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 JMF  Vol.3 No.4 , November 2013
An Extension of Some Results Due to Cox and Leland
Abstract: We investigate an optimal portfolio allocation problem between a risky and a risk-free asset, as in [1]. They obtained explicit conditions for path-independence and optimality of allocation strategies when the price of the risky asset follows a geometric Brownian motion with constant asset characteristics. This paper analyzes and extends their results for dynamic investment strategies by allowing for non-constant returns and volatility. We adopt a continuous-time approach and appeal to well established results in stochastic calculus for doing so.
Cite this paper: A. Leung and W. Shi, "An Extension of Some Results Due to Cox and Leland," Journal of Mathematical Finance, Vol. 3 No. 4, 2013, pp. 416-425. doi: 10.4236/jmf.2013.34043.
References

[1]   J. C. Cox and H. E. Leland, “On Dynamic Investment Strategies,” Journal of Economic Dynamics and Control, Vol. 24, No. 11-12, 2000, pp. 1859-1880.
http://dx.doi.org/10.1016/S0165-1889(99)00095-0

[2]   R. C. Merton, “Optimum Consumption and Portfolio Rules in a Continuoustime Model,” Journal of Economic Theory, Vol. 3, No. 4, 1971, pp. 373-413.
http://dx.doi.org/10.1016/0022-0531(71)90038-X

[3]   F. Black and M. Scholes, “The Pricing of Options and Corporate Liabilities,” The Journal of Political Economy, Vol. 81, No. 3, 1973, pp. 637-654.
http://dx.doi.org/10.1086/260062

[4]   J. C. Cox and S. A. Ross, “The Valuation of Options for Alternative Stochastic Processes,” Journal of Financial Economics, Vol. 3, No. 1-2, 1976, pp. 145-166.
http://dx.doi.org/10.1016/0304-405X(76)90023-4

[5]   P. Carr, H. Geman and D. Madan, “The Fine Structure of Asset Returns: An Empirical Investigation,” Journal of Business, Vol. 75, No. 2, 2002, pp. 305-332.
http://dx.doi.org/10.1086/338705

[6]   S. Hodges and R. Clarkson, “Dynamic Asset Allocation: Insights from Theory,” Philosophical Transactions: Physical Sciences and Engineering, Vol. 347, No. 1684, 1994, pp. 587-598.

[7]   R. C. Merton, “Lifetime Consumption under Uncertaintiy: The Continuous Time Case,” The Review of Economics and Statistics, Vol. 51, No. 3, 1969, pp. 247-257.
http://dx.doi.org/10.2307/1926560

[8]   S. A. Ross, “Mutual Fund Separation in Financial Theory—The Separating Distributions,” Journal of Economic Theory, Vol. 17, No. 2, 1978, pp. 254-286.
http://dx.doi.org/10.1016/0022-0531(78)90073-X

[9]   G. S. Amin and H. M. Kat, “Hedge Fund Performance 1990-2000: Do the ‘Money Machines’ Really Add Value?” Journal of Financial and Quantitative Analysis, Vol. 38, No. 2, 2003, pp. 251-274.
http://dx.doi.org/10.2307/4126750

[10]   S. Vanduffel, A. Chernih, M. Maj and W. Schoutens, “A Note on the Suboptimality of Path-Dependent Pay-Offs in Lévy Markets,” Applied Mathematical Finance, Vol. 16, No. 4, 2009, pp. 315-330.
http://dx.doi.org/10.1080/13504860802639360

[11]   S. Kassberger and T. Liebmann, “When Are Path-Dependent Payoffs Suboptimal?” 2009.

[12]   F. C. Klebaner, “Introduction to Stochastic Calculus with Applications,” Imperial College Press, London, 2005.
http://dx.doi.org/10.1142/p386

[13]   O. C. Ibe, “Markov Processes for Stochastic Modeling,” Elsevier Academic Press, Waltham, 2009.

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

[15]   J. Cvitanic and V. Polimenis, “Optimal Portfolio Allocation with Higher Mo Moments,” Annals of Finance, 2008.

[16]   S. P. Sethi, “A note on Mertons Optimum Consumption and Portfolio Rules in a continuous-Time Model,” Journal of Economic Theory, Vol. 46, No. 2, 1988, pp. 395401. http://dx.doi.org/10.1016/0022-0531(88)90138-X

[17]   A. Meucci, “Review of Dynamic Allocation Strategies: Utility Maximization, Option Replication, Insurance, Drawdown Control, Convex-Concave Management,” 2010.

[18]   K. Tanaka, “Dynamic Asset Allocation under Stochastic Interest Rate and Market Price of Risk,” 2009.

[19]   C. Chiarella, C. Hsaio and W. Semmler, “Intertemporal Investment Strategies under Inflation Risk,” University of Technology, Sydney, 2007.

[20]   S. E. Dreyfus, “Dynamic Programming and the Calculus of Variations,” Academic Press, Norwell, 1965.

[21]   S. Janson and J. Tysk, “Feynman-Kac Formulas for BlackScholes Type Operators,” Bulletin London Mathematical Society, Vol. 38, No. 2, 2006, pp. 269-282.
http://dx.doi.org/10.1112/S0024609306018194

 
 
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