Portfolio choice, portfolio liquidation, and portfolio transition under drift uncertainty

Abstract

This paper presents several models addressing optimal portfolio choice, optimal portfolio liquidation, and optimal portfolio transition issues, in which the expected returns of risky assets are unknown. Our approach is based on a coupling between Bayesian learning and dynamic programming techniques that leads to partial differential equations. It enables to recover the well-known results of Karatzas and Zhao in a framework à la Merton, but also to deal with cases where martingale methods are no longer available. In particular, we address optimal portfolio choice, portfolio liquidation, and portfolio transition problems in a framework à la Almgren–Chriss, and we build therefore a model in which the agent takes into account in his decision process both the liquidity of assets and the uncertainty with respect to their expected return.

This is a preview of subscription content, log in to check access.

Fig. 1
Fig. 2
Fig. 3

Notes

  1. 1.

    Markowitz was awarded the Nobel Prize in 1990 for his work. For a brief history of portfolio theory, see [35].

  2. 2.

    This problem in continuous time is now referred to as Merton’s problem.

  3. 3.

    It is noteworthy that this approach can be carried out in the frequentist case as well.

  4. 4.

    Guéant et al. also used the Almgren–Chriss framework to tackle the pricing, hedging, and execution issues of Accelerated Share Repurchase contracts—see [20, 23].

  5. 5.

    Almgren and Lorenz used Bayesian techniques in optimal execution (see [4]), but they considered myopic agents with respect to learning.

  6. 6.

    This assumption can be slightly relaxed, but we consider this simple one to simplify the statement of our results.

  7. 7.

    Because we are dealing with asset returns, the class of compactly supported distributions is sufficient, from a financial point of view, to deal with almost all relevant cases. Gaussian distributions are not in that class but Gaussian priors are approximations of real-life beliefs that are used mainly for their convenience in computations.

  8. 8.

    We omit the proofs in this subsection. They are similar to those presented in Sect. 3.

  9. 9.

    It is clear from the form of Eq. (78) that the solution is a polynomial of degree 2 in \(\beta -r\vec {1}\).

  10. 10.

    We omit the proofs in this subsection. They are similar to those presented in Sect. 3.

  11. 11.

    Unlike in the previous sections where we used the classical Black–Scholes (log-normal) dynamics, we consider here the Bachelier dynamics. This dynamics is indeed standard in the optimal execution literature, although it raises the problem of negative prices.

  12. 12.

    The results we obtain in this section can be generalized if the process is only piecewise continuous.

  13. 13.

    This process can be set to very small values for modelling the night.

  14. 14.

    It is a relaxed form of the classical optimal liquidation problem.

  15. 15.

    It is a relaxed form of optimal transition problem.

  16. 16.

    The function d should not be confused with the number d of risky assets.

  17. 17.

    The matrix A is a scalar in the one-asset case.

References

  1. 1.

    Almgren, R., Chriss, N.: Value under liquidation. Risk 12(12), 61–63 (1999)

    Google Scholar 

  2. 2.

    Almgren, R., Chriss, N.: Optimal execution of portfolio transactions. J. Risk 3, 5–40 (2001)

    Article  Google Scholar 

  3. 3.

    Almgren, R., Li, T.M.: Option hedging with smooth market impact. Mark. Microstruct. Liq. 2(1), 1650002 (2016)

    Article  Google Scholar 

  4. 4.

    Almgren, R., Lorenz, J.: Bayesian adaptive trading with a daily cycle. J. Trading 1(4), 38–46 (2006)

    Article  Google Scholar 

  5. 5.

    Bain, A., Crisan, D.: Fundamentals of Stochastic Filtering. Springer, Berlin (2009)

    Google Scholar 

  6. 6.

    Björk, T., Davis, M., Landén, C.: Optimal investment under partial information. Math. Methods Oper. Res. 71(2), 371–399 (2010)

    MathSciNet  MATH  Article  Google Scholar 

  7. 7.

    Black, F., Litterman, R.: Global portfolio optimization. Financ. Anal. J. 48(5), 28–43 (1992)

    Article  Google Scholar 

  8. 8.

    Brendle, S.: Portfolio selection under incomplete information. Stoch. Process. Appl. 116(5), 701–723 (2006)

    MathSciNet  MATH  Article  Google Scholar 

  9. 9.

    Casgrain, P., Jaimungal, S.: Trading algorithms with learning in latent alpha models. Working paper (2017)

  10. 10.

    Chris, L., Rogers, G.: The relaxed investor and parameter uncertainty. Financ. Stoch. 5(2), 131–154 (2001)

    MathSciNet  MATH  Article  Google Scholar 

  11. 11.

    Cvitanić, J., Karatzas, I.: Convex duality in constrained portfolio optimization. Ann. Appl. Probab. 2, 767–818 (1992)

    MathSciNet  MATH  Article  Google Scholar 

  12. 12.

    Cvitanić, J., Lazrak, A., Martellini, L., Zapatero, F.: Dynamic portfolio choice with parameter uncertainty and the economic value of analysts’ recommendations. Rev. Financ. Stud. 19(4), 1113–1156 (2006)

    Article  Google Scholar 

  13. 13.

    Danilova, A., Monoyios, M., Ng, A.: Optimal investment with inside information and parameter uncertainty. Math. Financ. Econ. 3(1), 13–38 (2010)

    MathSciNet  MATH  Article  Google Scholar 

  14. 14.

    Davis, M., Lleo, S.: Black Litterman in continuous time: the case for filtering. Quant. Financ. Lett. 1(1), 30–35 (2013)

    Article  Google Scholar 

  15. 15.

    Ekström, E., Vaicenavicius, J.: Optimal liquidation of an asset under drift uncertainty. SIAM J. Financ. Math. 7(1), 357–381 (2016)

    MathSciNet  MATH  Article  Google Scholar 

  16. 16.

    Fernandez-Tapia, J.: High-frequency trading with on-line learning. Working paper (2015)

  17. 17.

    Fouque, J.-P., Papanicolaou, A., Sircar, R.: Filtering and portfolio optimization with stochastic unobserved drift in asset returns. Commun. Math. Sci. 13(4), 935–953 (2015)

    MathSciNet  MATH  Article  Google Scholar 

  18. 18.

    Fouque, J.-P., Papanicolaou, A., Sircar, R.: Perturbation analysis for investment portfolios under partial information with expert opinions. SIAM J. Control Optim. 55(3), 1534–1566 (2017)

    MathSciNet  MATH  Article  Google Scholar 

  19. 19.

    Friedman, A.: Partial Differential Equations of Parabolic Type. Courier Dover Publications, Mineola (2008)

    Google Scholar 

  20. 20.

    Guéant, O.: Optimal execution of asr contracts with fixed notional. J. Risk 19(3), 77–99 (2017)

    MathSciNet  Article  Google Scholar 

  21. 21.

    Guéant, O.: Optimal execution and block trade pricing: a general framework. Appl. Math. Financ. 22(4), 336–365 (2015)

    MathSciNet  MATH  Article  Google Scholar 

  22. 22.

    Guéant, O., Jiang, P.: Option pricing and hedging with execution costs and market impact. Math. Financ. 27(3), 803–831 (2017)

    MathSciNet  MATH  Article  Google Scholar 

  23. 23.

    Guéant, O., Jiang, P., Royer, G.: Accelerated share repurchase: pricing and execution strategy. Int. J. Theor. Appl. Financ. 18(3), 1550019 (2015)

    MathSciNet  MATH  Article  Google Scholar 

  24. 24.

    Guéant, O.: The Financial Mathematics of Market Liquidity: From Optimal Execution to Market Making. CRC Press, Boca Raton (2016)

    Google Scholar 

  25. 25.

    Honda, T.: Optimal portfolio choice for unobservable and regime-switching mean returns. J. Econ. Dyn. Control 28(1), 45–78 (2003)

    MathSciNet  MATH  Article  Google Scholar 

  26. 26.

    Karatzas, I., Lehoczky, J.P., Shreve, S.E.: Optimal portfolio and consumption decisions for a “small investor” on a finite horizon. SIAM J. Control Optim. 25(6), 1557–1586 (1987)

    MathSciNet  MATH  Article  Google Scholar 

  27. 27.

    Karatzas, I., Shreve, S.E.: Brownian Motion and Stochastic Calculus, vol. 113. Springer, Berlin (2012)

    Google Scholar 

  28. 28.

    Karatzas, I., Zhao, X.: Bayesian adaptive portfolio optimization. Columbia University, Preprint (1998)

  29. 29.

    Lakner, P.: Utility maximization with partial information. Stoch. Process. Appl. 56(2), 247–273 (1995)

    MathSciNet  MATH  Article  Google Scholar 

  30. 30.

    Lakner, P.: Optimal trading strategy for an investor the case of partial information. Stoch. Process. Appl. 76(1), 77–97 (1998)

    MathSciNet  MATH  Article  Google Scholar 

  31. 31.

    Laruelle, S., Lehalle, C.-A., Pagès, G.: Optimal posting price of limit orders: learning by trading. Math. Financ. Econ. 7(3), 359–403 (2013)

    MathSciNet  MATH  Article  Google Scholar 

  32. 32.

    Li, Y., Qiao, H., Wang, S., Zhang, L.: Time-consistent investment strategy under partial information. Insur. Math. Econ. 65(C), 187–197 (2015)

    MathSciNet  MATH  Article  Google Scholar 

  33. 33.

    Liptser, R., Shiryaev, A.N.: Statistics of Stochastics Processes, vol. 1,2. Springer, Berlin (2001)

    Google Scholar 

  34. 34.

    Markowitz, H.M.: Portfolio selection. J. Financ. 7(1), 77–91 (1952)

    Google Scholar 

  35. 35.

    Markowitz, H.M.: The early history of portfolio theory: 1600–1960. Financ. Anal. J. 55(4), 5–16 (1999)

    Article  Google Scholar 

  36. 36.

    Merton, R.C.: Lifetime portfolio selection under uncertainty: the continuous-time case. Rev. Econ. Stat. 51, 247–257 (1969)

    Article  Google Scholar 

  37. 37.

    Merton, R.C.: Optimum consumption and portfolio rules in a continuous-time model. J. Econ. Theory 3(4), 373–413 (1971)

    MathSciNet  MATH  Article  Google Scholar 

  38. 38.

    Monoyios, M.: Optimal investment and hedging under partial and inside information. Radon Ser. Comput. Appl. Math. 8, 371–410 (2009)

    MathSciNet  MATH  Google Scholar 

  39. 39.

    Putschögl, W., Sass, J.: Optimal consumption and investment under partial information. Decis. Econ. Financ. 31(2), 137–170 (2008)

    MathSciNet  MATH  Article  Google Scholar 

  40. 40.

    Rieder, U., Bäuerle, N.: Portfolio optimization with unobservable Markov-modulated drift process. J. Appl. Probab. 42, 362–378 (2005)

    MathSciNet  MATH  Article  Google Scholar 

  41. 41.

    Rishel, R.: Optimal portfolio management with partial observations and power utility function. In: Stochastic Analysis, Control, Optimization and Applications, pp. 605–619. Springer (1999)

  42. 42.

    Samuelson, P.A.: Lifetime portfolio selection by dynamic stochastic programming. Rev. Econ. Stat. 51, 239–246 (1969)

    Article  Google Scholar 

  43. 43.

    Sass, J., Haussmann, U.: Optimizing the terminal wealth under partial information: the drift process as a continuous time Markov chain. Financ. Stoch. 8(4), 553–577 (2004)

    MathSciNet  MATH  Article  Google Scholar 

  44. 44.

    Sass, J., Westphal, D., Wunderlich, R.: Expert opinions and logarithmic utility maximization for multivariate stock returns with Gaussian drift. Int. J. Theor. Appl. Financ. 20(04), 1750022 (2017)

    MathSciNet  MATH  Article  Google Scholar 

  45. 45.

    Tobin, J.: Liquidity preference as behavior towards risk. Rev. Econ. Stud. 25(2), 65–86 (1958)

    Article  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Olivier Guéant.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This research has been conducted with the support of the Research Initiative “Modélisation des marchés actions et dérivés” financed by HSBC France under the aegis of the Europlace Institute of Finance. The authors would like to thank Rama Cont (Imperial College), Nicolas Grandchamp des Raux (HSBC France), Charles-Albert Lehalle (CFM and Imperial College), Jean-Michel Lasry (Institut Louis Bachelier), Huyên Pham (Université Paris-Diderot), and Christopher Ulph (HSBC London) for the conversations they had on the subject.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Bismuth, A., Guéant, O. & Pu, J. Portfolio choice, portfolio liquidation, and portfolio transition under drift uncertainty. Math Finan Econ 13, 661–719 (2019). https://doi.org/10.1007/s11579-019-00241-1

Download citation

Keywords

  • Optimal portfolio choice
  • Optimal execution
  • Optimal portfolio liquidation
  • Optimal portfolio transition
  • Bayesian learning
  • Online learning
  • Stochastic optimal control
  • Hamilton–Jacobi–Bellman equations

PACS

  • G110
  • C110
  • C180