IDEAS home Printed from https://ideas.repec.org/p/arx/papers/1110.6289.html
   My bibliography  Save this paper

Portfolio optimisation under non-linear drawdown constraints in a semimartingale financial model

Author

Listed:
  • Vladimir Cherny
  • Jan Obloj

Abstract

A drawdown constraint forces the current wealth to remain above a given function of its maximum to date. We consider the portfolio optimisation problem of maximising the long-term growth rate of the expected utility of wealth subject to a drawdown constraint, as in the original setup of Grossman and Zhou (1993). We work in an abstract semimartingale financial market model with a general class of utility functions and drawdown constraints. We solve the problem by showing that it is in fact equivalent to an unconstrained problem with a suitably modified utility function. Both the value function and the optimal investment policy for the drawdown problem are given explicitly in terms of their counterparts in the unconstrained problem.

Suggested Citation

  • Vladimir Cherny & Jan Obloj, 2011. "Portfolio optimisation under non-linear drawdown constraints in a semimartingale financial model," Papers 1110.6289, arXiv.org, revised Apr 2013.
    • Handle: RePEc:arx:papers:1110.6289
    as

    Download full text from publisher

    File URL: http://arxiv.org/pdf/1110.6289
    File Function: Latest version
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Sanford J. Grossman & Zhongquan Zhou, 1993. "Optimal Investment Strategies For Controlling Drawdowns," Mathematical Finance, Wiley Blackwell, vol. 3(3), pages 241-276, July.
    2. Romuald Elie & Nizar Touzi, 2008. "Optimal lifetime consumption and investment under a drawdown constraint," Finance and Stochastics, Springer, vol. 12(3), pages 299-330, July.
    3. W. H. Fleming & S. J. Sheu, 2000. "Risk‐Sensitive Control and an Optimal Investment Model," Mathematical Finance, Wiley Blackwell, vol. 10(2), pages 197-213, April.
    4. Huberman, Gur & Ross, Stephen, 1983. "Portfolio Turnpike Theorems, Risk Aversion, and Regularly Varying Utility Functions," Econometrica, Econometric Society, vol. 51(5), pages 1345-1361, September.
    5. Dumas, Bernard & Luciano, Elisa, 1991. "An Exact Solution to a Dynamic Portfolio Choice Problem under Transactions Costs," Journal of Finance, American Finance Association, vol. 46(2), pages 577-595, June.
    6. Bansal, Ravi & Lehmann, Bruce N., 1997. "Growth-Optimal Portfolio Restrictions On Asset Pricing Models," Macroeconomic Dynamics, Cambridge University Press, vol. 1(2), pages 333-354, June.
    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. David Landriault & Bin Li & Hongzhong Zhang, 2017. "A Unified Approach for Drawdown (Drawup) of Time-Homogeneous Markov Processes," Papers 1702.07786, arXiv.org.
    2. Sergio Ortobelli Lozza & Enrico Angelelli & Alda Ndoci, 2019. "Timing portfolio strategies with exponential Lévy processes," Computational Management Science, Springer, vol. 16(1), pages 97-127, February.
    3. 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.
    4. 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).
    5. Ola Mahmoud, 2015. "The Temporal Dimension of Risk," Papers 1501.01573, arXiv.org, revised Jun 2016.
    6. Stanislaus Maier-Paape & Andreas Platen & Qiji Jim Zhu, 2019. "A General Framework for Portfolio Theory. Part III: Multi-Period Markets and Modular Approach," Risks, MDPI, vol. 7(2), pages 1-31, June.
    7. Ankush Agarwal & Ronnie Sircar, 2016. "Portfolio Benchmarking under Drawdown Constraint and Stochastic Sharpe Ratio," Papers 1610.08558, arXiv.org.
    8. Leonie Violetta Brinker, 2021. "Minimal Expected Time in Drawdown through Investment for an Insurance Diffusion Model," Risks, MDPI, vol. 9(1), pages 1-18, January.
    9. Ankush Agarwal & Ronnie Sircar, 2017. "Portfolio Benchmarking under Drawdown Constraint and Stochastic Sharpe Ratio," Working Papers hal-01388399, HAL.
    10. Chen, Xinfu & Landriault, David & Li, Bin & Li, Dongchen, 2015. "On minimizing drawdown risks of lifetime investments," Insurance: Mathematics and Economics, Elsevier, vol. 65(C), pages 46-54.
    11. Paolo Guasoni & Jan Obłój, 2016. "The Incentives Of Hedge Fund Fees And High-Water Marks," Mathematical Finance, Wiley Blackwell, vol. 26(2), pages 269-295, April.
    12. 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.
    13. 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.
    14. 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.
    15. 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.
    16. Wang, Wenyuan & Chen, Ping & Li, Shuanming, 2020. "Generalized expected discounted penalty function at general drawdown for Lévy risk processes," Insurance: Mathematics and Economics, Elsevier, vol. 91(C), pages 12-25.
    17. Linus Wilson, 2016. "Discrete Portfolio Adjustment with Fixed Transaction Costs," The Review of Finance and Banking, Academia de Studii Economice din Bucuresti, Romania / Facultatea de Finante, Asigurari, Banci si Burse de Valori / Catedra de Finante, vol. 8(2), pages 055-060, December.

    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. 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.
    2. Vladimir Cherny & Jan Obloj, 2013. "Optimal portfolios of a long-term investor with floor or drawdown constraints," Papers 1305.6831, arXiv.org.
    3. Paolo Guasoni & Scott Robertson, 2012. "Portfolios and risk premia for the long run," Papers 1203.1399, arXiv.org.
    4. Paolo Guasoni & Gu Wang, 2015. "Hedge and mutual funds’ fees and the separation of private investments," Finance and Stochastics, Springer, vol. 19(3), pages 473-507, July.
    5. T. Arun, 2012. "The Merton Problem with a Drawdown Constraint on Consumption," Papers 1210.5205, arXiv.org.
    6. Hongzhong Zhang, 2018. "Stochastic Drawdowns," World Scientific Books, World Scientific Publishing Co. Pte. Ltd., number 10078, August.
    7. Paolo Guasoni & Jan Obłój, 2016. "The Incentives Of Hedge Fund Fees And High-Water Marks," Mathematical Finance, Wiley Blackwell, vol. 26(2), pages 269-295, April.
    8. Michael J. Klass & Krzysztof Nowicki, 2010. "On The Consumption/Distribution Theorem Under The Long-Run Growth Criterion Subject To A Drawdown Constraint," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 13(06), pages 931-957.
    9. Traian A. Pirvu & Gordan Zitkovic, 2007. "Maximizing the Growth Rate under Risk Constraints," Papers 0706.0480, arXiv.org.
    10. Wang, Wenyuan & Chen, Ping & Li, Shuanming, 2020. "Generalized expected discounted penalty function at general drawdown for Lévy risk processes," Insurance: Mathematics and Economics, Elsevier, vol. 91(C), pages 12-25.
    11. Paolo Guasoni & Gu Wang, 2012. "Hedge and Mutual Funds' Fees and the Separation of Private Investments," Papers 1208.4799, arXiv.org, revised Oct 2014.
    12. Ren Liu & Johannes Muhle-Karbe & Marko H. Weber, 2014. "Rebalancing with Linear and Quadratic Costs," Papers 1402.5306, arXiv.org, revised Sep 2017.
    13. Jean-Pierre Fouque & Ruimeng Hu, 2016. "Asymptotic Optimal Strategy for Portfolio Optimization in a Slowly Varying Stochastic Environment," Papers 1603.03538, arXiv.org, revised Nov 2016.
    14. Ankush Agarwal & Ronnie Sircar, 2017. "Portfolio Benchmarking under Drawdown Constraint and Stochastic Sharpe Ratio," Working Papers hal-01388399, HAL.
    15. Zabarankin, Michael & Pavlikov, Konstantin & Uryasev, Stan, 2014. "Capital Asset Pricing Model (CAPM) with drawdown measure," European Journal of Operational Research, Elsevier, vol. 234(2), pages 508-517.
    16. Leonie Violetta Brinker, 2021. "Minimal Expected Time in Drawdown through Investment for an Insurance Diffusion Model," Risks, MDPI, vol. 9(1), pages 1-18, January.
    17. Bahman Angoshtari & Erhan Bayraktar & Virginia R. Young, 2018. "Optimal Dividend Distribution Under Drawdown and Ratcheting Constraints on Dividend Rates," Papers 1806.07499, arXiv.org, revised Mar 2019.
    18. Scott Robertson & Hao Xing, 2014. "Long Term Optimal Investment in Matrix Valued Factor Models," Papers 1408.7010, arXiv.org.
    19. Chen, Xinfu & Landriault, David & Li, Bin & Li, Dongchen, 2015. "On minimizing drawdown risks of lifetime investments," Insurance: Mathematics and Economics, Elsevier, vol. 65(C), pages 46-54.
    20. Stefan Gerhold & Paolo Guasoni & Johannes Muhle-Karbe & Walter Schachermayer, 2014. "Transaction costs, trading volume, and the liquidity premium," Finance and Stochastics, Springer, vol. 18(1), pages 1-37, January.

    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:1110.6289. 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.