Annals of Finance

, Volume 15, Issue 1, pp 1–28 | Cite as

Optimal risk-averse timing of an asset sale: trending versus mean-reverting price dynamics

  • Tim LeungEmail author
  • Zheng Wang
Research Article


This paper studies the optimal risk-averse timing to sell a risky asset. The investor’s risk preference is described by the exponential, power, or log utility. Two stochastic models are considered for the asset price— the geometric Brownian motion and exponential Ornstein–Uhlenbeck models—to account for, respectively, the trending and mean-reverting price dynamics. In all cases, we derive the optimal thresholds and certainty equivalents to sell the asset, and compare them across models and utilities, with emphasis on their dependence on asset price, risk aversion, and quantity. We find that the timing option may render the investor’s value function and certainty equivalent non-concave in price. Numerical results are provided to illustrate the investor’s strategies and the premium associated with optimally timing to sell.


Asset sale Optimal stopping Certainty equivalent Variational inequality 

JEL Classification

C41 G11 G12 


  1. Alili, L., Patie, P., Pedersen, J.: Representations of the first hitting time density of an Ornstein–Uhlenbeck process. Stoch Models 21(4), 967–980 (2005)CrossRefGoogle Scholar
  2. Borodin, A., Salminen, P.: Handbook of Brownian Motion: Facts and Formulae, 2nd edn. Basel: Birkhauser(2002)Google Scholar
  3. Ekström, E., Vaicenavicius, J.: Optimal liquidation of an asset under drift uncertainty. Working Paper (2016)Google Scholar
  4. Ekström, E., Lindberg, C., Tysk, J.: Optimal liquidation of a pairs trade. In: Nunno, G.D., Øksendal, B. (eds.) Advanced Mathematical Methods for Finance, Chap 9, pp. 247–255. Berlin: Springer (2011)Google Scholar
  5. Erdélyi, A., Magnus, W., Oberhettinger, F., Tricomi, F.G.: Higher Transcendental Functions, vol. 2. New York: McGraw-Hill (1953)Google Scholar
  6. Evans, J., Henderson, V., Hobson, D.: Optimal timing for an indivisible asset sale. Math Finance 18(4), 545–567 (2008)CrossRefGoogle Scholar
  7. Ewald, C.O., Wang, W.K.: Irreversible investment with Cox–Ingersoll–Ross type mean reversion. Math Soc Sci 59(3), 314–318 (2010)CrossRefGoogle Scholar
  8. Henderson, V.: Valuing the option to invest in an incomplete market. Math Financ Econ 1(2), 103–128 (2007)CrossRefGoogle Scholar
  9. Henderson, V.: Prospect theory, liquidation, and the disposition effect. Manag Sci 58(2), 445–460 (2012)CrossRefGoogle Scholar
  10. Henderson, V., Hobson, D.: Optimal liquidation of derivative portfolios. Math Finance 21(3), 365–382 (2011)CrossRefGoogle Scholar
  11. Leung, T., Li, X.: Optimal mean reversion trading with transaction costs and stop-loss exit. Int J Theor Appl Finance 18(3), 15,500 (2015)CrossRefGoogle Scholar
  12. Leung, T., Ludkovski, M.: Accounting for risk aversion in derivatives purchase timing. Math Financ Econ 6(4), 363–386 (2012)CrossRefGoogle Scholar
  13. Leung, T., Shirai, Y.: Optimal derivative liquidation timing under path-dependent risk penalties. J Financ Eng 2(1), 1550,004 (2015)CrossRefGoogle Scholar
  14. Leung, T., Li, X., Wang, Z.: Optimal starting–stopping and switching of a CIR process with fixed costs. Risk Decis Anal 5(2), 149–161 (2014)Google Scholar
  15. Leung, T., Li, X., Wang, Z.: Optimal multiple trading times under the exponential OU model with transaction costs. Stoch Models 31(4), 554–587 (2015)CrossRefGoogle Scholar
  16. Øksendal, B.: Stochastic Differential Equations: An Introduction with Applications. Berlin: Springer (2003)Google Scholar
  17. Pedersen, J.L., Peskir, G.: Optimal mean-variance selling strategies. Math Financ Econ 10(2), 203–220 (2016)CrossRefGoogle Scholar
  18. Peskir, G., Shiryaev, A.N.: Optimal Stopping and Free-boundary problems. Lectures in Mathematics, Birkhauser, ETH Zurich (2006)Google Scholar
  19. Yang, Z., Ewald, C.O.: Utility based pricing and exercising of real options under geometric mean reversion and risk aversion toward idiosyncratic risk. Math Methods Oper Res 68(1), 97–123 (2008)CrossRefGoogle Scholar
  20. Zervos, M., Johnson, T., Alazemi, F.: Buy-low and sell-high investment strategies. Math Finance 23(3), 560–578 (2013)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Applied Mathematics DepartmentUniversity of WashingtonSeattleUSA
  2. 2.IEOR DepartmentColumbia UniversityNew YorkUSA

Personalised recommendations