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On the equivalence of the static and dynamic asset allocation problems

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  • Robert Kohn
  • Oana Papazoglu-Statescu
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    Abstract

    A classic dynamic asset allocation problem optimizes the expected final-time utility of wealth, for an individual who can invest in a risky stock and a risk-free bond, trading continuously in time. Recently, several authors considered the corresponding static asset allocation problem in which the individual cannot trade but can invest in options as well as the underlying. The optimal static strategy can never do better than the optimal dynamic one. Surprisingly, however, for some market models the two approaches are equivalent. When this happens the static strategy is clearly preferable, since it avoids any impact of market frictions. This paper examines the question: when, exactly, are the static and dynamic approaches equivalent? We give an easily tested necessary and sufficient condition, and many non-trivial examples. Our analysis assumes that the stock follows a scalar diffusion process, and uses the completeness of the resulting market model. A simple special case is when the drift and volatility depend only on time; then the two approaches are equivalent precisely if (μ (t)- r)/σ2(t) is constant. This is not the Sharpe ratio or the market price of risk, but rather a nondimensional ratio of excess return to squared volatility that arises naturally in portfolio optimization problems.

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    File URL: http://www.tandfonline.com/doi/abs/10.1080/14697680600580946
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    Bibliographic Info

    Article provided by Taylor & Francis Journals in its journal Quantitative Finance.

    Volume (Year): 6 (2006)
    Issue (Month): 2 ()
    Pages: 173-183

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    Handle: RePEc:taf:quantf:v:6:y:2006:i:2:p:173-183

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    Web page: http://www.tandfonline.com/RQUF20

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    Related research

    Keywords: Asset allocation; Portfolio optimization;

    References

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    1. M. B. Haugh & A. W. Lo, 2001. "Asset allocation and derivatives," Quantitative Finance, Taylor & Francis Journals, vol. 1(1), pages 45-72.
    2. David G. Hobson & L. C. G. Rogers, 1998. "Complete Models with Stochastic Volatility," Mathematical Finance, Wiley Blackwell, vol. 8(1), pages 27-48.
    3. Peter Carr & Jin Xing & Madam Dilip, 2001. "Optimal Investment in Derivative Securities," Working Papers wpn01-01, Warwick Business School, Finance Group.
    4. Merton, Robert C., 1995. "Financial innovation and the management and regulation of financial institutions," Journal of Banking & Finance, Elsevier, vol. 19(3-4), pages 461-481, June.
    5. Liu, Jun & Pan, Jun, 2003. "Dynamic derivative strategies," Journal of Financial Economics, Elsevier, vol. 69(3), pages 401-430, September.
    6. Green, Richard C. & Jarrow, Robert A., 1987. "Spanning and completeness in markets with contingent claims," Journal of Economic Theory, Elsevier, vol. 41(1), pages 202-210, February.
    7. Merton, Robert C, 1969. "Lifetime Portfolio Selection under Uncertainty: The Continuous-Time Case," The Review of Economics and Statistics, MIT Press, vol. 51(3), pages 247-57, August.
    8. P. Carr & D. Madan, 2001. "Optimal positioning in derivative securities," Quantitative Finance, Taylor & Francis Journals, vol. 1(1), pages 19-37.
    9. Liu, Jun & Pan, Jun, 2003. "Dynamic Derivative Strategies," Working papers 4334-02, Massachusetts Institute of Technology (MIT), Sloan School of Management.
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    Cited by:
    1. Roger Bowden & Jennifer Zhu, 2010. "Multi-scale variation, path risk and long-term portfolio management," Quantitative Finance, Taylor & Francis Journals, vol. 10(7), pages 783-796.

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