Abstract
We study cautious stochastic choice (CSC) agents facing optimal timing decisions in a dynamic setting. In an expected utility setting, the optimal strategy is always a threshold strategy—to stop/sell the first time the price process exits an interval. In contrast, we show that in the CSC setting, where the agent has a family of utility functions and is concerned with the worst case certainty equivalent, the optimal strategy may be of non-threshold form and may involve randomization. We provide some carefully constructed examples, including one where we can solve explicitly for the optimal stopping rule and show it is a non-trivial mixture of threshold strategies. Our model is consistent with recent experimental evidence in dynamic setups whereby individuals do not play cut-off or threshold strategies.
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References
Agranov, M., Ortoleva, P.: Stochastic choice and preferences for randomization. J. Polit. Econ. 125(1), 40–68 (2017)
Agranov, M., Ortoleva, P.: Ranges of Randomization. Working paper (2020)
Barberis, N., Xiong, W.: Realization utility. J. Financ. Econ. 104, 251–271 (2012)
Bielecki, T.R., Crépey, S., Jeanblanc, M., Rutkowski, M.: Arbitrage pricing of defaultable game options with applications to convertible bonds. Quant. Finance 8, 795–810 (2008)
Cerreia-Vioglio, S., Dillenberger, D., Ortoleva, P.: Cautious expected utility and the certainty effect. Econometrica 83, 693–728 (2015)
Cerreia-Vioglio, S., Dillenberger, D., Ortoleva, P., Riella, G.: Deliberately stochastic. Am. Econ. Rev. 109(7), 2425–2445 (2019)
Cox, A.M.G., Hobson, D.G.: An optimal Skorokhod embedding for diffusions. Stoch. Process. Appl. 111, 17–39 (2004)
Dayanik, S., Karatzas, I.: On the optimal stopping problem for one-dimensional diffusions. Stoch. Process. Appl. 107(2), 173–212 (2003)
Dixit, A.K., Pindyck, R.S.: Investment Under Uncertainty. Princeton University Press, Princeton (1994)
Duraj, J.: Optimal Stopping with General Risk Preferences. Working paper (2019)
Dwenger, N., Kübler, D., Weizsäcker, G.: Flipping a coin: evidence from university applications. J. Public Econ. 167, 240–250 (2018)
Dynkin, E.B.: Game variant of a problem on optimal stopping. Soviet Math. Dokl. 10, 270–274 (1969)
Ebert, S., Strack, P.: Until the bitter end: on prospect theory in a dynamic context. Am. Econ. Rev. 105(4), 1618–1633 (2015)
Feldman, P., Rehbeck, J.: Revealing a preference for mixtures: an experimental study of risk. Quant. Econ. (2020) (to appear)
Fischbacher, U., Hoffmann, G., Schudy, S.: The causal effect of stop-loss and take-gain orders on the disposition effect. Rev. Financ. Stud. 30(6), 2110–2129 (2017)
Fudenberg, D., Strack, P., Strzalecki, T.: Speed, accuracy and the optimal timing of choices. Am. Econ. Rev. 108(12), 3651–3684 (2018)
Fudenberg, D., Tirole, J.: Game Theory. The MIT Press, Cambridge (1991)
Grenadier, S.R.: The strategic exercise of options: Development cascades and overbuilding in real estate markets. J. Finance 51, 1653–1679 (1996)
Gul, F., Pesendorfer, W.: Random expected utility. Econometrica 74, 121–146 (2006)
Hansen, L.P., Sargent, T.J.: Robustness. Princeton University Press, Princeton (2008)
He, X., Hu, S., Obloj, J., Zhou, X.Y.: Path dependent and randomized strategies in Barberis’ Casino Gambling model. Oper. Res. 65(1), 97–103 (2017)
Henderson, V.: Prospect theory, liquidation and the disposition effect. Manag. Sci. 58(2), 445–460 (2012)
Henderson, V., Hobson, D., Tse, A.S.L.: Probability weighting, stop-loss and the disposition effect. J. Econ. Theory 178, 360–397 (2018)
Henderson, V., Hobson, D., Tse, A.S.L.: Randomized Strategies and Prospect Theory in a Dynamic Context. Journal of Economic Theory 168, 287–300 (2017)
Henderson, V., Hobson, D., Zeng, M.: Optimal stopping and the sufficiency of randomized thresholds. Electron. Commun. Probab. 23, 1–11 (2018)
Hendricks, K., Weiss, A., Wilson, C.: The war of attrition in continuous time with complete information. Int. Econ. Rev. 29(4), 663–680 (1988)
Hey, J.D., Orme, C.: Investigating generalizations of expected utility theory using experimental data. Econometrica 62(6), 1291–1326 (1994)
Hobson, D.: The Skorokhod embedding problem and model-independent bounds for option prices. In: Paris-Princeton Lectures on Mathematical Finance 2010, pp. 267–318. Springer, Berlin (2011)
Huang, Y.J., Nguyen-Huu, A., Zhou, X.Y.: General stopping behaviors of Naive and non-committed sophisticated agents, with application to probability distortion. Math. Financ. 30(1), 310–340 (2020)
Ingersoll, J.E., Jin, L.: Realization utility with reference-dependent preferences. Rev. Financ. Stud. 26(3), 723–767 (2013)
Johnson, E.J., Ratcliff, R.: Computational and process models of decision making in psychology and behavioral economics. In: Glimcher, P.W., Fehr, E. (eds.) Neutroeconomics: Decision Making and the Brain, pp. 35–48. Academic Press, Cambridge (2013)
Karni, E., Safra, Z.: Behaviorally consistent optimal stopping rules. J. Econ. Theory 51(2), 391–402 (1990)
Kifer, Y.: Game options. Finance Stochast. 4, 443–463 (2000)
Kyle, A.S., Ou-Yang, H., Xiong, W.: Prospect Theory and Liquidation Decisions. J. Econ. Theory 129, 273–288 (2006)
Luce, R.D.: Individual Choice Behavior: A Theoretical Analysis. Wiley, New York (1959)
Maccheroni, F.: Maxmin under risk. Econ. Theor. 19, 823–831 (2002)
Machina, M.J.: Stochastic choice functions generated from deterministic preferences over lotteries. Econ. J. 95(379), 575–594 (1985)
Markowitz, H.: The utility of wealth. J. Political Econom. 60(2), 151–158 (1952)
McCall, J.J.: Economics of information and job search. Q. J. Econ. 84(1), 113–126 (1970)
McDonald, R., Siegel, D.: The value of waiting to invest. Q. J. Econ. 101, 707–728 (1986)
McKean, H.P.: A free boundary problem for the heat equation arising from a problem in mathematical economics. Ind. Manag. Rev. 6, 32–39 (1965)
Merton, R.C.: Theory of rational option pricing. Bell J. Econ. Manag. Sci. 4, 141–183 (1973)
Odean, T.: Are investors reluctant to realize their losses? J. Finance 53(5), 1775–1798 (1998)
Oprea, R., Friedman, D., Anderson, S.T.: Learning to wait: a laboratory investigation. Rev. Econ. Stud. 76(3), 1103–1124 (2009)
Pedersen, J., Pekir, G.: The Azema-Yor embedding in non-singular diffusions. Stoch. Process. Appl. 96, 305–312 (2001)
Permana, Y.: Why do people prefer randomisation? An experimental investigation. Theor. Decis. 88, 73–96 (2020)
Peskir, G., Shiryaev, A.: Optimal Stopping and Free-Boundary Problems. Birkhauser Verlag, Basel (2006)
Radzik, T., Raghavan, T.E.S.: Duels. In: Aumann, R.J., Hart, S. (eds.), Handbook of Game Theory with Economic Applications, Vol. 2, pp. 761–768 (1994)
Sirbu, M., Shreve, S.E.: A two-person game for pricing convertible bonds. SIAM J. Control. Optim. 45(4), 1508–1539 (2006)
Stigler, G.: Information in the labor market. J. Polit. Econ. 70(5), 94–105 (1962)
Strack, P., Viefers, P.: Too proud to stop: regret in dynamic decisions. J. Eur. Econ. Assoc. 19(1), 165–199 (2021)
Touzi, N.: Martingale inequalities, optimal martingale transport, and robust superhedging. ESAIM Proc. Surv. 45, 32–47 (2014)
Tversky, A.: Intransitivity of preferences. Psychol. Rev. 76, 31 (1969)
Tversky, A., Kahneman, D.: Advances in prospect theory: cumulative representation of uncertainty. J. Risk Uncertain. 5(4), 297–323 (1992)
Wakker, P.: Separating marginal utility and probabilistic risk aversion. Theor. Decis. 36(1), 1–44 (1994)
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We would like to thank participants at the 10th Oxford-Princeton workshop (May 25–26, 2017), the Stochastics of Financial Markets Seminar, HU/TU Berlin (November, 2017), the Mathematics of Behavioral Economics and Knightian Uncertainty in Financial Markets Workshop, Bielefeld (May, 2018) and the FUR conference York (June 2018) for helpful comments. We also thank an anonymous referee for their comments. Matthew Zeng was supported by a Chancellor’s International Scholarship at the University of Warwick.
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Henderson, V., Hobson, D. & Zeng, M. Cautious stochastic choice, optimal stopping and deliberate randomization. Econ Theory 75, 887–922 (2023). https://doi.org/10.1007/s00199-022-01428-2
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DOI: https://doi.org/10.1007/s00199-022-01428-2