IDEAS home Printed from https://ideas.repec.org/a/eee/insuma/v79y2018icp1-14.html
   My bibliography  Save this article

Pricing insurance drawdown-type contracts with underlying Lévy assets

Author

Listed:
  • Palmowski, Zbigniew
  • Tumilewicz, Joanna

Abstract

In this paper we consider some insurance policies related to drawdown and drawup events of log-returns for an underlying asset modeled by a spectrally negative geometric Lévy process. We consider four contracts, three of which were introduced in Zhang (2013) for a geometric Brownian motion. The first one is an insurance contract where the protection buyer pays a constant premium until the drawdown of fixed size of log-returns occurs. In return he/she receives a certain insured amount at the drawdown epoch. The next insurance contract provides protection from any specified drawdown with a drawup contingency. This contract expires early if a certain fixed drawup event occurs prior to the fixed drawdown. The last two contracts are extensions of the previous ones by an additional cancellation feature which allows the investor to terminate the contract earlier. We focus on two problems: calculating the fair premium p for the basic contracts and identifying the optimal stopping rule for the policies with the cancellation feature. To do this we solve some two-sided exit problems related to drawdown and drawup of spectrally negative Lévy processes, which is of independent mathematical interest. We also heavily rely on the theory of optimal stopping.

Suggested Citation

  • Palmowski, Zbigniew & Tumilewicz, Joanna, 2018. "Pricing insurance drawdown-type contracts with underlying Lévy assets," Insurance: Mathematics and Economics, Elsevier, vol. 79(C), pages 1-14.
    • Handle: RePEc:eee:insuma:v:79:y:2018:i:c:p:1-14
    DOI: 10.1016/j.insmatheco.2017.12.007
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0167668717300161
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.insmatheco.2017.12.007?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. Libor Pospisil & Jan Vecer, 2010. "Portfolio sensitivity to changes in the maximum and the maximum drawdown," Quantitative Finance, Taylor & Francis Journals, vol. 10(6), pages 617-627.
    2. Hongzhong Zhang & Olympia Hadjiliadis, 2010. "Drawdowns and Rallies in a Finite Time-horizon," Methodology and Computing in Applied Probability, Springer, vol. 12(2), pages 293-308, June.
    3. Peter Carr & Hongzhong Zhang & Olympia Hadjiliadis, 2011. "Maximum Drawdown Insurance," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 14(08), pages 1195-1230.
    4. Sanford J. Grossman & Zhongquan Zhou, 1993. "Optimal Investment Strategies For Controlling Drawdowns," Mathematical Finance, Wiley Blackwell, vol. 3(3), pages 241-276, July.
    5. 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.
    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.
    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. Li, Shu & Zhou, Xiaowen, 2022. "The Parisian and ultimate drawdowns of Lévy insurance models," Insurance: Mathematics and Economics, Elsevier, vol. 107(C), pages 140-160.
    2. Landriault, David & Li, Bin & Lkabous, Mohamed Amine, 2021. "On the analysis of deep drawdowns for the Lévy insurance risk model," Insurance: Mathematics and Economics, Elsevier, vol. 100(C), pages 147-155.
    3. Zbigniew Palmowski & Joanna Tumilewicz, 2018. "Drawdown insurance contracts for the Lévy-type model with the phase-type jump distribution and general reward function," Collegium of Economic Analysis Annals, Warsaw School of Economics, Collegium of Economic Analysis, issue 51, pages 255-270.

    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. 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).
    2. David Landriault & Bin Li & Hongzhong Zhang, 2014. "On the Frequency of Drawdowns for Brownian Motion Processes," Papers 1403.1183, arXiv.org.
    3. Ola Mahmoud, 2015. "The Temporal Dimension of Risk," Papers 1501.01573, arXiv.org, revised Jun 2016.
    4. Zbigniew Palmowski & Joanna Tumilewicz, 2017. "Pricing insurance drawdown-type contracts with underlying L\'evy assets," Papers 1701.01891, arXiv.org, revised Oct 2017.
    5. David Landriault & Bin Li & Hongzhong Zhang, 2017. "A Unified Approach for Drawdown (Drawup) of Time-Homogeneous Markov Processes," Papers 1702.07786, arXiv.org.
    6. Hongzhong Zhang, 2018. "Stochastic Drawdowns," World Scientific Books, World Scientific Publishing Co. Pte. Ltd., number 10078, December.
    7. 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.
    8. 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.
    9. 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.
    10. Damiano Rossello & Silvestro Lo Cascio, 2021. "A refined measure of conditional maximum drawdown," Risk Management, Palgrave Macmillan, vol. 23(4), pages 301-321, December.
    11. Landriault, David & Li, Bin & Li, Shu, 2015. "Analysis of a drawdown-based regime-switching Lévy insurance model," Insurance: Mathematics and Economics, Elsevier, vol. 60(C), pages 98-107.
    12. Zbigniew Palmowski & Joanna Tumilewicz, 2018. "Drawdown insurance contracts for the Lévy-type model with the phase-type jump distribution and general reward function," Collegium of Economic Analysis Annals, Warsaw School of Economics, Collegium of Economic Analysis, issue 51, pages 255-270.
    13. Landriault, David & Li, Bin & Lkabous, Mohamed Amine, 2021. "On the analysis of deep drawdowns for the Lévy insurance risk model," Insurance: Mathematics and Economics, Elsevier, vol. 100(C), pages 147-155.
    14. 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.
    15. Zhenyu Cui & Duy Nguyen, 2018. "Magnitude and Speed of Consecutive Market Crashes in a Diffusion Model," Methodology and Computing in Applied Probability, Springer, vol. 20(1), pages 117-135, March.
    16. Li, Shu & Zhou, Xiaowen, 2022. "The Parisian and ultimate drawdowns of Lévy insurance models," Insurance: Mathematics and Economics, Elsevier, vol. 107(C), pages 140-160.
    17. Vladimir Cherny & Jan Obłój, 2013. "Portfolio optimisation under non-linear drawdown constraints in a semimartingale financial model," Finance and Stochastics, Springer, vol. 17(4), pages 771-800, October.
    18. 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.
    19. Mendes, Beatriz Vaz de Melo & Lavrado, Rafael Coelho, 2017. "Implementing and testing the Maximum Drawdown at Risk," Finance Research Letters, Elsevier, vol. 22(C), pages 95-100.
    20. Long Bai & Peng Liu, 2019. "Drawdown and Drawup for Fractional Brownian Motion with Trend," Journal of Theoretical Probability, Springer, vol. 32(3), pages 1581-1612, September.

    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:insuma:v:79:y:2018:i:c:p:1-14. 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/locate/inca/505554 .

    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.