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Primal-dual methods for the computation of trading regions under proportional transaction costs

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Abstract

Portfolio optimization problems on a finite time horizon under proportional transaction costs are considered. The objective is to maximize the expected utility of the terminal wealth. The ensuing non-smooth time-dependent Hamilton–Jacobi–Bellman equation is solved by regularization and the application of a semi-smooth Newton method. Discretization in space is carried out by finite differences or finite elements. Computational results for one and two risky assets are provided.

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References

  • Akian M, Menaldi JL, Sulem A (1996) On an investment-consumption model with transaction costs. SIAM J Control Optim 34: 329–364

    Article  MathSciNet  MATH  Google Scholar 

  • Akian M, Séquier P, Sulem A (1995) A finite horizon multidimensional portfolio selection problem with singular transactions. In: Proceedings of the 34th conference on decision and control, vol 3, pp 2193–2198

  • Almgren R, Tourin A (2004) Optimal soaring with Hamilton–Jacobi–Bellman equations (preprint). URL: http://cims.nyu.edu/~almgren/papers/optsoar.pdf

  • Bertsekas DP (1996) Constrained optimization and lagrange multiplier methods. Athena Scientific, Belmont

    Google Scholar 

  • Dai M, Zhong Y (2010) Penalty methods for continuous-time portfolio selection with proportional transaction costs. J Comput Financ 13(3): 1–31

    MathSciNet  Google Scholar 

  • Dai Min, Yi Fahuai (2009) Finite-horizon optimal investment with transaction costs: a parabolic double obstacle problem. J Differ Equ 246(4):1445–1469. ISSN 0022-0396. doi:10.1016/j.jde.2008.11.003

    Google Scholar 

  • Davis MHA, Norman AR (1990) Portfolio selection with transaction costs. Math Oper Res 15: 676–713

    Article  MathSciNet  MATH  Google Scholar 

  • Evans G (1979) A second order elliptic equation with gradient constraint. Commun Partial Differ Equ 4: 555–572

    Article  MATH  Google Scholar 

  • Griesse R, Kunisch K (2009) A semismooth Newton method for solving elliptic equations with gradient constraints. ESAIM M2AN: Math Model Numer Anal 43(2): 209–238. doi:10.1051/m2an:2008049

    Article  MathSciNet  MATH  Google Scholar 

  • Hintermüller M, Kunisch K (2006) Feasible and non-interior path-following in constrained minimization with low multiplier regularity. SIAM J Control Optim 45: 1198–1221. doi:10.1137/050637480

    Article  MathSciNet  MATH  Google Scholar 

  • Hodder JE, Tourin A, Zariphopoulou T (2001) Numerical schemes for variational inequalities arising in international asset pricing. Comput Econ 17: 43–80

    Article  MATH  Google Scholar 

  • Irle A, Sass J (2006) Optimal portfolio policies under fixed and proportional transaction costs. Adv Appl Probab 38: 916–942

    MathSciNet  MATH  Google Scholar 

  • Ito K, Kunisch K (2008) Lagrange multiplier approach to variational problems and applications volume 15 of advances in design and control. Society for Industrial and Applied Mathematics (SIAM), Philadelphia

    Book  Google Scholar 

  • Kunisch K, Sass J (2007) Trading regions under proportional transaction costs. In: Waldmann K-H, Stocker UM (eds) Operations research proceedings. Springer, New York, pp 563–568

    Google Scholar 

  • Li W, Wang S (2009) Penalty approach to the HJB equation arising in European stock option pricing with proportional transaction costs. J Optim Theory Appl 143(2):279–293. ISSN 0022-3239. doi:10.1007/s10957-009-9559-7

  • Merton RC (1969) Lifetime portfolio selection under uncertainty: the continuous-time case. Rev Econ Stat 51(3): 247–257

    Article  Google Scholar 

  • Muthuraman K (2007) A computational scheme for optimal investment-consumption with proportional transaction costs. J Econ Dyn Control 31(4): 1132–1159. doi:10.1016/j.jedc.2006.04.005

    Article  MathSciNet  MATH  Google Scholar 

  • Muthuraman K, Kumar S (2006) Multidimensional portfolio optimization with proportional transaction costs. Math Financ Int J Math Stat Financ Econ 16(2): 301–335. doi:10.1111/j.1467-9965.2006.00273.x

    MathSciNet  MATH  Google Scholar 

  • Roos H-G, Stynes M, Tobiska L (1996) Numerical methods for singularly perturbed differential equations. Springer, Berlin

    MATH  Google Scholar 

  • Shreve S, Soner HM (1994) Optimal investment and consumption with transaction costs. Ann Appl Probab 4: 609–692

    Article  MathSciNet  MATH  Google Scholar 

  • Zakamouline VI (2005) A unified approach to portfolio optimization with linear transaction costs. Math Methods Oper Res 62(2):319–343. ISSN 1432-2994. doi:10.1007/s00186-005-0005-9

    Google Scholar 

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Authors and Affiliations

  1. Faculty of Mathematics, Chemnitz University of Technology, 09107, Chemnitz, Germany

    Roland Herzog

  2. Institute for Mathematics and Scientific Computing, Karl Franzens University Graz, Heinrichstrasse 36, 8010, Graz, Austria

    Karl Kunisch

  3. Department of Mathemaics, University of Kaiserslautern, P.O. Box 3049, 67663, Kaiserslautern, Germany

    Jörn Sass

Authors
  1. Roland Herzog
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  2. Karl Kunisch
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  3. Jörn Sass
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Corresponding author

Correspondence to Roland Herzog.

Additional information

Roland Herzog: This research was carried out in part at the Johann Radon Institute for Computational and Applied Mathematics (RICAM), Austrian Academy of Sciences, Linz, Austria; Karl Kunisch: Research supported in part by the Fonds zur Förderung der wissenschaftlichen Forschung (FWF) under SFB 32, Mathematical Optimization and Applications in the Biomedical Sciences; Jörn Sass: J. Sass gratefully acknowledges financial support by Deutsche Forschungsgemeinschaft (DFG).

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Herzog, R., Kunisch, K. & Sass, J. Primal-dual methods for the computation of trading regions under proportional transaction costs. Math Meth Oper Res 77, 101–130 (2013). https://doi.org/10.1007/s00186-012-0416-3

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  • DOI: https://doi.org/10.1007/s00186-012-0416-3

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