Skip to main content

Advertisement

Springer Nature Link
Log in

Portfolio optimization in stochastic markets

  • Original Article
  • Published:
Mathematical Methods of Operations Research Aims and scope Submit manuscript

Abstract

We consider a multiperiod mean-variance model where the model parameters change according to a stochastic market. The mean vector and covariance matrix of the random returns of risky assets all depend on the state of the market during any period where the market process is assumed to follow a Markov chain. Dynamic programming is used to solve an auxiliary problem which, in turn, gives the efficient frontier of the mean-variance formulation. An explicit expression is obtained for the efficient frontier and an illustrative example is given to demonstrate the application 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

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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

  • Asmussen S (2000) Ruin probabilities. World Scientific, Singapore

    Google Scholar 

  • Bäuerle N, Rieder U (2004) Portfolio optimization with Markov-modulated stock prices and interest rates. IEEE Trans Autom Control 49:442–447

    Article  Google Scholar 

  • Bielecki T, Hernández-Hernández D, Pliska SR (1999) Risk sensitive control of finite state Markov chains in discrete time, with applications to portfolio management. Math Methods Oper Res 50:167–188

    Article  MATH  MathSciNet  Google Scholar 

  • Bielecki TR, Pliska SR (1999) Risk sensitive dynamic asset management. J Appl Math Optim 39:337–360

    Article  MATH  MathSciNet  Google Scholar 

  • Bodily SE, White CC (1982) Optimal consumption and portfolio strategies in a discrete time model with summary-dependent preferences. J Financ Quant Anal 17:1–14

    Article  Google Scholar 

  • Chen AH, Jen FC, Zionts S (1971) The optimal portfolio revision policy. J Bus 44:51–61

    Article  Google Scholar 

  • Çılar E, Özekici S (1987) Reliability of complex devices in random environments. Probab Eng Inform Sci 1:97–115

    Article  Google Scholar 

  • Dumas B, Luciano E (1991) An exact solution to a dynamic portfolio choice problem under transaction costs. J Finance 46:577–595

    Article  Google Scholar 

  • Ehrlich I, Hamlen WA (1995) Optimal portfolio and consumption decisions in a stochastic environment with precommitment. J Econ Dyn Control 19:457–480

    Article  MATH  Google Scholar 

  • Elliott RJ, Mamon RS (2003) A complete yield curve description of a Markov interest rate model. Int J Theor Appl Finance 6:317–326

    Article  MATH  MathSciNet  Google Scholar 

  • Elton EJ, Gruber MJ (1974) On the optimality of some multiperiod portfolio selection criteria. J Bus 47:231–243

    Article  Google Scholar 

  • Fleming WH, Hernández-Hernández D (2003) An optimal consumption model with stochastic volatility. Finance Stoch 7:245–262

    Article  MATH  MathSciNet  Google Scholar 

  • Hernández-Hernández D, Marcus SI (1999) Existence of risk sensitive optimal stationary policies for controlled Markov processes. Appl Math Optim 40:273–285

    Article  MATH  MathSciNet  Google Scholar 

  • Korn R, Kraft H (2001) A stochastic control approach to portfolio problems with stochastic interest rates. SIAM J Control Optim 40:1250–1269

    Article  MATH  MathSciNet  Google Scholar 

  • Li D, Chan TF, Ng WL (1998) Safety-first dynamic portfolio selection. Dyn Contin Discrete Impuls Syst 4:585–600

    MATH  MathSciNet  Google Scholar 

  • Li D, Ng WL (2000) Optimal dynamic portfolio selection: Multiperiod mean-variance formulation. Math Finance 10:387–406

    Article  MATH  MathSciNet  Google Scholar 

  • Markowitz H (1952) Portfolio selection. J Finance 7:77–91

    Article  Google Scholar 

  • Di Massi GB, Stettner L (1999) Risk sensitive control of discrete time partially observed Markov processes with infinite horizon. Stoch Stoch Rep 67:309–322

    Google Scholar 

  • Mossin J (1968) Optimal multiperiod portfolio policies. J Bus 41:215–229

    Article  Google Scholar 

  • Nagai H, Peng S (2002) Risk-sensitive dynamic portfolio optimization with partial information on infinite time horizon. Ann Appl Probab 12:173–195

    Article  MATH  MathSciNet  Google Scholar 

  • Norberg R (1995) A continuous-time Markov chain interest model with applications to insurance. J Appl Stoch Models Data Anal 11:245–256

    Article  MATH  Google Scholar 

  • Özekici S (1996) Complex systems in random environments. In: Özekici S (ed). Reliability and maintenance of complex systems (Berlin), NATO ASI series volume F154. Springer, Berlin Heidelberg New York, pp. 137–157

    Google Scholar 

  • Prabhu NU, Zhu Y (1989) Markov-modulated queueing systems. Queueing Syst 5:215–246

    Article  MATH  MathSciNet  Google Scholar 

  • Rolski T, Schmidli H, Schmidt V, Teugels J (1999) Stochastic processes for insurance and finance. Wiley, Chichester

    MATH  Google Scholar 

  • Samuelson PA (1969) Lifetime portfolio selection by dynamic stochastic programming. Rev Econ Statist 51:239–246

    Article  Google Scholar 

  • Song JS, Zipkin P (1993) Inventory control in fluctuating demand environment. Oper Res 41:351–370

    Article  MATH  Google Scholar 

  • Steinbach MC (2001) Markowitz revisited: mean-variance models in financial portfolio analysis. Soc Ind Appl Math Rev 43:31–85

    MATH  MathSciNet  Google Scholar 

  • Stettner L (1999) Risk sensitive portfolio optimization. Math Methods Oper Res 50:463–474

    Article  MATH  MathSciNet  Google Scholar 

  • Stettner L (2004) Risk-sensitive portfolio optimization with completely and partially observed factors. IEEE Trans Autom Control 49:457–464

    Article  MathSciNet  Google Scholar 

  • Yin G, Zhou XY (2004) Markowitz’s mean-variance portfolio selection with regime switching: from discrete-time models to their continuous-time limits. IEEE Trans Autom Control 49:349–360

    Article  MathSciNet  Google Scholar 

  • Zariphopoulou T (2001) A solution approach to valuation with unhedgeable risks. Finance Stoch 5:61–82

    Article  MATH  MathSciNet  Google Scholar 

  • Zhang Q (2001) Stock trading: an optimal selling rule. SIAM J Control Optim 40:64–87

    Article  MATH  MathSciNet  Google Scholar 

  • Zhou XY, Li D (2000) Continuous-time mean-variance portfolio selection: a stochastic LQ framework. Appl Math Optim 42:19–33

    Article  MATH  MathSciNet  Google Scholar 

  • Zhu S-S, Li D, Wang S-Y (2004) Risk control over bankruptcy in dynamic portfolio selection: A generalized mean-variance formulation. IEEE Trans Autom Control 49:447–457

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta, GA, 30332-0205, USA

    U. Çakmak

  2. Department of Industrial Engineering, Koç University, 34450, Sarıyer-İstanbul, Turkey

    S. Özekici

Authors
  1. U. Çakmak
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. S. Özekici
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to S. Özekici.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Çakmak, U., Özekici, S. Portfolio optimization in stochastic markets. Math Meth Oper Res 63, 151–168 (2006). https://doi.org/10.1007/s00186-005-0020-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00186-005-0020-x

Keywords