IDEAS home Printed from https://ideas.repec.org/p/arx/papers/0902.1328.html
   My bibliography  Save this paper

On Az\'ema-Yor processes, their optimal properties and the Bachelier-drawdown equation

Author

Listed:
  • Laurent Carraro
  • Nicole El Karoui
  • Jan Ob{l}'oj

Abstract

We study the class of Az\'ema-Yor processes defined from a general semimartingale with a continuous running maximum process. We show that they arise as unique strong solutions of the Bachelier stochastic differential equation which we prove is equivalent to the drawdown equation. Solutions of the latter have the drawdown property: they always stay above a given function of their past maximum. We then show that any process which satisfies the drawdown property is in fact an Az\'ema-Yor process. The proofs exploit group structure of the set of Az\'ema-Yor processes, indexed by functions, which we introduce. We investigate in detail Az\'ema-Yor martingales defined from a nonnegative local martingale converging to zero at infinity. We establish relations between average value at risk, drawdown function, Hardy-Littlewood transform and its inverse. In particular, we construct Az\'ema-Yor martingales with a given terminal law and this allows us to rediscover the Az\'ema-Yor solution to the Skorokhod embedding problem. Finally, we characterize Az\'ema-Yor martingales showing they are optimal relative to the concave ordering of terminal variables among martingales whose maximum dominates stochastically a given benchmark.

Suggested Citation

  • Laurent Carraro & Nicole El Karoui & Jan Ob{l}'oj, 2009. "On Az\'ema-Yor processes, their optimal properties and the Bachelier-drawdown equation," Papers 0902.1328, arXiv.org, revised Sep 2012.
  • Handle: RePEc:arx:papers:0902.1328
    as

    Download full text from publisher

    File URL: http://arxiv.org/pdf/0902.1328
    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.
    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. He, Xue Dong & Hu, Sang & Obłój, Jan & Zhou, Xun Yu, 2019. "Two explicit Skorokhod embeddings for simple symmetric random walk," Stochastic Processes and their Applications, Elsevier, vol. 129(9), pages 3431-3445.
    2. Alfred Galichon & Pierre Henri-Labordère & Nizar Touzi, 2014. "A stochastic control approach to No-Arbitrage bounds given marginals, with an application to Lookback options," Post-Print hal-03460952, HAL.
    3. Hongzhong Zhang, 2018. "Stochastic Drawdowns," World Scientific Books, World Scientific Publishing Co. Pte. Ltd., number 10078, April.
    4. Alfred Galichon & Pierre Henri-Labordère & Nizar Touzi, 2014. "A stochastic control approach to No-Arbitrage bounds given marginals, with an application to Lookback options," SciencePo Working papers Main hal-03460952, HAL.
    5. Vladimir Cherny & Jan Obloj, 2013. "Optimal portfolios of a long-term investor with floor or drawdown constraints," Papers 1305.6831, arXiv.org.
    6. Tongseok Lim, 2023. "Replication of financial derivatives under extreme market models given marginals," Papers 2307.00807, arXiv.org.
    7. Avram, Florin & Vu, Nhat Linh & Zhou, Xiaowen, 2017. "On taxed spectrally negative Lévy processes with draw-down stopping," Insurance: Mathematics and Economics, Elsevier, vol. 76(C), pages 69-74.
    8. Cox, Alexander M.G. & Obłój, Jan, 2015. "On joint distributions of the maximum, minimum and terminal value of a continuous uniformly integrable martingale," Stochastic Processes and their Applications, Elsevier, vol. 125(8), pages 3280-3300.
    9. Najnudel, Joseph & Nikeghbali, Ashkan, 2012. "On some universal σ-finite measures related to a remarkable class of submartingales," Stochastic Processes and their Applications, Elsevier, vol. 122(4), pages 1582-1600.
    10. 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.
    11. 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.
    12. A. Galichon & P. Henry-Labord`ere & N. Touzi, 2014. "A stochastic control approach to no-arbitrage bounds given marginals, with an application to lookback options," Papers 1401.3921, arXiv.org.
    13. Constantinos Kardaras & Jan Obłój & Eckhard Platen, 2017. "The Numéraire Property And Long-Term Growth Optimality For Drawdown-Constrained Investments," Mathematical Finance, Wiley Blackwell, vol. 27(1), pages 68-95, January.
    14. Alfred Galichon & Pierre Henri-Labordère & Nizar Touzi, 2014. "A stochastic control approach to No-Arbitrage bounds given marginals, with an application to Lookback options," SciencePo Working papers hal-03460952, HAL.
    15. T. Arun, 2012. "The Merton Problem with a Drawdown Constraint on Consumption," Papers 1210.5205, arXiv.org.

    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. Neofytos Rodosthenous & Hongzhong Zhang, 2020. "When to sell an asset amid anxiety about drawdowns," Mathematical Finance, Wiley Blackwell, vol. 30(4), pages 1422-1460, October.
    2. Hongzhong Zhang, 2018. "Stochastic Drawdowns," World Scientific Books, World Scientific Publishing Co. Pte. Ltd., number 10078, April.
    3. 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.
    4. Alexander, Gordon J. & Baptista, Alexandre M., 2006. "Portfolio selection with a drawdown constraint," Journal of Banking & Finance, Elsevier, vol. 30(11), pages 3171-3189, November.
    5. Kentaro Imajo & Kentaro Minami & Katsuya Ito & Kei Nakagawa, 2020. "Deep Portfolio Optimization via Distributional Prediction of Residual Factors," Papers 2012.07245, arXiv.org.
    6. Drenovak, Mikica & Ranković, Vladimir & Urošević, Branko & Jelic, Ranko, 2022. "Mean-Maximum Drawdown Optimization of Buy-and-Hold Portfolios Using a Multi-objective Evolutionary Algorithm," Finance Research Letters, Elsevier, vol. 46(PA).
    7. 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.
    8. Sami Attaoui & Vincent Lacoste, 2013. "A scenario-based description of optimal American capital guaranteed strategies," Finance, Presses universitaires de Grenoble, vol. 34(2), pages 65-119.
    9. David Landriault & Bin Li & Hongzhong Zhang, 2014. "On the Frequency of Drawdowns for Brownian Motion Processes," Papers 1403.1183, arXiv.org.
    10. 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.
    11. Koichi Matsumoto, 2007. "Portfolio Insurance with Liquidity Risk," Asia-Pacific Financial Markets, Springer;Japanese Association of Financial Economics and Engineering, vol. 14(4), pages 363-386, December.
    12. Auh, Jun Kyung & Cho, Wonho, 2023. "Factor-based portfolio optimization," Economics Letters, Elsevier, vol. 228(C).
    13. Steven D. Moffitt, 2018. "Why Markets are Inefficient: A Gambling "Theory" of Financial Markets For Practitioners and Theorists," Papers 1801.01948, arXiv.org.
    14. Hanauer, Matthias X. & Lauterbach, Jochim G., 2019. "The cross-section of emerging market stock returns," Emerging Markets Review, Elsevier, vol. 38(C), pages 265-286.
    15. 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.
    16. Dangl, Thomas & Randl, Otto & Zechner, Josef, 2016. "Risk control in asset management: Motives and concepts," CFS Working Paper Series 546, Center for Financial Studies (CFS).
    17. 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.
    18. Anna Ananova & Rama Cont & Renyuan Xu, 2020. "Model-free Analysis of Dynamic Trading Strategies," Papers 2011.02870, arXiv.org, revised Mar 2025.
    19. Ren Liu & Johannes Muhle-Karbe & Marko H. Weber, 2014. "Rebalancing with Linear and Quadratic Costs," Papers 1402.5306, arXiv.org, revised Sep 2017.
    20. Antje Mahayni & Judith C. Schneider, 2016. "Minimum return guarantees, investment caps, and investment flexibility," Review of Derivatives Research, Springer, vol. 19(2), pages 85-111, July.

    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:0902.1328. 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.