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Portfolio with consumption and terminal gains under loss aversion

  • 1.

    School of Mathematical Sciences, University of Science and Technology of China, Hefei 230026, China

  • 2.

    Department of Statistics and Finance, University of Science and Technology of China, Hefei 230026, China

Cite this:
https://doi.org/10.3969/j.issn.0253-2778.2016.11.006
  • Received Date: 08 October 2015
  • Accepted Date: 10 April 2016
  • Rev Recd Date: 10 April 2016
  • Publish Date: 30 November 2016
    Abstract

  • Abstract

    A continuous-time portfolio selection problem with consumption and terminal gains was considered in the framework of prospect theory. The Inada conditions for the utility functions were discarded by assuming a regularity condition on the terminal utility. First, the problem with the reference point depending on the wealth was considered, and the corresponding Hamilton-Jacobi-Bellman (HJB) equation was derived. Then, by assuming that the terminal utility relies on the gains process, a new model with the reference point as part of the control was established. This makes the optimal control problem non-Markovian. To deal with this problem, the idea for transforming the Asian option pricing problem into a Markov problem was used. A singular Markov control problem was yielded, and then acorresponding HJB variational inequality was derived.

    Abstract

    A continuous-time portfolio selection problem with consumption and terminal gains was considered in the framework of prospect theory. The Inada conditions for the utility functions were discarded by assuming a regularity condition on the terminal utility. First, the problem with the reference point depending on the wealth was considered, and the corresponding Hamilton-Jacobi-Bellman (HJB) equation was derived. Then, by assuming that the terminal utility relies on the gains process, a new model with the reference point as part of the control was established. This makes the optimal control problem non-Markovian. To deal with this problem, the idea for transforming the Asian option pricing problem into a Markov problem was used. A singular Markov control problem was yielded, and then acorresponding HJB variational inequality was derived.
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    • References(18)
  • [1]
    BARBERIS N, HUANG M, SANTOS T.Prospect theory and asset prices[J].Quarterly Journal of Economics, 2001, 116: 1-53.
    [2]
    BERKELAAR A, KOUWENBERG R, POST T.Optimal portfolio choice under loss aversion[J].The Review of Economics and Statistics, 2004, 86(4): 973-987.
    [3]
    BUCKDAHN R, LI J.Stochastic differential games and viscosity solutions of Hamilton-Jacobi-Bellman-Isaacs equations[J].SIAM Journal on Control and Optimization, 2004, 47(1):444-475.
    [4]
    CHEUNG H L A.Utility maximisation: Non-concave utility and non-linear expectation [D].Oxford: Mathematical Institute, University of Oxford, 2011.
    [5]
    FLEMING W H, SONER H M.Controlled Markov Processes and Viscosity Solutions [M].2nd edition.New York: Springer-Verlag, 2006.
    [6]
    JIN H, ZHOU X Y.Behavioral portfolio selection in continuous time[J].Mathematical Finance, 2008, 18(3): 385-426.
    [7]
    KAHNEMAN D, TVERSKY A.Prospect theory: An analysis of decision under risk[J].Econometrica, 1979, 47: 263-290.
    [8]
    KARATZAS I, SHREVE S E.Methods of Mathematical Finance[M].New York: Springer-Verlag, 1998.
    [9]
    KARATZAS I, LEHOCZKY J P, SHREVE S E.Optimal portfolio and consumption decisions for a “small investor” on a finite horizon[J].SIAM Journal of Control and Optimizaiton, 1987, 25(6): 1 557-1 586.
    [10]
    KUSHNER H J, DUPUIS P G.Numerical Methods for Stochastic Control Problems in Continuous Time[M].New York: Springer-Verlag, 1992.
    [11]
    MI H, ZHANG S G.Continuous-time portfolio selection with loss aversion in an incomplete market [J].Operations Research Transactions, 2012, 16(1): 1-12.
    [12]
    PHAM H.Continuous-time Stochastic Control and Optimization with Financial Applications [M].New York: Springer-Verlag, 2009.
    [13]
    SHREVE S E.Stochastic Calculus for Finance Ⅱ: Continuous-Time Models [M].New York: Springer, 2004.
    [14]
    TVERSKY A, KAHNEMAN D.Advances in prospect theory: Cumulative representation of uncertainty [J].Journal of Risk and Uncertainty, 1992, 5: 297-323.
    [15]
    YONG J M, ZHOU X Y.Stochastic Controls: Hamiltonian Systems and HJB Equations [M].New York: Springer-Verlag, 1999.
    [16]
    ZHOU X Y.Mathematicalising behavioural finance [C]// Proceedings of the International Congress of Mathematicians, Hyderabad, India.Delhi: Hindustan Book Agency, 2010.
    [17]
    张松.行为金融学中的资产组合选择问题[D].北京：北京大学，2011.
    Zhang S. Portfolio selection in behavioral finance[D].Beijing: Peking University, 2011.
    [18]
    ZHANG S, JIN H Q, ZHOU X Y.Behavioral portfolio selection with loss control[J].Acta Mathematica Sinica, English Series, 2011, 27(2): 255-274.
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    [1]
    BARBERIS N, HUANG M, SANTOS T.Prospect theory and asset prices[J].Quarterly Journal of Economics, 2001, 116: 1-53.
    [2]
    BERKELAAR A, KOUWENBERG R, POST T.Optimal portfolio choice under loss aversion[J].The Review of Economics and Statistics, 2004, 86(4): 973-987.
    [3]
    BUCKDAHN R, LI J.Stochastic differential games and viscosity solutions of Hamilton-Jacobi-Bellman-Isaacs equations[J].SIAM Journal on Control and Optimization, 2004, 47(1):444-475.
    [4]
    CHEUNG H L A.Utility maximisation: Non-concave utility and non-linear expectation [D].Oxford: Mathematical Institute, University of Oxford, 2011.
    [5]
    FLEMING W H, SONER H M.Controlled Markov Processes and Viscosity Solutions [M].2nd edition.New York: Springer-Verlag, 2006.
    [6]
    JIN H, ZHOU X Y.Behavioral portfolio selection in continuous time[J].Mathematical Finance, 2008, 18(3): 385-426.
    [7]
    KAHNEMAN D, TVERSKY A.Prospect theory: An analysis of decision under risk[J].Econometrica, 1979, 47: 263-290.
    [8]
    KARATZAS I, SHREVE S E.Methods of Mathematical Finance[M].New York: Springer-Verlag, 1998.
    [9]
    KARATZAS I, LEHOCZKY J P, SHREVE S E.Optimal portfolio and consumption decisions for a “small investor” on a finite horizon[J].SIAM Journal of Control and Optimizaiton, 1987, 25(6): 1 557-1 586.
    [10]
    KUSHNER H J, DUPUIS P G.Numerical Methods for Stochastic Control Problems in Continuous Time[M].New York: Springer-Verlag, 1992.
    [11]
    MI H, ZHANG S G.Continuous-time portfolio selection with loss aversion in an incomplete market [J].Operations Research Transactions, 2012, 16(1): 1-12.
    [12]
    PHAM H.Continuous-time Stochastic Control and Optimization with Financial Applications [M].New York: Springer-Verlag, 2009.
    [13]
    SHREVE S E.Stochastic Calculus for Finance Ⅱ: Continuous-Time Models [M].New York: Springer, 2004.
    [14]
    TVERSKY A, KAHNEMAN D.Advances in prospect theory: Cumulative representation of uncertainty [J].Journal of Risk and Uncertainty, 1992, 5: 297-323.
    [15]
    YONG J M, ZHOU X Y.Stochastic Controls: Hamiltonian Systems and HJB Equations [M].New York: Springer-Verlag, 1999.
    [16]
    ZHOU X Y.Mathematicalising behavioural finance [C]// Proceedings of the International Congress of Mathematicians, Hyderabad, India.Delhi: Hindustan Book Agency, 2010.
    [17]
    张松.行为金融学中的资产组合选择问题[D].北京：北京大学，2011.
    Zhang S. Portfolio selection in behavioral finance[D].Beijing: Peking University, 2011.
    [18]
    ZHANG S, JIN H Q, ZHOU X Y.Behavioral portfolio selection with loss control[J].Acta Mathematica Sinica, English Series, 2011, 27(2): 255-274.
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