Skip to main content
SpringerLink
Log in

Multiperiod Portfolio Optimization with Terminal Liability: Bounds for the Convex Case

  • Published:
Computational Optimization and Applications Aims and scope Submit manuscript

Abstract

This paper is concerned with an investor trading in multiple securities over many time periods in order to meet an outstanding liability at some future date. The investor is concerned with maximizing the expected profits from portfolio rebalancing under an initial wealth restriction to meet the future liabilities. We formulate the problem as a discrete-time stochastic optimization model and allow asset prices to have continuous probability distributions on compact domains. For the case of Markovian price uncertainty and convex terminal liability, we develop a simplicial approximation, under which bounds on the problem can be computed efficiently. Computations only require evaluating a dynamic programming recursion, which thus, allows its application to problems with a large number of trading periods. The bounds are tight in that they are exact in certain cases. Numerical results are given to demonstrate the computational efficiency of the procedure.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. C.D. Aliprantis, D.J. Brown, and J. Werner, “Minimum-cost portfolio insurance,” Journal of Economic Dynamics and Control, vol. 24, no. 11, pp. 1703–1719, 2000.

    Article  Google Scholar 

  2. E.J. Anderson and P. Nash, Linear Programming in Infinite-Dimensional Spaces, John Wiley and Sons, 1987.

  3. J.R. Birge and R.J-B. Wets, “Designing approximation schemes for stochastic optimization problems, in particular for stochastic programs with recourse,” Mathematical Programming Study, vol. 27, pp. 54–102, 1986.

    Google Scholar 

  4. J.R. Birge and R.J-B. Wets, “Computing Bounds for Stochastic Programming Problems by Means of a Generalized Moment Problem,” Mathematics of Operations Research, vol. 12, pp. 149–162, 1987.

    Google Scholar 

  5. Black, F. and M. Scholes, The Pricing of Options and Corporate Liabilities, Journal of Political Economy, 81, pp. 637–654, 1973.

    Article  Google Scholar 

  6. J.C. Cox, S. Ross, and M. Rubinstein, “Options pricing: A simplified approach,” J. Financial Economics, vol. 7, pp. 229–263, 1979.

    Article  Google Scholar 

  7. G.B. Dantzig and P.W. Glynn, “Parallel processors for planning under uncertainty,” Annals of Operations Research, vol. 22, pp. 1–21, 1990.

    Article  Google Scholar 

  8. M.A.H. Dempster, “Introduction to stochastic programming,” in Stochastic Programming, M.A.H. Dempster (ed.), Academic Press: London, 1980, pp. 3–59.

    Google Scholar 

  9. N.C.P. Edirisinghe, “Arbitrage-free pricing under transaction costs via generalized moment problems,” Presentation at the Tenth International Conference on Stochastic Programming, Oct 11–15, Tucson, AZ, USA, 2004.

    Google Scholar 

  10. C. Edirisinghe, V. Naik, and R. Uppal, “Optimal replication of options with transactions costs and trading restrictions,” Journal of Financial and Quantitative Analysis, vol. 28, pp. 117–138, 1993.

    Google Scholar 

  11. N.C.P. Edirisinghe and G.-M. You, “Second-order Scenario Approximation and Refinement in Optimization under Uncertainty,” Annals of Operations Research, vol. 19, pp. 314–340, 1996.

    Google Scholar 

  12. N.C.P. Edirisinghe and W.T. Ziemba, “Tight Bounds For Stochastic Convex Programs,” Operations Research, vol. 40, pp. 660–677, 1992.

    Google Scholar 

  13. N.C.P. Edirisinghe and W.T. Ziemba, “Bounds For Two-Stage Stochastic Programs With Fixed Recourse,” Mathematics of Operations Research, vol. 19, pp. 292–313, 1994a.

    Google Scholar 

  14. N.C.P. Edirisinghe and W.T. Ziemba, “Bounding the Expectation of a Saddle Function, With Application to Stochastic Programming,” Mathematics of Operations Research, vol. 19, pp. 314–340, 1994b.

    Google Scholar 

  15. Y. Ermoliev and A. Gaivoronsky, “Stochastic quasigradient methods and their implementation,” in Numerical techniques for stochastic optimization, Y. Ermoliev and R.J-B. Wets (Eds.), Springer-Verlag, 1988.

  16. Y. Ermoliev and R. J-B. Wets, Numerical Techniques for Stochastic Optimization, Springer-Verlag, 1988.

  17. K. Frauendorfer, “SLP Recourse Problems with Arbitrary Multivariate Distributions—the Dependent Case,” Mathematics of Operations Research, vol. 13, pp. 377–394, 1988.

    Google Scholar 

  18. K. Frauendorfer, Stochastic Two-Stage Programming, Lecture Notes in Economics and Mathematical Systems, Springer-Verlag, 1992.

  19. H. Gassmann and W.T. Ziemba, “A Tight Upper Bound for the Expectation of a Convex Function of a Multivariate Random Variable,” Mathematical Programming Study, vol. 27, pp. 39–53, 1986.

    Google Scholar 

  20. H. He and N.D. Pearson, “Consumption and portfolio policies with incomplete markets and short-sale constraints: The infinite dimensional case,” Journal of Economic Theory, vol. 54, pp. 259–304, 1991.

    Article  Google Scholar 

  21. J.L. Higle and S. Sen, “Stochastic decomposition: An algorithm for two-stage stochastic linear programs with recourse,” Mathematics of Operations Research, vol. 16, pp. 650–669, 1991.

    Google Scholar 

  22. J. Hull, “Options, futures, and other derivative securities,” Prentice Hall: Englewood Cliffs, N.J., 1989.

    Google Scholar 

  23. P. Kall, “Stochastic Programming with Recourse: Upper Bounds and Moment Problems—A Review. in Advances in Mathematical Optimization and Related Topics. J. Guddat, P. Kall, K. Lommatzsch, M. Vlach, and K. Zimmermann (Eds.) Akademie-Verlag: Berlin, 1987.

    Google Scholar 

  24. J. Kemperman, “The General Moment Problem. A Geometric Approach,” Annals of Mathematical Statistics, vol. 39, pp. 93–112, 1968.

    Google Scholar 

  25. A.J. King, “Duality and martingales: A stochastic programming perspective on contingent claims,” Mathematical Programming, vol. 91, pp. 543–562, 2002.

    Article  Google Scholar 

  26. A. Madansky, “Bounds on the expectation of a convex function of a multivariate random variable,” Annals of Mathematical Statistics, vol. 30, pp. 743–746, 1959.

    Google Scholar 

  27. J. Mulvey and H. Vladimirou, “Stochastic network programming for financial planning under uncertainty,” Management Science, vol. 18, pp. 1642–1664, 1992.

    Google Scholar 

  28. C. Munk, “Optimal consumption/investment policies with undiversifiable income risk and liquidity constraints,” Journal of Economic Dynamics and Control, vol. 24, no. 9, pp. 1315–1343, 2000.

    Article  Google Scholar 

  29. V. Naik and R. Uppal, “Leverage Constraints and the Optimal Hedging of Stock and Bond Options,” Journal of Financial and Quantitative Optimization, vol. 29, pp. 199–222, 1994.

    Google Scholar 

  30. P.H. Ritchken, “On option pricing bounds,” Journal of Finance, vol. 11, no. 4, pp. 1219–1233, 1985.

    Google Scholar 

  31. P.H. Ritchken and S. Kuo, “Option Bounds With Finite Revision Opportunities,” Journal of Finance, vol. 18, pp. 301–308, 1988.

    Google Scholar 

  32. W.T. Ziemba and J. Mulvey, World Wide Asset and Liability Modeling, Cambridge University Press, 1998.

Download references

Author information

Authors and Affiliations

  1. Management Science Program, College of Business Administration, University of Tennessee, Knoxville, TN, 37996, USA

    N. C. P. Edirisinghe

Authors
  1. N. C. P. Edirisinghe
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to N. C. P. Edirisinghe.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Edirisinghe, N.C.P. Multiperiod Portfolio Optimization with Terminal Liability: Bounds for the Convex Case. Comput Optim Applic 32, 29–59 (2005). https://doi.org/10.1007/s10589-005-2053-8

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10589-005-2053-8

Keywords

Search

Navigation