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

  • Alexis Bismuth
  • Olivier GuéantEmail author
  • Jiang Pu
Article
  • 24 Downloads

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.

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 

Notes

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)Google Scholar
  3. 3.
    Almgren, R., Li, T.M.: Option hedging with smooth market impact. Mark. Microstruct. Liq. 2(1), 1650002 (2016)Google Scholar
  4. 4.
    Almgren, R., Lorenz, J.: Bayesian adaptive trading with a daily cycle. J. Trading 1(4), 38–46 (2006)Google Scholar
  5. 5.
    Bain, A., Crisan, D.: Fundamentals of Stochastic Filtering. Springer, Berlin (2009)zbMATHGoogle Scholar
  6. 6.
    Björk, T., Davis, M., Landén, C.: Optimal investment under partial information. Math. Methods Oper. Res. 71(2), 371–399 (2010)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Black, F., Litterman, R.: Global portfolio optimization. Financ. Anal. J. 48(5), 28–43 (1992)Google Scholar
  8. 8.
    Brendle, S.: Portfolio selection under incomplete information. Stoch. Process. Appl. 116(5), 701–723 (2006)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Casgrain, P., Jaimungal, S.: Trading algorithms with learning in latent alpha models. Working paper (2017)Google Scholar
  10. 10.
    Chris, L., Rogers, G.: The relaxed investor and parameter uncertainty. Financ. Stoch. 5(2), 131–154 (2001)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Cvitanić, J., Karatzas, I.: Convex duality in constrained portfolio optimization. Ann. Appl. Probab. 2, 767–818 (1992)MathSciNetzbMATHGoogle 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)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)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Davis, M., Lleo, S.: Black Litterman in continuous time: the case for filtering. Quant. Financ. Lett. 1(1), 30–35 (2013)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)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Fernandez-Tapia, J.: High-frequency trading with on-line learning. Working paper (2015)Google Scholar
  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)MathSciNetzbMATHGoogle 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)MathSciNetzbMATHGoogle 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)MathSciNetGoogle Scholar
  21. 21.
    Guéant, O.: Optimal execution and block trade pricing: a general framework. Appl. Math. Financ. 22(4), 336–365 (2015)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Guéant, O., Jiang, P.: Option pricing and hedging with execution costs and market impact. Math. Financ. 27(3), 803–831 (2017)MathSciNetzbMATHGoogle 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)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Guéant, O.: The Financial Mathematics of Market Liquidity: From Optimal Execution to Market Making. CRC Press, Boca Raton (2016)zbMATHGoogle Scholar
  25. 25.
    Honda, T.: Optimal portfolio choice for unobservable and regime-switching mean returns. J. Econ. Dyn. Control 28(1), 45–78 (2003)MathSciNetzbMATHGoogle 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)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Karatzas, I., Shreve, S.E.: Brownian Motion and Stochastic Calculus, vol. 113. Springer, Berlin (2012)zbMATHGoogle Scholar
  28. 28.
    Karatzas, I., Zhao, X.: Bayesian adaptive portfolio optimization. Columbia University, Preprint (1998)Google Scholar
  29. 29.
    Lakner, P.: Utility maximization with partial information. Stoch. Process. Appl. 56(2), 247–273 (1995)MathSciNetzbMATHGoogle Scholar
  30. 30.
    Lakner, P.: Optimal trading strategy for an investor the case of partial information. Stoch. Process. Appl. 76(1), 77–97 (1998)MathSciNetzbMATHGoogle 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)MathSciNetzbMATHGoogle 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)MathSciNetzbMATHGoogle 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)Google Scholar
  36. 36.
    Merton, R.C.: Lifetime portfolio selection under uncertainty: the continuous-time case. Rev. Econ. Stat. 51, 247–257 (1969)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)MathSciNetzbMATHGoogle Scholar
  38. 38.
    Monoyios, M.: Optimal investment and hedging under partial and inside information. Radon Ser. Comput. Appl. Math. 8, 371–410 (2009)MathSciNetzbMATHGoogle Scholar
  39. 39.
    Putschögl, W., Sass, J.: Optimal consumption and investment under partial information. Decis. Econ. Financ. 31(2), 137–170 (2008)MathSciNetzbMATHGoogle Scholar
  40. 40.
    Rieder, U., Bäuerle, N.: Portfolio optimization with unobservable Markov-modulated drift process. J. Appl. Probab. 42, 362–378 (2005)MathSciNetzbMATHGoogle 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)Google Scholar
  42. 42.
    Samuelson, P.A.: Lifetime portfolio selection by dynamic stochastic programming. Rev. Econ. Stat. 51, 239–246 (1969)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)MathSciNetzbMATHGoogle 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)MathSciNetzbMATHGoogle Scholar
  45. 45.
    Tobin, J.: Liquidity preference as behavior towards risk. Rev. Econ. Stud. 25(2), 65–86 (1958)Google Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Centre d’Economie de la SorbonneUniversité Paris 1 Panthéon-SorbonneParisFrance
  2. 2.Den-Service de thermo-hydraulique et de mécanique des fluides - Laboratoire de Génie Logiciel pour la Simulation (DEN/STMF/LGLS), CEAUniversité Paris-SaclayGif-sur-YvetteFrance
  3. 3.Institut Europlace de FinanceParisFrance

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