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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

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 & 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.  doi: 10.1137/080742889.  Google Scholar

[2]

T. C. Johnson and M. Zervos, A discretionary stopping problem with applications to the optimal timing of investment decisions,, Working paper, (2011).   Google Scholar

[3]

H. T. Kong, G. Yin and Q. Zhang, A trend following strategy: Conditions for optimality,, Automatica J. IFAC, 47 (2011), 661.  doi: 10.1016/j.automatica.2011.01.039.  Google Scholar

[4]

A. Løkka and M. Zervos, A model for the long-term optimal capacity level of an investment project,, Working paper, (2011).   Google Scholar

[5]

A. Merhi and M. Zervos, A model for reversible investment capacity expansion,, SIAM Journal on Control and Optimization, 46 (2007), 839.  doi: 10.1137/050640758.  Google Scholar

[6]

D. Nguyen, Ph.D. Dissertation,, 2013., ().   Google Scholar

[7]

D. Nguyen, J. Tie and Q. Zhang, An optimal trading rule under switchable mean-reversion model,, Journal of Optimization Theory and Applications, ().   Google Scholar

[8]

B. Øksendal, "Stochastic Differential Equations. An Introduction with Applications,", Sixth edition, (2003).  doi: 10.1007/978-3-642-14394-6.  Google Scholar

[9]

W. J. O'Neil, "How to Make Money in Stocks,", Second edition, (1995).   Google Scholar

[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.  doi: 10.3934/mcrf.2012.2.81.  Google Scholar

[11]

H. Zhang and Q. Zhang, Trading a mean-reverting asset: Buy low and sell high,, Automatica J. IFAC, 44 (2008), 1511.  doi: 10.1016/j.automatica.2007.11.003.  Google Scholar

show all references

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.  doi: 10.1137/080742889.  Google Scholar

[2]

T. C. Johnson and M. Zervos, A discretionary stopping problem with applications to the optimal timing of investment decisions,, Working paper, (2011).   Google Scholar

[3]

H. T. Kong, G. Yin and Q. Zhang, A trend following strategy: Conditions for optimality,, Automatica J. IFAC, 47 (2011), 661.  doi: 10.1016/j.automatica.2011.01.039.  Google Scholar

[4]

A. Løkka and M. Zervos, A model for the long-term optimal capacity level of an investment project,, Working paper, (2011).   Google Scholar

[5]

A. Merhi and M. Zervos, A model for reversible investment capacity expansion,, SIAM Journal on Control and Optimization, 46 (2007), 839.  doi: 10.1137/050640758.  Google Scholar

[6]

D. Nguyen, Ph.D. Dissertation,, 2013., ().   Google Scholar

[7]

D. Nguyen, J. Tie and Q. Zhang, An optimal trading rule under switchable mean-reversion model,, Journal of Optimization Theory and Applications, ().   Google Scholar

[8]

B. Øksendal, "Stochastic Differential Equations. An Introduction with Applications,", Sixth edition, (2003).  doi: 10.1007/978-3-642-14394-6.  Google Scholar

[9]

W. J. O'Neil, "How to Make Money in Stocks,", Second edition, (1995).   Google Scholar

[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.  doi: 10.3934/mcrf.2012.2.81.  Google Scholar

[11]

H. Zhang and Q. Zhang, Trading a mean-reverting asset: Buy low and sell high,, Automatica J. IFAC, 44 (2008), 1511.  doi: 10.1016/j.automatica.2007.11.003.  Google Scholar

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