IDEAS home Printed from https://ideas.repec.org/a/eee/spapps/v126y2016i7p2038-2061.html
   My bibliography  Save this article

Perpetual American options in diffusion-type models with running maxima and drawdowns

Author

Listed:
  • Gapeev, Pavel V.
  • Rodosthenous, Neofytos

Abstract

We study perpetual American option pricing problems in an extension of the Black–Merton–Scholes model in which the dividend and volatility rates of the underlying risky asset depend on the running values of its maximum and maximum drawdown. The optimal exercise times are shown to be the first times at which the underlying asset hits certain boundaries depending on the running values of the associated maximum and maximum drawdown processes. We obtain closed-form solutions to the equivalent free-boundary problems for the value functions with smooth fit at the optimal stopping boundaries and normal reflection at the edges of the state space of the resulting three-dimensional Markov process. The optimal exercise boundaries for the perpetual American options on the maximum of the market depth with fixed and floating strikes are determined as the minimal solutions of certain first-order nonlinear ordinary differential equations.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:spapps:v:126:y:2016:i:7:p:2038-2061
    DOI: 10.1016/j.spa.2016.01.003
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0304414916000041
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.spa.2016.01.003?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. Forde, Martin, 2011. "A diffusion-type process with a given joint law for the terminal level and supremum at an independent exponential time," Stochastic Processes and their Applications, Elsevier, vol. 121(12), pages 2802-2817.
    2. Pospisil, Libor & Vecer, Jan & Hadjiliadis, Olympia, 2009. "Formulas for stopped diffusion processes with stopping times based on drawdowns and drawups," Stochastic Processes and their Applications, Elsevier, vol. 119(8), pages 2563-2578, August.
    3. Peskir, Goran, 2012. "Optimal detection of a hidden target: The median rule," Stochastic Processes and their Applications, Elsevier, vol. 122(5), pages 2249-2263.
    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. Baurdoux, E.J. & Kyprianou, A.E. & Pardo, J.C., 2011. "The Gapeev-Kühn stochastic game driven by a spectrally positive Lévy process," Stochastic Processes and their Applications, Elsevier, vol. 121(6), pages 1266-1289, June.
    6. Mijatović, Aleksandar & Pistorius, Martijn R., 2012. "On the drawdown of completely asymmetric Lévy processes," Stochastic Processes and their Applications, Elsevier, vol. 122(11), pages 3812-3836.
    7. Guo, Xin & Zervos, Mihail, 2010. "[pi] options," Stochastic Processes and their Applications, Elsevier, vol. 120(7), pages 1033-1059, July.
    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. Hongzhong Zhang, 2018. "Stochastic Drawdowns," World Scientific Books, World Scientific Publishing Co. Pte. Ltd., number 10078, January.
    2. 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.
    3. 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.
    4. 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.
    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., 2020. "Optimal stopping problems for running minima with positive discounting rates," Statistics & Probability Letters, Elsevier, vol. 167(C).
    7. 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.

    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. 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.
    2. 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.
    3. 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.
    4. Baurdoux, E.J. & Palmowski, Z. & Pistorius, M.R., 2017. "On future drawdowns of Lévy processes," Stochastic Processes and their Applications, Elsevier, vol. 127(8), pages 2679-2698.
    5. 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.
    6. Zhang, Gongqiu & Li, Lingfei, 2023. "A general method for analysis and valuation of drawdown risk," Journal of Economic Dynamics and Control, Elsevier, vol. 152(C).
    7. 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.
    8. 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.
    9. Gapeev, Pavel V., 2020. "Optimal stopping problems for running minima with positive discounting rates," Statistics & Probability Letters, Elsevier, vol. 167(C).
    10. Salminen, Paavo & Vallois, Pierre, 2020. "On the maximum increase and decrease of one-dimensional diffusions," Stochastic Processes and their Applications, Elsevier, vol. 130(9), pages 5592-5604.
    11. David Landriault & Bin Li & Hongzhong Zhang, 2017. "A Unified Approach for Drawdown (Drawup) of Time-Homogeneous Markov Processes," Papers 1702.07786, arXiv.org.
    12. Giovanni Masala & Filippo Petroni, 2023. "Drawdown risk measures for asset portfolios with high frequency data," Annals of Finance, Springer, vol. 19(2), pages 265-289, June.
    13. Hongzhong Zhang, 2018. "Stochastic Drawdowns," World Scientific Books, World Scientific Publishing Co. Pte. Ltd., number 10078, January.
    14. Zbigniew Palmowski & Joanna Tumilewicz, 2017. "Fair valuation of L\'evy-type drawdown-drawup contracts with general insured and penalty functions," Papers 1712.04418, arXiv.org, revised Feb 2018.
    15. David Landriault & Bin Li & Hongzhong Zhang, 2014. "On the Frequency of Drawdowns for Brownian Motion Processes," Papers 1403.1183, arXiv.org.
    16. Hongzhong Zhang & Olympia Hadjiliadis, 2012. "Drawdowns and the Speed of Market Crash," Methodology and Computing in Applied Probability, Springer, vol. 14(3), pages 739-752, September.
    17. Zhang, Hongzhong & Leung, Tim & Hadjiliadis, Olympia, 2013. "Stochastic modeling and fair valuation of drawdown insurance," Insurance: Mathematics and Economics, Elsevier, vol. 53(3), pages 840-850.
    18. Cui, Zhenyu & Nguyen, Duy, 2016. "Omega diffusion risk model with surplus-dependent tax and capital injections," Insurance: Mathematics and Economics, Elsevier, vol. 68(C), pages 150-161.
    19. Zhenyu Cui, 2014. "Omega risk model with tax," Papers 1403.7680, arXiv.org.
    20. 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.

    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:spapps:v:126:y:2016:i:7:p:2038-2061. 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/505572/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.