IDEAS home Printed from https://ideas.repec.org/p/arx/papers/1610.00999.html
   My bibliography  Save this paper

Exponential utility maximization under model uncertainty for unbounded endowments

Author

Listed:
  • Daniel Bartl

Abstract

We consider the robust exponential utility maximization problem in discrete time: An investor maximizes the worst case expected exponential utility with respect to a family of nondominated probabilistic models of her endowment by dynamically investing in a financial market, and statically in available options. We show that, for any measurable random endowment (regardless of whether the problem is finite or not) an optimal strategy exists, a dual representation in terms of (calibrated) martingale measures holds true, and that the problem satisfies the dynamic programming principle (in case of no options). Further it is shown that the value of the utility maximization problem converges to the robust superhedging price as the risk aversion parameter gets large, and examples of nondominated probabilistic models are discussed.

Suggested Citation

  • Daniel Bartl, 2016. "Exponential utility maximization under model uncertainty for unbounded endowments," Papers 1610.00999, arXiv.org, revised Feb 2019.
  • Handle: RePEc:arx:papers:1610.00999
    as

    Download full text from publisher

    File URL: http://arxiv.org/pdf/1610.00999
    File Function: Latest version
    Download Restriction: no

    References listed on IDEAS

    as
    1. Anis Matoussi & Dylan Possamaï & Chao Zhou, 2015. "Robust Utility Maximization In Nondominated Models With 2bsde: The Uncertain Volatility Model," Mathematical Finance, Wiley Blackwell, vol. 25(2), pages 258-287, April.
    2. Daniel Lacker, 2015. "Liquidity, risk measures, and concentration of measure," Papers 1510.07033, arXiv.org, revised Oct 2015.
    3. Patrick Cheridito & Michael Kupper & Ludovic Tangpi, 2016. "Duality formulas for robust pricing and hedging in discrete time," Papers 1602.06177, arXiv.org, revised Sep 2017.
    4. Fabio Maccheroni & Massimo Marinacci & Aldo Rustichini, 2006. "Ambiguity Aversion, Robustness, and the Variational Representation of Preferences," Econometrica, Econometric Society, vol. 74(6), pages 1447-1498, November.
    5. Merton, Robert C., 1971. "Optimum consumption and portfolio rules in a continuous-time model," Journal of Economic Theory, Elsevier, vol. 3(4), pages 373-413, December.
    6. David G. Hobson, 1998. "Robust hedging of the lookback option," Finance and Stochastics, Springer, vol. 2(4), pages 329-347.
    7. Marco Frittelli, 2000. "The Minimal Entropy Martingale Measure and the Valuation Problem in Incomplete Markets," Mathematical Finance, Wiley Blackwell, vol. 10(1), pages 39-52, January.
    8. Anne Gundel, 2005. "Robust utility maximization for complete and incomplete market models," Finance and Stochastics, Springer, vol. 9(2), pages 151-176, April.
    9. Gilboa, Itzhak & Schmeidler, David, 1989. "Maxmin expected utility with non-unique prior," Journal of Mathematical Economics, Elsevier, vol. 18(2), pages 141-153, April.
    10. B. Acciaio & M. Beiglböck & F. Penkner & W. Schachermayer, 2016. "A Model-Free Version Of The Fundamental Theorem Of Asset Pricing And The Super-Replication Theorem," Mathematical Finance, Wiley Blackwell, vol. 26(2), pages 233-251, April.
    11. Freddy Delbaen & Peter Grandits & Thorsten Rheinländer & Dominick Samperi & Martin Schweizer & Christophe Stricker, 2002. "Exponential Hedging and Entropic Penalties," Mathematical Finance, Wiley Blackwell, vol. 12(2), pages 99-123, April.
    12. Marcel Nutz, 2016. "Utility Maximization Under Model Uncertainty In Discrete Time," Mathematical Finance, Wiley Blackwell, vol. 26(2), pages 252-268, April.
    13. Ying Hu & Peter Imkeller & Matthias Muller, 2005. "Utility maximization in incomplete markets," Papers math/0508448, arXiv.org.
    14. J. Jacod & A.N. Shiryaev, 1998. "Local martingales and the fundamental asset pricing theorems in the discrete-time case," Finance and Stochastics, Springer, vol. 2(3), pages 259-273.
    15. Michael Mania & Martin Schweizer, 2005. "Dynamic exponential utility indifference valuation," Papers math/0508489, arXiv.org.
    16. Bruno Bouchard & Marcel Nutz, 2013. "Arbitrage and duality in nondominated discrete-time models," Papers 1305.6008, arXiv.org, revised Mar 2015.
    17. Mathias Beiglböck & Pierre Henry-Labordère & Friedrich Penkner, 2013. "Model-independent bounds for option prices—a mass transport approach," Finance and Stochastics, Springer, vol. 17(3), pages 477-501, July.
    18. Mathias Beiglbock & Pierre Henry-Labord`ere & Friedrich Penkner, 2011. "Model-independent Bounds for Option Prices: A Mass Transport Approach," Papers 1106.5929, arXiv.org, revised Feb 2013.
    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. Daniel Bartl & Michael Kupper & Ariel Neufeld, 2020. "Duality Theory for Robust Utility Maximization," Papers 2007.08376, arXiv.org.
    2. Romain Blanchard & Laurence Carassus, 2019. "No-arbitrage with multiple-priors in discrete time," Papers 1904.08780, arXiv.org, revised Oct 2019.
    3. Ariel Neufeld & Mario Šikić, 2019. "Nonconcave robust optimization with discrete strategies under Knightian uncertainty," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 90(2), pages 229-253, October.
    4. Mikl'os R'asonyi & Andrea Meireles-Rodrigues, 2018. "On Utility Maximisation Under Model Uncertainty in Discrete-Time Markets," Papers 1801.06860, arXiv.org, revised Jul 2020.
    5. Huy N. Chau & Miklos Rasonyi, 2018. "Robust utility maximization in markets with transaction costs," Papers 1803.04213, arXiv.org, revised Dec 2018.
    6. Daniel Bartl & Michael Kupper & David J. Prömel & Ludovic Tangpi, 2019. "Duality for pathwise superhedging in continuous time," Finance and Stochastics, Springer, vol. 23(3), pages 697-728, July.
    7. Kerem Ugurlu, 2019. "Robust Utility Maximization with Drift and Volatility Uncertainty," Papers 1909.05335, arXiv.org.
    8. Shuoqing Deng & Xiaolu Tan & Xiang Yu, 2020. "Utility Maximization with Proportional Transaction Costs Under Model Uncertainty," Mathematics of Operations Research, INFORMS, vol. 45(4), pages 1210-1236, November.
    9. Huy N. Chau & Miklós Rásonyi, 2019. "Robust utility maximisation in markets with transaction costs," Finance and Stochastics, Springer, vol. 23(3), pages 677-696, July.
    10. Daniel Bartl & Michael Kupper & David J. Promel & Ludovic Tangpi, 2017. "Duality for pathwise superhedging in continuous time," Papers 1705.02933, arXiv.org, revised Apr 2019.
    11. Daniel Bartl, 2016. "Conditional nonlinear expectations," Papers 1612.09103, arXiv.org, revised Mar 2019.
    12. Bartl, Daniel, 2020. "Conditional nonlinear expectations," Stochastic Processes and their Applications, Elsevier, vol. 130(2), pages 785-805.
    13. Daniel Bartl & Stephan Eckstein & Michael Kupper, 2020. "Limits of random walks with distributionally robust transition probabilities," Papers 2007.08815, arXiv.org.
    14. Derek Singh & Shuzhong Zhang, 2020. "Robust Arbitrage Conditions for Financial Markets," Papers 2004.09432, arXiv.org.
    15. Blanchard, Romain & Carassus, Laurence, 2020. "No-arbitrage with multiple-priors in discrete time," Stochastic Processes and their Applications, Elsevier, vol. 130(11), pages 6657-6688.

    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. Erhan Bayraktar & Gu Wang, 2018. "Quantile Hedging in a semi-static market with model uncertainty," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 87(2), pages 197-227, April.
    2. Sebastian Herrmann & Johannes Muhle-Karbe & Frank Thomas Seifried, 2017. "Hedging with small uncertainty aversion," Finance and Stochastics, Springer, vol. 21(1), pages 1-64, January.
    3. Romain Blanchard & Laurence Carassus, 2017. "Convergence of utility indifference prices to the superreplication price in a multiple-priors framework," Papers 1709.09465, arXiv.org, revised Oct 2020.
    4. Daniel Bartl & Michael Kupper & David J. Prömel & Ludovic Tangpi, 2019. "Duality for pathwise superhedging in continuous time," Finance and Stochastics, Springer, vol. 23(3), pages 697-728, July.
    5. Matteo Burzoni & Marco Frittelli & Zhaoxu Hou & Marco Maggis & Jan Ob{l}'oj, 2016. "Pointwise Arbitrage Pricing Theory in Discrete Time," Papers 1612.07618, arXiv.org, revised Feb 2018.
    6. Sebastian Herrmann & Johannes Muhle-Karbe & Frank Thomas Seifried, 2016. "Hedging with Small Uncertainty Aversion," Papers 1605.06429, arXiv.org.
    7. Sebastian Herrmann & Johannes Muhle-Karbe, 2017. "Model uncertainty, recalibration, and the emergence of delta–vega hedging," Finance and Stochastics, Springer, vol. 21(4), pages 873-930, October.
    8. Sebastian Herrmann & Florian Stebegg, 2017. "Robust Pricing and Hedging around the Globe," Papers 1707.08545, arXiv.org, revised Apr 2019.
    9. Matteo Burzoni & Marco Frittelli & Zhaoxu Hou & Marco Maggis & Jan Obłój, 2019. "Pointwise Arbitrage Pricing Theory in Discrete Time," Mathematics of Operations Research, INFORMS, vol. 44(3), pages 1034-1057, August.
    10. Beatrice Acciaio & Mathias Beiglboeck & Gudmund Pammer, 2020. "Weak Transport for Non-Convex Costs and Model-independence in a Fixed-Income Market," Papers 2011.04274, arXiv.org.
    11. Sebastian Herrmann & Johannes Muhle-Karbe, 2017. "Model Uncertainty, Recalibration, and the Emergence of Delta-Vega Hedging," Papers 1704.04524, arXiv.org.
    12. Huy N. Chau & Masaaki Fukasawa & Miklos Rasonyi, 2021. "Super-replication with transaction costs under model uncertainty for continuous processes," Papers 2102.02298, arXiv.org.
    13. Daniel Bartl & Michael Kupper & David J. Promel & Ludovic Tangpi, 2017. "Duality for pathwise superhedging in continuous time," Papers 1705.02933, arXiv.org, revised Apr 2019.
    14. Daniel Bartl & Patrick Cheridito & Michael Kupper, 2017. "Robust expected utility maximization with medial limits," Papers 1712.07699, arXiv.org, revised Nov 2018.
    15. Sergey Badikov & Mark H. A. Davis & Antoine Jacquier, 2018. "Perturbation analysis of sub/super hedging problems," Papers 1806.03543, arXiv.org, revised Nov 2019.
    16. Roger J. A. Laeven & Mitja Stadje, 2014. "Robust Portfolio Choice and Indifference Valuation," Mathematics of Operations Research, INFORMS, vol. 39(4), pages 1109-1141, November.
    17. David Hobson & Anthony Neuberger, 2017. "Model uncertainty and the pricing of American options," Finance and Stochastics, Springer, vol. 21(1), pages 285-329, January.
    18. Sergey Nadtochiy & Jan Obłój, 2017. "Robust Trading Of Implied Skew," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 20(02), pages 1-41, March.
    19. Pierre Henry-Labordère & Nizar Touzi, 2016. "An explicit martingale version of the one-dimensional Brenier theorem," Finance and Stochastics, Springer, vol. 20(3), pages 635-668, July.
    20. Patrick Cheridito & Matti Kiiski & David J. Promel & H. Mete Soner, 2019. "Martingale optimal transport duality," Papers 1904.04644, arXiv.org, revised Nov 2020.

    More about this item

    NEP fields

    This paper has been announced in the following NEP Reports:

    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:1610.00999. See general information about how to correct material in RePEc.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: (arXiv administrators). General contact details of provider: http://arxiv.org/ .

    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 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.

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service hosted by the Research Division of the Federal Reserve Bank of St. Louis . RePEc uses bibliographic data supplied by the respective publishers.