps And Spreads," Journal of Financial Research, Southern Finance Association;Southwestern Finance Association, vol. 17(4), pages 561-584, December.
    19. Chiarella, Carl & Ziogas, Andrew, 2005. "Evaluation of American strangles," Journal of Economic Dynamics and Control, Elsevier, vol. 29(1-2), pages 31-62, January.
    20. Masahiko Egami & Rusudan Kevkhishvili, 2017. "A Direct Solution Method for Pricing Options in Regime-switching Models," Papers 1711.08883, arXiv.org, revised Sep 2018.

    More about this item

    Statistics

    Access and download statistics

    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:arx:papers:1707.06138. 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: arXiv administrators (email available below). General contact details of provider: http://arxiv.org/ .

    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.