Stochastic Control for Asset Management

Affiliation(s)

Department of International Business, Ming Chuan University, Taipei, Taiwan.

Department of Economics, Hong Kong Baptist University, Hong Kong, China.

Department of Finance, Providence University, Taichung, Taiwan.

Department of International Business, Ming Chuan University, Taipei, Taiwan.

Department of Economics, Hong Kong Baptist University, Hong Kong, China.

Department of Finance, Providence University, Taichung, Taiwan.

ABSTRACT

An investor is often faced with the investment situation in which he/she has to decide how to allocate his/her limited funds optimally among different assets to maximize his/her expected utility over the holding period. To this end, this study sets up a dynamic model driven by three assets to characterize the stochastic nature of the securities market and uses stochastic control to derive an explicit formula for the optimal fraction invested in each of the three assets for an investor with a power utility and a holding period of 10 years. Using estimated parameter values as inputs and implicit finite difference method, we determine numerically the optimal percentages invested in the three assets at each time over the holding period for both less risk-averse and more risk-averse investors.

Cite this paper

J. Kung, W. Wong and E. Wu, "Stochastic Control for Asset Management,"*Journal of Mathematical Finance*, Vol. 3 No. 1, 2013, pp. 59-69. doi: 10.4236/jmf.2013.31005.

J. Kung, W. Wong and E. Wu, "Stochastic Control for Asset Management,"

References

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[14] D. Easley and M. O’Hara, “Time and the Process of Security Price Adjustment,” Journal of Finance, Vol. 47, No. 2, 1992, pp. 576-605. doi:10.1111/j.1540-6261.1992.tb04402.x

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[24] O. A. Vasicek, “An Equilibrium Characterization of the Term Structure,” Journal of Financial Economics, Vol. 5, No. 2, 1977, pp. 177-188. doi:10.1016/0304-405X(77)90016-2

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[27] N. Barberis, “Investing for the Long Run when Returns Are Predictable,” Journal of Finance, Vol. 55, No. 1, 2000, pp. 225-264. doi:10.1111/0022-1082.00205

[28] M. Brandt, “Estimating Portfolio and Consumption Choice: A Conditional Euler Equations Approach,” Journal of Finance, Vol. 54, No. 5, 1999, pp. 1609-1645. doi:10.1111/0022-1082.00162

[29] A. Graflund and B. Nilsson, “Dynamic Portfolio Selection: the Relevance of Switching Regimes and Investment Horizon,” European Financial Management, Vol. 9, No. 2, 2003, pp. 179-200. doi:10.1111/1468-036X.00215

[30] R. S. Tsay, “Analysis of Financial Time Series,” 2nd Edition, John Wiley & Sons, New York, 2005. doi:10.1002/0471746193

[31] M. J. Brennan and E. S. Schwartz, “Finite Difference Method and Jump Processes Arising in the Pricing of Contingent Claims,” Journal of Financial & Quantitative Analysis, Vol. 13, No. 3, 1978, pp. 461-474. doi:10.2307/2330152

[32] G. Courtadon, “A More Accurate Finite Difference Approximation for the Valuation of Options,” Journal of Financial & Quantitative Analysis, Vol. 17, No. 5, 1982, pp. 697-705. doi:10.2307/2330857

[33] J. C. Hull, “Options, Futures and Other Derivatives,” 6th Edition, Prentice-Hall, Englewood Cliffs, 2006.

[34] M. J. Brennan, E. S. Schwartz and R. Lagnado, “Strategic Asset Allocation,” Journal of Economic Dynamics & Control, Vol. 21, No. 8-9, 1997, pp. 1377-1403. doi:10.1016/S0165-1889(97)00031-6

[1] H. M. Markowitz, “Portfolio Selection,” Journal of Finance, Vol. 7, No. 1, 1952, pp. 77-91.

[2] H. M. Markowitz, “Mean-Variance Analysis in Portfolio Choice and Capital Markets,” Blackwell, New York, 1987.

[3] A. D. Roy, “Safety First and the Holding of Assets,” Econometrica, Vol. 20, No. 3, 1952, pp. 431-439. doi:10.2307/1907413

[4] T. E. Copeland, J. F. Weston and K. Shastri, “Financial Theory and Corporate Policy,” 4th Edition, Addison Wesley, Boston, 2005.

[5] M. Grinblatt and S. Titman, “Financial Markets and Corporate Strategy,” 2nd Edition, McGraw-Hill, Boston, 2002.

[6] B. G. Malkiel, “A Random Walk down Wall Street,” Norton and Company, New York, 2003.

[7] F. K. Reilly and E. A. Norton, “Investments,” 6th Edition, South-Western, Cincinnati, 2003.

[8] W. F. Sharpe, G. J. Alexander and J. V. Bailey, “Investments,” 6th Edition, Prentice-Hall, Englewood Cliffs, 1999.

[9] J. J. Siegel, “Stocks for the Long Run,” 3rd Edition, McGraw-Hill, New York, 2002.

[10] G. N. Mankiw and J. A. Miron, “Changing Behavior of the Term Structure of Interest Rates,” Quarterly Journal of Economics, Vol. 101, No. 2, 1986, pp. 211-228. doi:10.2307/1891113

[11] G. S. Oldfield and R. J. Roglaski, “The Stochastic Properties of Term Structure Movements,” Journal of Monetary Economics, Vol. 19, No. 2, 1987, pp. 229-254. doi:10.1016/0304-3932(87)90048-1

[12] C. E. Walsh, “Interest Rate Volatility and Monetary Policy,” Journal of Money, Credit and Banking, Vol. 16, No. 2, 1984, pp. 133-150. doi:10.2307/1992540

[13] J. Conrad and G. Kaul, “Long-Term Market Overreaction or Biases in Computed Returns,” Journal of Finance, Vol. 48, No. 1, 1993, pp. 39-63. doi:10.1111/j.1540-6261.1993.tb04701.x

[14] D. Easley and M. O’Hara, “Time and the Process of Security Price Adjustment,” Journal of Finance, Vol. 47, No. 2, 1992, pp. 576-605. doi:10.1111/j.1540-6261.1992.tb04402.x

[15] E. Fama, “Efficient Capital Markets II,” Journal of Finance, Vol. 46, No. 5, 1991, pp. 1575-1617. doi:10.1111/j.1540-6261.1991.tb04636.x

[16] W. Ferson and C. Harvey, “Sources of Predictability in Portfolio Returns,” Financial Analysts Journal, Vol. 47, No. 3, 1991, pp. 49-56. doi:10.2469/faj.v47.n3.49

[17] W. Ferson and C. Harvey, “The Variation of Economic Risk Premiums,” Journal of Political Economy, Vol. 99, No. 2, 1991, pp. 385-415. doi:10.1086/261755

[18] N. Jegadeesh, “Seasonality in Stock Price Mean Reversion: Evidence from the US and the UK,” Journal of Finance, Vol. 46, No. 4, 1991, pp. 1427-1444. doi:10.1111/j.1540-6261.1991.tb04624.x

[19] A. W. Lo and A. C. MacKinlay, “A Non-Random Walk down Wall Street,” Princeton University Press, Princeton, 1999.

[20] R. Litterman and J. Scheinkman, “Common Factors Affecting Bond Returns,” Journal of Fixed Income, Vol. 1, No. 1, 1991, pp. 54-61. doi:10.3905/jfi.1991.692347

[21] P. H. Dybvig, “Bond and Bond Option Pricing based on the Current Term Structure,” In: M. A. H. Dempster and S. R. Pliska (Eds.), Mathematics of Derivative Securities, Cambridge University Press, Cambridge, 1997, pp. 271293.

[22] T. Andersen and J. Lund, “Stochastic Volatility and Mean Drift in the Short Rate Diffusion: Sources of Steepness, Level and Curvature in the Yield Curve,” Manuscript, Northwestern University, Evanston, 1999.

[23] S. Heston and S. Nandi, “A Two-Factor Term Structure Model under GARCH Volatility,” Journal of Fixed Income, Vol. 13, No. 1, 2003, pp. 87-95. doi:10.3905/jfi.2003.319348

[24] O. A. Vasicek, “An Equilibrium Characterization of the Term Structure,” Journal of Financial Economics, Vol. 5, No. 2, 1977, pp. 177-188. doi:10.1016/0304-405X(77)90016-2

[25] J. Y. Campbell and L. M. Viceira, “Strategic Asset Allocation: Portfolio Choice for Long-Term Investors,” Oxford University Press, Oxford, 2002.

[26] P. Balduzzi and A. W. Lynch, “Transaction Costs and Predictability: Some Utility Cost Calculations,” Journal of Financial Economics, Vol. 52, No. 1, 1999, pp. 47-78. doi:10.1016/S0304-405X(99)00004-5

[27] N. Barberis, “Investing for the Long Run when Returns Are Predictable,” Journal of Finance, Vol. 55, No. 1, 2000, pp. 225-264. doi:10.1111/0022-1082.00205

[28] M. Brandt, “Estimating Portfolio and Consumption Choice: A Conditional Euler Equations Approach,” Journal of Finance, Vol. 54, No. 5, 1999, pp. 1609-1645. doi:10.1111/0022-1082.00162

[29] A. Graflund and B. Nilsson, “Dynamic Portfolio Selection: the Relevance of Switching Regimes and Investment Horizon,” European Financial Management, Vol. 9, No. 2, 2003, pp. 179-200. doi:10.1111/1468-036X.00215

[30] R. S. Tsay, “Analysis of Financial Time Series,” 2nd Edition, John Wiley & Sons, New York, 2005. doi:10.1002/0471746193

[31] M. J. Brennan and E. S. Schwartz, “Finite Difference Method and Jump Processes Arising in the Pricing of Contingent Claims,” Journal of Financial & Quantitative Analysis, Vol. 13, No. 3, 1978, pp. 461-474. doi:10.2307/2330152

[32] G. Courtadon, “A More Accurate Finite Difference Approximation for the Valuation of Options,” Journal of Financial & Quantitative Analysis, Vol. 17, No. 5, 1982, pp. 697-705. doi:10.2307/2330857

[33] J. C. Hull, “Options, Futures and Other Derivatives,” 6th Edition, Prentice-Hall, Englewood Cliffs, 2006.

[34] M. J. Brennan, E. S. Schwartz and R. Lagnado, “Strategic Asset Allocation,” Journal of Economic Dynamics & Control, Vol. 21, No. 8-9, 1997, pp. 1377-1403. doi:10.1016/S0165-1889(97)00031-6