Annals of Operations Research

, Volume 45, Issue 1, pp 349–372 | Cite as

Optimal consumption and arbitrage in incomplete, finite state security markets

  • Hiroshi Shirakawa
  • Hiromichi Kassai
Article

Abstract

We study a consistent treatment for both the multi-period portfolio selection problem and the option attainability problem by a dual approach. We assume that time is discrete, the horizon is finite, the sample space is finite and the number of securities is less than that of the possible securities price transitions, i.e. an incomplete security market. The investor is prohibited from investing stocks more than given linear investment amount constraints at any time and he maximizes an expected additive utility function for the consumption process. First we give a set of budget feasibility conditions so that a consumption process is attainable by an admissible portfolio process. To establish this relation, we used an algorithmic approach which has a close connection with the linear programming duality. Then we prove the unique existence of a primal optimal solution from the budget feasibility conditions. Finally, we formulate a dual control problem and establish the duality between primal and dual control problems.

Keywords

Incomplete market trading constraints linear programming optimal portfolio duality 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    J.M. Bisumt, An introductory approach to duality in optimal stochastic control, SIAM Rev. 20 (1978) 62–78.Google Scholar
  2. [2]
    V. Chvătal,Linear Programming (W.H. Freeman, 1983).Google Scholar
  3. [3]
    J. Cox and C.F. Huang, Optimal consumption and portfolio policies when asset prices follow a diffusion process, J. Econ. Theory 49 (1989) 33–83.Google Scholar
  4. [4]
    J. Cvitanić and I. Karatzas, Convex duality in constrained portfolio optimization, Ann. Appl. Prob. 2 (1992) 767–818.Google Scholar
  5. [5]
    M.U. Dothan,Prices in Financial Markets (Oxford University Press, 1990).Google Scholar
  6. [6]
    N.H. Hakansson, Optimal investment and consumption strategies under risk for a class of utility functions, Econometrica 38 (1970) 587–607.Google Scholar
  7. [7]
    N.H. Hakansson, On optimal myopic portfolio policies with and without serial correlation yields, J. Bus. 44 (1971) 324–334.Google Scholar
  8. [8]
    J.M. Harrison and D.M. Kreps, Martingales and arbitrage in multiperiod securities markets, J. Econ. Theory 20 (1979) 381–408.Google Scholar
  9. [9]
    J.M. Harrison and S.R. Pliska, Martingales and stochastic integrals in the theory of continuous trading, Stoch. Proc. Appl. 11 (1981) 215–260.Google Scholar
  10. [10]
    J.M. Harrison and S.R. Pliska, A stochastic calculus model of continuous trading: Complete markets, Stoch. Proc. Appl. 15 (1983) 313–316.Google Scholar
  11. [11]
    H. He and N.D. Pearson, Consumption and portfolio policies with incomplete markets and short-sale constraints: The finite dimensional case, Math. Fin. 1 (1991) 1–10.Google Scholar
  12. [12]
    H. He and N.D. Pearson, Consumption and portfolio policies with incomplete markets and short-sale constraints: The infinite dimensional case, J. Econ. Theory (1989), to appear.Google Scholar
  13. [13]
    S.D. Jacka, A martingale representation result and an application to incomplete financial markets, Math. Fin. 2 (1992) 239–250.Google Scholar
  14. [14]
    I. Karatzas, Optimization problems in the theory of continuous trading, SIAM J. Control Optim. 27 (1989) 1221–1259.Google Scholar
  15. [15]
    I. Karatzas,Martingale and Duality Methods for Stochastic Control Problems in Financial Economics, Lecture Note (1990).Google Scholar
  16. [16]
    I. Karatzas, J.P. Lehoczky and S.E. Shreve, Optimal portfolio and consumption decisions for a “small investor” on a finite horizon, SIAM J. Control Optim. 25 (1987) 1557–1586.Google Scholar
  17. [17]
    I. Karatzas, J.P. Lehoczky, S.E. Shreve and G.L. Xu, Martingale and duality methods for utility maximization in an incomplete market, SIAM J. Control Optim. 29 (1991) 702–730.Google Scholar
  18. [18]
    D.G. Luenberger,Optimization by Vector Space Methods (Wiley, 1969).Google Scholar
  19. [19]
    D.G. Luenberger,Introduction to Linear and Nonlinear Programming (Addison-Wesley, Reading, MA, 1973).Google Scholar
  20. [20]
    O.L. Mangasarian,Nonlinear Programming (McGraw-Hill, New York, 1969).Google Scholar
  21. [21]
    R.C. Merton, Optimum consumption and portfolio rules in a continuous-time model, J. Econ. Theory 3(1971)373–413; Erratum, ibid. 6 (1971) 213–214.Google Scholar
  22. [22]
    J. Mossin, Optimal multiperiod portfolio policies, J. Bus. 41 (1968) 215–229.Google Scholar
  23. [23]
    G.L. Nemhauser and L.A. Wolsey,Integer and Combinational Optimization (Wiley, 1988).Google Scholar
  24. [24]
    S. Pliska, A stochastic calculus model of continuous trading: Optimal portfolios, Math. Oper. Res. 11 (1986) 371–382.Google Scholar
  25. [25]
    R.T. Rockafellar,Convex Analysis (Princeton University Press, 1972).Google Scholar
  26. [26]
    R.T. Rockafellar and R.J.B. Wets, Nonanticipativity andL 1-martingales in stochastic optimization problems, Math. Progr. Study 6 (1976) 170–187.Google Scholar
  27. [27]
    R.T. Rockafellar and J.B. Wets, The optimal resource problem in discrete time:L 1-multipliers for inequality constraints, SIAM J. Control Optim. 16 (1978) 16–36.Google Scholar
  28. [28]
    H. Shirakawa, Optimal consumption and portfolio selection with incomplete markets and upper and lower bound constraints, to appear in Math. Fin.Google Scholar
  29. [29]
    S.E. Shreve,Optimal Portfolio and Consumption Decisions under the Constant Coefficients, Portfolio Cone Constraints, Lecture Note (1991).Google Scholar
  30. [30]
    M. Taqqu and W. Willinger, The analysis of finite security markets using martingales, Adv. Appl. Prob. 19 (1987) 1–25.Google Scholar
  31. [31]
    W. Willinger and M. Taqqu, Pathwise stochastic integration and applications to the theory of continuous trading, Stoch. Proc. Appl. 32 (1989) 253–280.Google Scholar
  32. [32]
    G.L. Xu and S.E. Shreve, A duality method for optimal consumption and investment under short-selling prohibition. I. General market coefficients, Ann. Appl. Prob. 2 (1992) 87–112.Google Scholar
  33. [33]
    G.L. Xu and S.E. Shreve, A duality method for optimal consumption and investment under short-selling prohibition. II. Constant market coefficients, Ann. Appl. Prob. 2 (1992) 314–328.Google Scholar

Copyright information

© J.C. Baltzer AG, Science Publishers 1993

Authors and Affiliations

  • Hiroshi Shirakawa
    • 1
  • Hiromichi Kassai
    • 2
  1. 1.Institute of Socio-Economic PlanningUniversity of TsukubaIbarakiJapan
  2. 2.Financial Engineering DepartmentThe Industrial Bank of Japan, Ltd.Japan

Personalised recommendations