Stock trading rules under a switchable market

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June 2013, 3(2): 209-231. doi: 10.3934/mcrf.2013.3.209

Stock trading rules under a switchable market

Duy Nguyen ^1, , Jingzhi Tie ^1, and Qing Zhang ^1,

1.	Department of Mathematics, University of Georgia, Athens, GA 30602, United States, United States

Received January 2012 Revised September 2012 Published March 2013

Abstract
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This work provides an optimal trading rule that allows buying and selling of an asset sequentially over time. The asset price follows a regime switching model involving a geometric Brownian motion and a mean reversion model. The objective is to determine a sequence of trading times to maximize an overall return. The corresponding value functions are characterized by a set of quasi variational inequalities. Closed-form solutions are obtained. The sequence of trading times can be given in terms of various threshold levels. Numerical examples are given to demonstrate the results.

Keywords: optimal stopping, Mean-reverting process, quasi-variational inequalities..

Mathematics Subject Classification: Primary: 93E20, 91B26; Secondary: 49L2.

Citation: Duy Nguyen, Jingzhi Tie, Qing Zhang. Stock trading rules under a switchable market. Mathematical Control and Related Fields, 2013, 3 (2) : 209-231. doi: 10.3934/mcrf.2013.3.209

References:

[1]	M. Dai, Q. Zhang and Q. Zhu, Trend following trading under a regime switching model, SIAM Journal on Financial Mathematics, 1 (2010), 780-810. doi: 10.1137/080742889.
[2]	T. C. Johnson and M. Zervos, A discretionary stopping problem with applications to the optimal timing of investment decisions, Working paper, (2011).
[3]	H. T. Kong, G. Yin and Q. Zhang, A trend following strategy: Conditions for optimality, Automatica J. IFAC, 47 (2011), 661-667. doi: 10.1016/j.automatica.2011.01.039.
[4]	A. Løkka and M. Zervos, A model for the long-term optimal capacity level of an investment project, Working paper, (2011).
[5]	A. Merhi and M. Zervos, A model for reversible investment capacity expansion, SIAM Journal on Control and Optimization, 46 (2007), 839-876. doi: 10.1137/050640758.
[6]	D. Nguyen, Ph.D. Dissertation, 2013.
[7]	D. Nguyen, J. Tie and Q. Zhang, An optimal trading rule under switchable mean-reversion model, Journal of Optimization Theory and Applications, in press.
[8]	B. Øksendal, "Stochastic Differential Equations. An Introduction with Applications," Sixth edition, Universitext, Springer-Verlag, Berlin, 2003. doi: 10.1007/978-3-642-14394-6.
[9]	W. J. O'Neil, "How to Make Money in Stocks," Second edition, McGraw Hill, Inc., New York, 1995.
[10]	J. Yu and Q. Zhang, Optimal trend-following trading rules under a three-state regime switching model, Mathematical Control and Related Fields, 2 (2012), 81-100. doi: 10.3934/mcrf.2012.2.81.
[11]	H. Zhang and Q. Zhang, Trading a mean-reverting asset: Buy low and sell high, Automatica J. IFAC, 44 (2008), 1511-1518. doi: 10.1016/j.automatica.2007.11.003.

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References:

[1]	M. Dai, Q. Zhang and Q. Zhu, Trend following trading under a regime switching model, SIAM Journal on Financial Mathematics, 1 (2010), 780-810. doi: 10.1137/080742889.
[2]	T. C. Johnson and M. Zervos, A discretionary stopping problem with applications to the optimal timing of investment decisions, Working paper, (2011).
[3]	H. T. Kong, G. Yin and Q. Zhang, A trend following strategy: Conditions for optimality, Automatica J. IFAC, 47 (2011), 661-667. doi: 10.1016/j.automatica.2011.01.039.
[4]	A. Løkka and M. Zervos, A model for the long-term optimal capacity level of an investment project, Working paper, (2011).
[5]	A. Merhi and M. Zervos, A model for reversible investment capacity expansion, SIAM Journal on Control and Optimization, 46 (2007), 839-876. doi: 10.1137/050640758.
[6]	D. Nguyen, Ph.D. Dissertation, 2013.
[7]	D. Nguyen, J. Tie and Q. Zhang, An optimal trading rule under switchable mean-reversion model, Journal of Optimization Theory and Applications, in press.
[8]	B. Øksendal, "Stochastic Differential Equations. An Introduction with Applications," Sixth edition, Universitext, Springer-Verlag, Berlin, 2003. doi: 10.1007/978-3-642-14394-6.
[9]	W. J. O'Neil, "How to Make Money in Stocks," Second edition, McGraw Hill, Inc., New York, 1995.
[10]	J. Yu and Q. Zhang, Optimal trend-following trading rules under a three-state regime switching model, Mathematical Control and Related Fields, 2 (2012), 81-100. doi: 10.3934/mcrf.2012.2.81.
[11]	H. Zhang and Q. Zhang, Trading a mean-reverting asset: Buy low and sell high, Automatica J. IFAC, 44 (2008), 1511-1518. doi: 10.1016/j.automatica.2007.11.003.

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