Advertisement

Optimal Investment

  • Ernst Eberlein
  • Jan Kallsen
Chapter
Part of the Springer Finance book series (FINANCE)

Abstract

As an investor in a securities market you have to decide how to arrange your portfolio. In this chapter we consider the natural situation that you want to maximise your profits. It is not entirely obvious how to formalise this goal because the payoff of investments is typically random.

References

  1. 6.
    I. Bajeux-Besnainou, R. Portait, Dynamic asset allocation in a mean-variance framework. Manag. Sci. 44(11), S79–S95 (1998)zbMATHCrossRefGoogle Scholar
  2. 13.
    S. Basak, G. Chabakauri, Dynamic mean-variance asset allocation. Rev. Financ. Stud. 23(8), 2970–3016 (2010)CrossRefGoogle Scholar
  3. 21.
    F. Bellini, M. Frittelli, On the existence of minimax martingale measures. Math. Finance 12(1), 1–21 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 22.
    G. Benedetti, L. Campi, J. Kallsen, J. Muhle-Karbe, On the existence of shadow prices. Finance Stochast. 17(4), 801–818 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 24.
    S. Biagini, M. Frittelli, A unified framework for utility maximization problems: an Orlicz space approach. Ann. Appl. Probab. 18(3), 929–966 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 27.
    J.-M. Bismut, Growth and optimal intertemporal allocation of risks. J. Econom. Theory 10(2), 239–257 (1975)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 29.
    T. Björk, Arbitrage Theory in Continuous Time, 3rd edn. (Oxford Univ. Press, Oxford, 2009)zbMATHGoogle Scholar
  8. 35.
    T. Björk, A. Murgoci, X. Zhou, Mean-variance portfolio optimization with state-dependent risk aversion. Math. Finance 24(1), 1–24 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 55.
    A. Černý, J. Kallsen, On the structure of general mean-variance hedging strategies. Ann. Probab. 35(4), 1479–1531 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 65.
    J. Cvitanić, I. Karatzas, Hedging and portfolio optimization under transaction costs: a martingale approach. Math. Finance 6(2), 133–165 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 66.
    C. Czichowsky, Time-consistent mean-variance portfolio selection in discrete and continuous time. Finance Stochast. 17(2), 227–271 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 67.
    C. Czichowsky, W. Schachermayer, Duality theory for portfolio optimisation under transaction costs. Ann. Appl. Probab. 26(3), 1888–1941 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 68.
    C. Czichowsky, J. Muhle-Karbe, W. Schachermayer, Transaction costs, shadow prices, and duality in discrete time. SIAM J. Financial Math. 5(1), 258–277 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 98.
    R. Elliott, Stochastic Calculus and Applications (Springer, New York, 1982)zbMATHGoogle Scholar
  15. 125.
    T. Goll, J. Kallsen, Optimal portfolios for logarithmic utility. Stoch. Process. Appl. 89(1), 31–48 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 126.
    T. Goll, J. Kallsen, A complete explicit solution to the log-optimal portfolio problem. Ann. Appl. Probab. 13(2), 774–799 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 127.
    T. Goll, L. Rüschendorf, Minimax and minimal distance martingale measures and their relationship to portfolio optimization. Finance Stochast. 5(4), 557–581 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 136.
    H. He, N. Pearson, Consumption and portfolio policies with incomplete markets and short-sale constraints: the finite-dimensional case. Math. Finance 1(3), 1–10 (1991)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 137.
    H. He, N. Pearson, Consumption and portfolio policies with incomplete markets and short-sale constraints: The infinite dimensional case. J. Econom. Theory 54(2), 259–304 (1991)MathSciNetzbMATHCrossRefGoogle Scholar
  20. 146.
    Y. Hu, P. Imkeller, M. Müller, Utility maximization in incomplete markets. Ann. Appl. Probab. 15(3), 1691–1712 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  21. 148.
    F. Hubalek, J. Kallsen, L. Krawczyk, Variance-optimal hedging for processes with stationary independent increments. Ann. Appl. Probab. 16(2), 853–885 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  22. 153.
    J. Jacod, P. Protter, Probability Essentials, 2nd edn. (Springer, Berlin, 2004)zbMATHCrossRefGoogle Scholar
  23. 154.
    J. Jacod, A. Shiryaev, Limit Theorems for Stochastic Processes, 2nd edn. (Springer, Berlin, 2003)zbMATHCrossRefGoogle Scholar
  24. 161.
    E. Jouini, H. Kallal, Martingales and arbitrage in securities markets with transaction costs. J. Econom. Theory 66(1), 178–197 (1995)MathSciNetzbMATHCrossRefGoogle Scholar
  25. 162.
    D. Kahneman, Thinking, Fast and Slow (Farrar, Straus and Giroux, New York, 2011)Google Scholar
  26. 163.
    J. Kallsen, Semimartingale Modelling in Finance. PhD thesis, University of Freiburg, 1998Google Scholar
  27. 164.
    J. Kallsen, A utility maximization approach to hedging in incomplete markets. Math. Methods Oper. Res. 50(2), 321–338 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  28. 165.
    J. Kallsen, Optimal portfolios for exponential Lévy processes. Math. Methods Oper. Res. 51(3), 357–374 (2000)MathSciNetCrossRefGoogle Scholar
  29. 167.
    J. Kallsen, Utility-based derivative pricing in incomplete markets, in Mathematical Finance—Bachelier Congress 2000, ed. by H. Geman, D. Madan, S. Pliska, T. Vorst (Springer, Berlin, 2002), pp. 313–338CrossRefGoogle Scholar
  30. 173.
    J. Kallsen, J. Muhle-Karbe, On using shadow prices in portfolio optimization with transaction costs. Ann. Appl. Probab. 20(4), 1341–1358 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  31. 174.
    J. Kallsen, J. Muhle-Karbe. Utility maximization in models with conditionally independent increments. Ann. Appl. Probab. 20(6), 2162–2177 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  32. 175.
    J. Kallsen, J. Muhle-Karbe, Existence of shadow prices in finite probability spaces. Math. Methods Oper. Res. 73(2), 251–262 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  33. 176.
    J. Kallsen, J. Muhle-Karbe, Option pricing and hedging with small transaction costs. Math. Finance 25(4), 702–723 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  34. 177.
    J. Kallsen, J. Muhle-Karbe, The general structure of optimal investment and consumption with small transaction costs. Math. Finance 27(3), 659–703 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
  35. 180.
    J. Kallsen, T. Rheinländer, Asymptotic utility-based pricing and hedging for exponential utility. Stat. Decis. 28(1), 17–36 (2011)MathSciNetzbMATHGoogle Scholar
  36. 187.
    I. Karatzas, S. Shreve, Methods of Mathematical Finance (Springer, Berlin, 1998)zbMATHCrossRefGoogle Scholar
  37. 188.
    I. Karatzas, G. Žitković, Optimal consumption from investment and random endowment in incomplete semimartingale markets. Ann. Probab. 31(4), 1821–1858 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  38. 189.
    I. Karatzas, J. Lehoczky, S. Shreve, G. Xu, Martingale and duality methods for utility maximization in an incomplete market. SIAM J. Control Optim. 29(3), 702–730 (1991)MathSciNetzbMATHCrossRefGoogle Scholar
  39. 197.
    D. Kramkov, W. Schachermayer, The asymptotic elasticity of utility functions and optimal investment in incomplete markets. Ann. Appl. Probab. 9(3), 904–950 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  40. 205.
    J. Laurent, H. Pham, Dynamic programming and mean-variance hedging. Finance Stochast. 3(1), 83–110 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  41. 208.
    M. Loewenstein, On optimal portfolio trading strategies for an investor facing transactions costs in a continuous trading market. J. Math. Econom. 33(2), 209–228 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  42. 213.
    H. Markowitz, Portfolio selection. J. Finance 7(1), 77–91 (1952)Google Scholar
  43. 214.
    R. Martin, Optimal trading under proportional transaction costs. Risk, 54–59 (2014)Google Scholar
  44. 215.
    R. Martin, T. Schöneborn, Mean reversion pays, but costs. arXiv preprint arXiv:1103.4934 (2011)Google Scholar
  45. 217.
    R. Merton, Lifetime portfolio selection under uncertainty: The continuous-time case. Rev. Econ. Stat. 51(3), 247–257 (1969)CrossRefGoogle Scholar
  46. 218.
    R. Merton, Optimum consumption and portfolio rules in a continuous-time model. J. Econom. Theory 3(4), 373–413 (1971)MathSciNetzbMATHCrossRefGoogle Scholar
  47. 222.
    J. Mossin, Optimal multiperiod portfolio policies. J. Bus. 41(2), 215–229 (1968)CrossRefGoogle Scholar
  48. 225.
    M. Nutz, Power utility maximization in constrained exponential Lévy models. Math. Finance 22(4), 690–709 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  49. 226.
    B. Øksendal, Stochastic Differential Equations, 6th edn. (Springer, Berlin, 2003)zbMATHCrossRefGoogle Scholar
  50. 227.
    B. Øksendal, A. Sulem, Applied Stochastic Control of Jump Diffusions, 2nd edn. (Springer, Berlin, 2007)zbMATHCrossRefGoogle Scholar
  51. 234.
    H. Pham, Continuous-Time Stochastic Control and Optimization with Financial Applications (Springer, Berlin, 2009)zbMATHCrossRefGoogle Scholar
  52. 235.
    H. Pham, T. Rheinländer, M. Schweizer, Mean-variance hedging for continuous processes: new proofs and examples. Finance Stochast. 2(2), 173–198 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  53. 236.
    E. Platen, D. Heath , A Benchmark Approach to Quantitative Finance (Springer, Berlin, 2006)zbMATHCrossRefGoogle Scholar
  54. 243.
    T. Rheinländer, G. Steiger, Utility indifference hedging with exponential additive processes. Asia-Pac. Financ. Mark. 17(2), 151–169 (2010)zbMATHCrossRefGoogle Scholar
  55. 244.
    H. Richardson, A minimum variance result in continuous trading portfolio optimization. Manag. Sci. 35(9), 1045–1055 (1989)MathSciNetzbMATHCrossRefGoogle Scholar
  56. 248.
    C. Rogers, The relaxed investor and parameter uncertainty. Finance Stochast. 5(2), 131–154 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  57. 250.
    C. Rogers, Optimal Investment (Springer, Heidelberg, 2013)zbMATHCrossRefGoogle Scholar
  58. 253.
    R. Rouge, N. El Karoui, Pricing via utility maximization and entropy. Math. Finance 10(2), 259–276 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  59. 258.
    P. Samuelson, Lifetime portfolio selection by dynamic stochastic programming. Rev. Econ. Stat. 51, 239–246 (1969)CrossRefGoogle Scholar
  60. 260.
    W. Schachermayer, M. Sîrbu, E. Taflin, In which financial markets do mutual fund theorems hold true? Finance Stochast. 13(1), 49–77 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  61. 266.
    M. Schweizer, Risk-minimality and orthogonality of martingales. Stochastics Stochastics Rep. 30(2), 123–131 (1990)MathSciNetzbMATHCrossRefGoogle Scholar
  62. 267.
    M. Schweizer, Option hedging for semimartingales. Stoch. Process. Appl. 37(2), 339–363 (1991)MathSciNetzbMATHCrossRefGoogle Scholar
  63. 268.
    M. Schweizer, Approximating random variables by stochastic integrals. Ann. Probab. 22(3), 1536–1575 (1994)MathSciNetzbMATHCrossRefGoogle Scholar
  64. 269.
    M. Schweizer, Approximation pricing and the variance-optimal martingale measure. Ann. Probab. 24(1), 206–236 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  65. 273.
    R. Shiller, Online data—Robert Shiller. url: http://www.econ.yale.edu/\~shiller/data.htm. Accessed: 1 May 2018Google Scholar
  66. 287.
    N. Touzi, Optimal Stochastic Control, Stochastic Target Problems, and Backward SDE (Springer, New York, 2013)zbMATHCrossRefGoogle Scholar
  67. 290.
    J. von Neumann, O. Morgenstern, Theory of Games and Economic Behavior, 2nd edn. (Princeton Univ. Press, Princeton, 1947)zbMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Ernst Eberlein
    • 1
  • Jan Kallsen
    • 2
  1. 1.Department of Mathematical StochasticsUniversity of FreiburgFreiburg im BreisgauGermany
  2. 2.Department of MathematicsKiel UniversityKielGermany

Personalised recommendations