Electronic Communications in Probability

Continuous-Time Portfolio Optimisation for a Behavioural Investor with Bounded Utility on Gains

Miklós Rásonyi and Andrea Meireles Rodrigues

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This paper examines an optimal investment problem in a continuous-time (essentially) complete financial market with a finite horizon. We deal with an investor who behaves consistently with principles of Cumulative Prospect Theory, and whose utility function on gains is bounded above. The well-posedness of the optimisation problemis trivial, and a necessary condition for the existence of an optimal trading strategyis derived. This condition requires that the investor’s probability distortion function on losses does not tend to 0 near 0 faster than a given rate, which is determined by the utility function. Under additional assumptions, we show that this condition is indeed the borderline for attainability, in the sense that for slower convergence of the distortion function there does exist an optimal portfolio.

Article information

Electron. Commun. Probab., Volume 19 (2014), paper no. 38, 13 pp.

Accepted: 23 June 2014
First available in Project Euclid: 7 June 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 91G10: Portfolio theory
Secondary: 49J55: Problems involving randomness [See also 93E20] 60H30: Applications of stochastic analysis (to PDE, etc.) 93E20: Optimal stochastic control

Behavioural finance Bounded utility Choquet integral Continuous-time models Market completeness Non-concave utility Optimal portfolio Probability distortion

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Rásonyi, Miklós; Meireles Rodrigues, Andrea. Continuous-Time Portfolio Optimisation for a Behavioural Investor with Bounded Utility on Gains. Electron. Commun. Probab. 19 (2014), paper no. 38, 13 pp. doi:10.1214/ECP.v19-2990. https://projecteuclid.org/euclid.ecp/1465316740

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  • Arrow, Kenneth J. Alternative approaches to the theory of choice in risk-taking situtations. Econometrica 19, (1951). 404–437.
  • Arrow, Kenneth J. Essays in the theory of risk-bearing. North-Holland Publishing Co., Amsterdam-London, 1970. vii+278 pp.
  • Arrow, Kenneth J. The use of unbounded utility functions in expected-utility maximization: response, Quart. J. Econom. 88 (1974), no. 1, 136–138.
  • Berkelaar, A.B.; Kouwenberg, R.; Post, T. Optimal portfolio choice under loss aversion, Rev. Econom. Statist. 86 (2004), no. 4, 973–987.
  • Carlier, G.; Dana, R.-A. Optimal demand for contingent claims when agents have law invariant utilities. Math. Finance 21 (2011), no. 2, 169–201.
  • Cvitanić, JakÅ¡a; Karatzas, Ioannis. Hedging and portfolio optimization under transaction costs: a martingale approach. Math. Finance 6 (1996), no. 2, 133–165.
  • Föllmer, Hans; Schied, Alexander. Stochastic finance. An introduction in discrete time. Second revised and extended edition. de Gruyter Studies in Mathematics, 27. Walter de Gruyter & Co., Berlin, 2004. xii+459 pp. ISBN: 3-11-018346-3
  • Jin, Hanqing; Zhou, Xun Yu. Behavioral portfolio selection in continuous time. Math. Finance 18 (2008), no. 3, 385–426.
  • Kahneman, D.; Tversky, A. Prospect theory: An analysis of decision under risk, Econometrica 47 (1979), no. 2, 263–292.
  • Kramkov, D.; Schachermayer, W. The asymptotic elasticity of utility functions and optimal investment in incomplete markets. Ann. Appl. Probab. 9 (1999), no. 3, 904–950.
  • Markowitz, H.M. Investment for the long run: new evidence for an old rule, J. Finance 31 (1976), no. 5, 1273–1286.
  • Menger, K. Das Unsicherheitsmoment in der Wertlehre, Zeitschrift für Nationalükonomie 5 (1934), no. 4, 459–485.
  • Menger, M. The role of uncertainty in economics (Das Unsicherheitsmoment in der Wertlehre), Essays in mathematical economics in honor of Oscar Morgenstern, Princeton University Press, 1967, pp. 211–231.
  • Muraviev, Roman; Rogers, L. C. G. Utilities bounded below. Ann. Finance 9 (2013), no. 2, 271–289.
  • Prelec, Drazen. The probability weighting function. Econometrica 66 (1998), no. 3, 497–527.
  • Rásonyi, Miklós; Rodrigues, Andrea M. Optimal portfolio choice for a behavioural investor in continuous-time markets. Ann. Finance 9 (2013), no. 2, 291–318.
  • Rásonyi, Miklós; Stettner, Lukasz. On utility maximization in discrete-time financial market models. Ann. Appl. Probab. 15 (2005), no. 2, 1367–1395.
  • Reichlin, C. Behavioural portfolio selection: asymptotics and stability along a sequence of models, Math. Finance (2013).
  • Ryan, T.M. The use of unbounded utility functions in expected-utility maximization: comment, Quart. J. Econom. 88 (1974), no. 1, 133–135.
  • Samuelson, P.A. St. Petersburg paradoxes: defanged, dissected, and historically described, Journal of Economic Literature 15 (1977), no. 1, 24–55.
  • Savage, Leonard J. The foundations of statistics. John Wiley & Sons, Inc., New York; Chapman & Hill, Ltd., London, 1954. xv+294 pp.
  • Tversky, A.; Kahneman, D. Advances in prospect theory: cumulative representation of uncertainty, Journal of Risk and Uncertainty 5 (1992), no. 4, 297–323.
  • von Neumann, John; Morgenstern, Oskar. Theory of Games and Economic Behavior. Princeton University Press, Princeton, New Jersey, 1944. xviii+625 pp.