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Pricing insurance drawdown-type contracts with underlying Lévy assets

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  • 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
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    References listed on IDEAS

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    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.
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    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.

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