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An explicit solution for an optimal stopping/optimal control problem which models an asset sale

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  • Vicky Henderson
  • David Hobson

Abstract

In this article we study an optimal stopping/optimal control problem which models the decision facing a risk-averse agent over when to sell an asset. The market is incomplete so that the asset exposure cannot be hedged. In addition to the decision over when to sell, the agent has to choose a control strategy which corresponds to a feasible wealth process. We formulate this problem as one involving the choice of a stopping time and a martingale. We conjecture the form of the solution and verify that the candidate solution is equal to the value function. The interesting features of the solution are that it is available in a very explicit form, that for some parameter values the optimal strategy is more sophisticated than might originally be expected, and that although the setup is based on continuous diffusions, the optimal martingale may involve a jump process. One interpretation of the solution is that it is optimal for the risk-averse agent to gamble.

Suggested Citation

  • Vicky Henderson & David Hobson, 2008. "An explicit solution for an optimal stopping/optimal control problem which models an asset sale," Papers 0806.4061, arXiv.org, revised Nov 2008.
  • Handle: RePEc:arx:papers:0806.4061
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    File URL: http://arxiv.org/pdf/0806.4061
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    References listed on IDEAS

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    1. Avinash K. Dixit & Robert S. Pindyck, 1994. "Investment under Uncertainty," Economics Books, Princeton University Press, edition 1, number 5474.
    2. Merton, Robert C, 1969. "Lifetime Portfolio Selection under Uncertainty: The Continuous-Time Case," The Review of Economics and Statistics, MIT Press, vol. 51(3), pages 247-257, August.
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    Cited by:

    1. Xiongfei Jian & Xun Li & Fahuai Yi, 2014. "Optimal Investment with Stopping in Finite Horizon," Papers 1406.6940, arXiv.org.
    2. Emilie Fabre & Guillaume Royer & Nizar Touzi, 2013. "Liquidation of an indivisible asset with independent investment," Papers 1312.2754, arXiv.org, revised Feb 2015.
    3. Sen, Rik & Tumarkin, Robert, 2015. "Stocking up: Executive optimism, option exercise, and share retention," Journal of Financial Economics, Elsevier, vol. 118(2), pages 399-430.
    4. 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.
    5. David Hobson & Yeqi Zhu, 2014. "Optimal consumption and sale strategies for a risk averse agent," Papers 1409.3394, arXiv.org.
    6. Jakub Trybu{l}a, 2014. "Merton problem with one additional indivisible asset," Papers 1403.3223, arXiv.org.
    7. Jason S. Anquandah & Leonid V. Bogachev, 2019. "Optimal Stopping and Utility in a Simple Model of Unemployment Insurance," Papers 1902.06175, arXiv.org, revised Sep 2019.
    8. Neofytos Rodosthenous & Hongzhong Zhang, 2020. "When to sell an asset amid anxiety about drawdowns," Papers 2006.00282, arXiv.org.
    9. Naveed Chehrazi & Peter W. Glynn & Thomas A. Weber, 2019. "Dynamic Credit-Collections Optimization," Management Science, INFORMS, vol. 67(6), pages 2737-2769, June.
    10. Jingtang Ma & Jie Xing & Harry Zheng, 2018. "Global Closed-form Approximation of Free Boundary for Optimal Investment Stopping Problems," Papers 1810.09397, arXiv.org.
    11. Erik Ekstrom & Juozas Vaicenavicius, 2015. "Optimal liquidation of an asset under drift uncertainty," Papers 1509.00686, arXiv.org.
    12. Jason S. Anquandah & Leonid V. Bogachev, 2019. "Optimal Stopping and Utility in a Simple Modelof Unemployment Insurance," Risks, MDPI, vol. 7(3), pages 1-41, September.

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