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Continuous-Time Markowitz's Model with Transaction Costs

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  • Min Dai
  • Zuo Quan Xu
  • Xun Yu Zhou
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Abstract

A continuous-time Markowitz's mean-variance portfolio selection problem is studied in a market with one stock, one bond, and proportional transaction costs. This is a singular stochastic control problem,inherently in a finite time horizon. With a series of transformations, the problem is turned into a so-called double obstacle problem, a well studied problem in physics and partial differential equation literature, featuring two time-varying free boundaries. The two boundaries, which define the buy, sell, and no-trade regions, are proved to be smooth in time. This in turn characterizes the optimal strategy, via a Skorokhod problem, as one that tries to keep a certain adjusted bond-stock position within the no-trade region. Several features of the optimal strategy are revealed that are remarkably different from its no-transaction-cost counterpart. It is shown that there exists a critical length in time, which is dependent on the stock excess return as well as the transaction fees but independent of the investment target and the stock volatility, so that an expected terminal return may not be achievable if the planning horizon is shorter than that critical length (while in the absence of transaction costs any expected return can be reached in an arbitrary period of time). It is further demonstrated that anyone following the optimal strategy should not buy the stock beyond the point when the time to maturity is shorter than the aforementioned critical length. Moreover, the investor would be less likely to buy the stock and more likely to sell the stock when the maturity date is getting closer. These features, while consistent with the widely accepted investment wisdom, suggest that the planning horizon is an integral part of the investment opportunities.

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File URL: http://arxiv.org/pdf/0906.0678
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Paper provided by arXiv.org in its series Papers with number 0906.0678.

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Date of creation: Jun 2009
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Publication status: Published in Siam Journal on Financial Mathematics, vol.1, 2010, pp.96-125
Handle: RePEc:arx:papers:0906.0678

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  1. Black, Fischer & Scholes, Myron S, 1973. "The Pricing of Options and Corporate Liabilities," Journal of Political Economy, University of Chicago Press, University of Chicago Press, vol. 81(3), pages 637-54, May-June.
  2. Magill, Michael J. P. & Constantinides, George M., 1976. "Portfolio selection with transactions costs," Journal of Economic Theory, Elsevier, Elsevier, vol. 13(2), pages 245-263, October.
  3. Basak, Suleyman & Chabakauri, Georgy, 2009. "Dynamic Mean-Variance Asset Allocation," CEPR Discussion Papers, C.E.P.R. Discussion Papers 7256, C.E.P.R. Discussion Papers.
  4. Kydland, Finn E & Prescott, Edward C, 1977. "Rules Rather Than Discretion: The Inconsistency of Optimal Plans," Journal of Political Economy, University of Chicago Press, University of Chicago Press, vol. 85(3), pages 473-91, June.
  5. Hong Liu & Mark Loewenstein, 2002. "Optimal Portfolio Selection with Transaction Costs and Finite Horizons," Review of Financial Studies, Society for Financial Studies, Society for Financial Studies, vol. 15(3), pages 805-835.
  6. Jianming Xia, 2005. "Mean-Variance Portfolio Choice: Quadratic Partial Hedging," Mathematical Finance, Wiley Blackwell, Wiley Blackwell, vol. 15(3), pages 533-538.
  7. Henry R. Richardson, 1989. "A Minimum Variance Result in Continuous Trading Portfolio Optimization," Management Science, INFORMS, INFORMS, vol. 35(9), pages 1045-1055, September.
  8. Hanqing Jin & Xun Yu Zhou, 2008. "Behavioral Portfolio Selection In Continuous Time," Mathematical Finance, Wiley Blackwell, Wiley Blackwell, vol. 18(3), pages 385-426.
  9. Merton, Robert C., 1972. "An Analytic Derivation of the Efficient Portfolio Frontier," Journal of Financial and Quantitative Analysis, Cambridge University Press, Cambridge University Press, vol. 7(04), pages 1851-1872, September.
  10. Gerard Gennotte & Alan Jung, 1994. "Investment Strategies under Transaction Costs: The Finite Horizon Case," Management Science, INFORMS, INFORMS, vol. 40(3), pages 385-404, March.
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