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Stochastic Modeling and Fair Valuation of Drawdown Insurance

Author

Listed:
  • Hongzhong Zhang
  • Tim Leung
  • Olympia Hadjiliadis

Abstract

This paper studies the stochastic modeling of market drawdown events and the fair valuation of insurance contracts based on drawdowns. We model the asset drawdown process as the current relative distance from the historical maximum of the asset value. We first consider a vanilla insurance contract whereby the protection buyer pays a constant premium over time to insure against a drawdown of a pre-specified level. This leads to the analysis of the conditional Laplace transform of the drawdown time, which will serve as the building block for drawdown insurance with early cancellation or drawup contingency. For the cancellable drawdown insurance, we derive the investor's optimal cancellation timing in terms of a two-sided first passage time of the underlying drawdown process. Our model can also be applied to insure against a drawdown by a defaultable stock. We provide analytic formulas for the fair premium and illustrate the impact of default risk.

Suggested Citation

  • Hongzhong Zhang & Tim Leung & Olympia Hadjiliadis, 2013. "Stochastic Modeling and Fair Valuation of Drawdown Insurance," Papers 1310.3860, arXiv.org.
  • Handle: RePEc:arx:papers:1310.3860
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    File URL: http://arxiv.org/pdf/1310.3860
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    References listed on IDEAS

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    1. William N. Goetzmann & Jonathan E. Ingersoll & Stephen A. Ross, 2003. "High-Water Marks and Hedge Fund Management Contracts," Journal of Finance, American Finance Association, vol. 58(4), pages 1685-1718, August.
    2. Vadim Linetsky, 2006. "Pricing Equity Derivatives Subject To Bankruptcy," Mathematical Finance, Wiley Blackwell, vol. 16(2), pages 255-282.
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    5. Olympia Hadjiliadis & Jan Vecer, 2006. "Drawdowns preceding rallies in the Brownian motion model," Quantitative Finance, Taylor & Francis Journals, vol. 6(5), pages 403-409.
    6. Vikas Agarwal & Naveen D. Daniel & Narayan Y. Naik, 2009. "Role of Managerial Incentives and Discretion in Hedge Fund Performance," Journal of Finance, American Finance Association, vol. 64(5), pages 2221-2256, October.
    7. 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.
    8. Merton, Robert C., 1976. "Option pricing when underlying stock returns are discontinuous," Journal of Financial Economics, Elsevier, vol. 3(1-2), pages 125-144.
    9. Larry Eisenberg & Thomas H. Noe, 2001. "Systemic Risk in Financial Systems," Management Science, INFORMS, vol. 47(2), pages 236-249, February.
    10. Sanford J. Grossman & Zhongquan Zhou, 1993. "Optimal Investment Strategies For Controlling Drawdowns," Mathematical Finance, Wiley Blackwell, vol. 3(3), pages 241-276.
    11. Moore, Kristen S., 2009. "Optimal surrender strategies for equity-indexed annuity investors," Insurance: Mathematics and Economics, Elsevier, vol. 44(1), pages 1-18, February.
    12. 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.
    13. Riccardo Rebonato & Valerio Gaspari, 2006. "Analysis of drawdowns and drawups in the US$ interest-rate market," Quantitative Finance, Taylor & Francis Journals, vol. 6(4), pages 297-326.
    14. Raphaël Douady & A.N. Shiryaev & Marc Yor, 2000. "On Probability Characteristics of "Downfalls" in a Standard Brownian Motion," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) hal-01477104, HAL.
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    Citations

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    Cited by:

    1. 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.
    2. repec:eee:insuma:v:79:y:2018:i:c:p:137-147 is not listed on IDEAS
    3. David Landriault & Bin Li & Hongzhong Zhang, 2017. "A Unified Approach for Drawdown (Drawup) of Time-Homogeneous Markov Processes," Papers 1702.07786, arXiv.org.
    4. repec:eee:spapps:v:127:y:2017:i:8:p:2679-2698 is not listed on IDEAS
    5. repec:eee:insuma:v:79:y:2018:i:c:p:1-14 is not listed on IDEAS
    6. Ola Mahmoud, 2015. "The Temporal Dimension of Risk," Papers 1501.01573, arXiv.org, revised Jun 2016.
    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. Zbigniew Palmowski & Joanna Tumilewicz, 2017. "Pricing insurance drawdown-type contracts with underlying L\'evy assets," Papers 1701.01891, arXiv.org, revised Oct 2017.
    9. David Landriault & Bin Li & Hongzhong Zhang, 2014. "On the Frequency of Drawdowns for Brownian Motion Processes," Papers 1403.1183, arXiv.org.
    10. Baurdoux, Erik J. & Palmowski, Z & Pistorius, Martijn R, 2017. "On future drawdowns of Lévy processes," LSE Research Online Documents on Economics 84342, London School of Economics and Political Science, LSE Library.

    More about this item

    JEL classification:

    • C61 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Optimization Techniques; Programming Models; Dynamic Analysis
    • G01 - Financial Economics - - General - - - Financial Crises
    • G13 - Financial Economics - - General Financial Markets - - - Contingent Pricing; Futures Pricing
    • G22 - Financial Economics - - Financial Institutions and Services - - - Insurance; Insurance Companies; Actuarial Studies

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