Advertisement

Optimal Portfolios and Pricing of Financial Derivatives Under Proportional Transaction Costs

  • Jörn SassEmail author
  • Manfred Schäl
Chapter
Part of the International Series in Operations Research & Management Science book series (ISOR, volume 248)

Abstract

A utility optimization problem is studied in discrete time 0 ≤ n ≤ N for a financial market with two assets, bond and stock. These two assets can be traded under transaction costs. A portfolio (Y n , Z n ) at time n is described by the values Y n and Z n of the stock account and the bank account, respectively. The choice of (Y n , Z n ) is controlled by a policy. Under concavity and homogeneity assumptions on the utility function U, the optimal policy has a simple cone structure. The final portfolio (Y N , Z N ) under the optimal policy has an important property. It can be used for the construction of a consistent price system for the underlying financial market.

Keywords

Numeraire portfolio Utility function Consistent price system Proportional transaction costs Dynamic programming 

References

  1. 1.
    N. Bäuerle, U. Rieder, Markov Decision Processes with Applications in Finance (Springer, Berlin, 2011)CrossRefGoogle Scholar
  2. 2.
    D. Becherer, The numeraire portfolio for unbounded semimartingales. Finance Stochast. 5, 327–341 (2001)CrossRefGoogle Scholar
  3. 3.
    H. Bühlmann, E. Platen, A discrete time benchmark approach for insurance and finance. ASTIN Bull. 33, 153–172 (2003)CrossRefGoogle Scholar
  4. 4.
    M.M. Christensen, K. Larsen, No arbitrage and the growth optimal portfolio. Stoch. Anal. Appl. 25, 255–280 (2007)CrossRefGoogle Scholar
  5. 5.
    G.M. Constantinides, Multiperiod consumption and investment behaviour with convex transaction costs. Manag. Sci. 25, 1127–1137 (1979)CrossRefGoogle Scholar
  6. 6.
    J. Cvitanić, I. Karatzas, Hedging and portfolio optimization under transaction costs: a martingale approach. Math. Financ. 6, 133–166 (1996)CrossRefGoogle Scholar
  7. 7.
    M.H.A. Davis, Option pricing in incomplete markets, in Mathematics of Derivative Securities, ed. By M. Dempster, S. Pliska (Cambridge University Press, Cambridge, 1997), pp. 216–226Google Scholar
  8. 8.
    M.H.A. Davis, A.R. Norman, Portfolio selection with transaction costs. Math. Oper. Res. 15, 676–713 (1990)CrossRefGoogle Scholar
  9. 9.
    T. Goll, J. Kallsen, A complete explicit solution to the log-optimal portfolio problem. Adv. Appl. Probab. 13, 774–779 (2003)CrossRefGoogle Scholar
  10. 10.
    E. Jouini, H. Kallal, Martingales and arbitrage in securities markets with transaction const. J. Econ. Theory 66, 178–197 (1995)CrossRefGoogle Scholar
  11. 11.
    J. Kallsen, J. Muhle-Karbe, On the existence of shadow prices in finite discrete time. Math. Meth. Oper. Res. 73, 251–262 (2011)CrossRefGoogle Scholar
  12. 12.
    J.H. Kamin, Optimal portfolio revision with a proportional transaction costs. Manag. Sci. 21, 1263–1271 (1975)CrossRefGoogle Scholar
  13. 13.
    I. Karatzas, C. Kardaras, The numéraire portfolio in semimartingale financial models. Finance Stochast. 11, 447–493 (2007)CrossRefGoogle Scholar
  14. 14.
    P.F. Koehl, H. Pham, N. Touzi, On super-replication in discrete time under transaction costs. Theory Probab. Appl. 45, 667–673 (2001)CrossRefGoogle Scholar
  15. 15.
    R. Korn, M. Schäl, On value preserving and growth optimal portfolios. Math. Meth. Oper. Res. 50, 189–218 (1999)CrossRefGoogle Scholar
  16. 16.
    R. Korn, M. Schäl, The numeraire portfolio in discrete time: existence, related concepts and applications. Radon Ser. Comput. Appl. Math. 8, 1–25 (2009). De GruyterGoogle Scholar
  17. 17.
    R. Korn, F. Oertel, M. Schäl, The numeraire portfolio in financial markets modeled by a multi-dimensional jump diffusion process. Decisions Econ. Finan. 26, 153–166 (2003)CrossRefGoogle Scholar
  18. 18.
    S. Kusuoka, Limit theorem on option replication with transaction costs. Ann. Appl. Probab. 5, 198–121 (1995)CrossRefGoogle Scholar
  19. 19.
    J. Long, The numeraire portfolio. J. Financ. 44, 205–209 (1990)Google Scholar
  20. 20.
    M.J.P. Magill, M. Constantinides, Portfolio selection with transaction costs. J. Econ. Theory 13, 245–263 (1976)CrossRefGoogle Scholar
  21. 21.
    E. Platen, A benchmark approach to finance. Math. Financ. 16, 131–151 (2006)CrossRefGoogle Scholar
  22. 22.
    J. Sass, Portfolio optimization under transaction costs in the CRR model. Math. Meth. Oper. Res. 61, 239–259 (2005)CrossRefGoogle Scholar
  23. 23.
    J. Sass, M. Schäl, Numerairs portfolios and utility-based price systems under proportional transaction costs. Decisions Econ. Finan. 37, 195–234 (2014)CrossRefGoogle Scholar
  24. 24.
    W. Schachermayer, The fundamental theorem of asset pricing under proportional transaction costs in finite discrete time. Math. Financ. 14, 19–48 (2004)CrossRefGoogle Scholar
  25. 25.
    M. Schäl, Portfolio optimization and martingale measures. Math. Financ. 10, 289–304 (2000)CrossRefGoogle Scholar
  26. 26.
    M. Schäl, Price systems constructed by optimal dynamic portfolios. Math. Meth. Oper. Res. 51, 375–397 (2000)CrossRefGoogle Scholar
  27. 27.
    S.E. Shreve, H.M. Soner, Optimal investment and consumption with transaction costs. Ann. Appl. Probab. 4, 609–692 (1994)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Fachbereich MathematikTU KaiserslauternKaiserslauternGermany
  2. 2.Institut für Angewandte MathematikUniversität BonnBonnGermany

Personalised recommendations