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September & December  2018, 8(3&4): 965-999. doi: 10.3934/mcrf.2018042

Switching between a pair of stocks: An optimal trading rule

Jingzhi Tie and  Qing Zhang

Department of Mathematics, University of Georgia, Athens, GA 30602, USA

Received  July 2017 Revised  June 2018 Published  September 2018

This paper is about a stock trading rule involving two stocks. The trader may have a long position in either stock or in cash. She may also switch between them any time. Her objective is to trade over time to maximize an expected return. In this paper, we reduce the problem to the optimal trading control problem under a geometric Brownian motion model with regime switching. We use a two-state Markov chain to capture the general market modes. In particular, a single market cycle consisting of a bull market followed by a bear market is considered. We also impose a fixed percentage cost on each transaction. We focus on simple threshold-type policies and study all possible combinations. We establish algebraic equations to characterize these threshold levels. We also present sufficient conditions that guarantee the optimality of these policies. Finally, some numerical examples are provided to illustrate our results.

Keywords: Switching trading, quasi-variational inequalities, geometric Brownian motions with regime switching.
Mathematics Subject Classification: Primary: 93E20, 91G80; Secondary: 49L20.
Citation: Jingzhi Tie, Qing Zhang. Switching between a pair of stocks: An optimal trading rule. Mathematical Control and Related Fields, 2018, 8 (3&4) : 965-999. doi: 10.3934/mcrf.2018042
References:
[1]

B. R. Barmish and J. A. Primbs, On a new paradigm for stock trading via a model-free feedback controller, IEEE Trans. Automatic Control, 61 (2016), 662-676.  doi: 10.1109/TAC.2015.2444078.

[2]

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: 10.1137/090770552.

[3]

M. H. A. Davis and A. R. Norman, Portfolio selection with transaction costs, Mathematics of Operations Research, 15 (1990), 676-713.  doi: 10.1287/moor.15.4.676.

[4]

X. Guo and Q. Zhang, Optimal selling rules in a regime switching model, IEEE Trans. Automatic Control, 50 (2005), 1450-1455.  doi: 10.1109/TAC.2005.854657.

[5]

Y. Hu and B. Øksendal, Optimal time to invest when the price processes are geometric Brownian motions, Finance and Stochastics, 2 (1998), 295-310.  doi: 10.1007/s007800050042.

[6]

S. Iwarere and B. R. Barmish, A confidence interval triggering method for stock trading via feedback control, Proc. American Control Conference, Baltimore, MD, (2010), 6910-6916. doi: 10.1109/ACC.2010.5531311.

[7]

A. Merhi and M. Zervos, A model for reversible investment capacity expansion, SIAM J. Control and Optimization, 46 (2007), 839-876.  doi: 10.1137/050640758.

[8]

D. Nguyen, J. Tie and Q. Zhang, An optimal trading rule under a switchable mean-reversion model, Journal of Optimization Theory and Applications, 161 (2014), 145-163.  doi: 10.1007/s10957-012-0260-x.

[9]

B. Øksendal, Stochastic Differential Equations, 6th Ed., Springer-Verlag, New York, 2003. doi: 10.1007/978-3-642-14394-6.

[10]

S. E. Shreve and H. M. Soner, Optimal investment and consumption with transaction costs, Annals of Applied Probability, 4 (1994), 609-692.  doi: 10.1214/aoap/1177004966.

[11]

Q. S. Song and Q. Zhang, An optimal pairs-trading rule, Automatica, 49 (2013), 3007-3014.  doi: 10.1016/j.automatica.2013.07.012.

[12]

E. M. Stein and R. Shakarchi, Functional Analysis: Introduction to Further Topics in Analysis, Princeton University Press, Princeton and Oxford, 2011.

[13]

J. Tie, H. Zhang and Q. Zhang, An Optimal strategy for pairs-trading under geometric Brownian motions, Journal of Optimization Theory and Applications, (2017), 1-22.  doi: 10.1007/s10957-017-1065-8.

[14]

H. Zhang and Q. Zhang, Trading a mean-reverting asset: Buy low and sell high, Automatica, 44 (2008), 1511-1518.  doi: 10.1016/j.automatica.2007.11.003.

[15]

Q. Zhang, Stock trading: An optimal selling rule, SIAM J. Control and Optimization, 40 (2001), 64-87.  doi: 10.1137/S0363012999356325.

show all references

References:
[1]

B. R. Barmish and J. A. Primbs, On a new paradigm for stock trading via a model-free feedback controller, IEEE Trans. Automatic Control, 61 (2016), 662-676.  doi: 10.1109/TAC.2015.2444078.

[2]

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: 10.1137/090770552.

[3]

M. H. A. Davis and A. R. Norman, Portfolio selection with transaction costs, Mathematics of Operations Research, 15 (1990), 676-713.  doi: 10.1287/moor.15.4.676.

[4]

X. Guo and Q. Zhang, Optimal selling rules in a regime switching model, IEEE Trans. Automatic Control, 50 (2005), 1450-1455.  doi: 10.1109/TAC.2005.854657.

[5]

Y. Hu and B. Øksendal, Optimal time to invest when the price processes are geometric Brownian motions, Finance and Stochastics, 2 (1998), 295-310.  doi: 10.1007/s007800050042.

[6]

S. Iwarere and B. R. Barmish, A confidence interval triggering method for stock trading via feedback control, Proc. American Control Conference, Baltimore, MD, (2010), 6910-6916. doi: 10.1109/ACC.2010.5531311.

[7]

A. Merhi and M. Zervos, A model for reversible investment capacity expansion, SIAM J. Control and Optimization, 46 (2007), 839-876.  doi: 10.1137/050640758.

[8]

D. Nguyen, J. Tie and Q. Zhang, An optimal trading rule under a switchable mean-reversion model, Journal of Optimization Theory and Applications, 161 (2014), 145-163.  doi: 10.1007/s10957-012-0260-x.

[9]

B. Øksendal, Stochastic Differential Equations, 6th Ed., Springer-Verlag, New York, 2003. doi: 10.1007/978-3-642-14394-6.

[10]

S. E. Shreve and H. M. Soner, Optimal investment and consumption with transaction costs, Annals of Applied Probability, 4 (1994), 609-692.  doi: 10.1214/aoap/1177004966.

[11]

Q. S. Song and Q. Zhang, An optimal pairs-trading rule, Automatica, 49 (2013), 3007-3014.  doi: 10.1016/j.automatica.2013.07.012.

[12]

E. M. Stein and R. Shakarchi, Functional Analysis: Introduction to Further Topics in Analysis, Princeton University Press, Princeton and Oxford, 2011.

[13]

J. Tie, H. Zhang and Q. Zhang, An Optimal strategy for pairs-trading under geometric Brownian motions, Journal of Optimization Theory and Applications, (2017), 1-22.  doi: 10.1007/s10957-017-1065-8.

[14]

H. Zhang and Q. Zhang, Trading a mean-reverting asset: Buy low and sell high, Automatica, 44 (2008), 1511-1518.  doi: 10.1016/j.automatica.2007.11.003.

[15]

Q. Zhang, Stock trading: An optimal selling rule, SIAM J. Control and Optimization, 40 (2001), 64-87.  doi: 10.1137/S0363012999356325.

Figure 1.  Switching Regions ($\alpha _t = 1$)
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Figure 2.  Value Functions
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Figure 3.  ${\bf{S}}^1$ = QQQ, ${\bf{S}}^2$ = SPY: The threshold levels ${b_1},{b_2}$ and the corresponding equity curve
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Figure 4.  ${\bf{S}}^1$ = QQQ, ${\bf{S}}^2$ = SPY: The threshold levels ${b_1},{b_2}$ and the corresponding equity curve
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Figure 5.  log(Daily closing prices) of KO and PEP
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Figure 6.  ${\bf{S}}^1$ = KO, ${\bf{S}}^2$ = PEP: The threshold levels $b_1,b_2,s_1,s_2$ and the corresponding equity curve
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Table 1.  $(b_1,b_2,s_1,s_2)$ with varying $\mu_1$
$\mu_1$ 0.1548 0.1648 0.1748 0.1848 0.1948
$s_1$ 2.0767 2.2457 2.4152 2.5851 2.7557
$b_2$ 1.9031 2.0666 2.2309 2.3961 2.5620
$b_1$ 1.8936 2.0566 2.2203 2.3849 2.5503
$s_2$ 1.6435 1.7812 1.9193 2.0576 2.1964
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$\mu_1$ 0.1548 0.1648 0.1748 0.1848 0.1948
$s_1$ 2.0767 2.2457 2.4152 2.5851 2.7557
$b_2$ 1.9031 2.0666 2.2309 2.3961 2.5620
$b_1$ 1.8936 2.0566 2.2203 2.3849 2.5503
$s_2$ 1.6435 1.7812 1.9193 2.0576 2.1964
Table 2.  $(b_1,b_2,s_1,s_2)$ with varying $\mu_2$
$\mu_2$ 0.0787 0.0887 0.0987 0.1087 0.1187
$s_1$ 3.4930 2.8570 2.4152 2.0905 1.8420
$b_2$ 3.2466 2.6468 2.2309 1.9261 1.6932
$b_1$ 3.2296 2.6336 2.2203 1.9173 1.6857
$s_2$ 2.7063 2.2459 1.9193 1.6753 1.4862
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$\mu_2$ 0.0787 0.0887 0.0987 0.1087 0.1187
$s_1$ 3.4930 2.8570 2.4152 2.0905 1.8420
$b_2$ 3.2466 2.6468 2.2309 1.9261 1.6932
$b_1$ 3.2296 2.6336 2.2203 1.9173 1.6857
$s_2$ 2.7063 2.2459 1.9193 1.6753 1.4862
Table 3.  $(b_1,b_2,s_1,s_2)$ with varying $\sigma_{11}$
$\sigma_{11}$ 0.2607 0.2707 0.2807 0.2907 0.3007
$s_1$ 2.4001 2.4076 2.4152 2.4227 2.4302
$b_2$ 2.2291 2.2300 2.2309 2.2320 2.2331
$b_1$ 2.2199 2.2201 2.2203 2.2206 2.2210
$s_2$ 1.9350 1.9270 1.9193 1.9116 1.9042
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$\sigma_{11}$ 0.2607 0.2707 0.2807 0.2907 0.3007
$s_1$ 2.4001 2.4076 2.4152 2.4227 2.4302
$b_2$ 2.2291 2.2300 2.2309 2.2320 2.2331
$b_1$ 2.2199 2.2201 2.2203 2.2206 2.2210
$s_2$ 1.9350 1.9270 1.9193 1.9116 1.9042
Table 4.  $(b_1,b_2,s_1,s_2)$ with varying $\sigma_{22}$
$\sigma_{22}$ 0.1071 0.1171 0.1271 0.1371 0.1471
$s_1$ 2.4132 2.4140 2.4152 2.4167 2.4187
$b_2$ 2.2307 2.2308 2.2309 2.2312 2.2314
$b_1$ 2.2203 2.2203 2.2203 2.2204 2.2205
$s_2$ 1.9213 1.9205 1.9193 1.9177 1.9157
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$\sigma_{22}$ 0.1071 0.1171 0.1271 0.1371 0.1471
$s_1$ 2.4132 2.4140 2.4152 2.4167 2.4187
$b_2$ 2.2307 2.2308 2.2309 2.2312 2.2314
$b_1$ 2.2203 2.2203 2.2203 2.2204 2.2205
$s_2$ 1.9213 1.9205 1.9193 1.9177 1.9157
Table 5.  $(b_1,b_2,s_1,s_2)$ with varying $\sigma_{12}( = \sigma_{21})$
$\sigma_{12}(=\sigma_{21})$ 0.0729 0.0829 0.0929 0.1029 0.1129
$s_1$ 2.4333 2.4242 2.4152 2.4064 2.3978
$b_2$ 2.2336 2.2322 2.2309 2.2298 2.2288
$b_1$ 2.2211 2.2207 2.2203 2.2201 2.2199
$s_2$ 1.9011 1.9102 1.9193 1.9283 1.9374
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$\sigma_{12}(=\sigma_{21})$ 0.0729 0.0829 0.0929 0.1029 0.1129
$s_1$ 2.4333 2.4242 2.4152 2.4064 2.3978
$b_2$ 2.2336 2.2322 2.2309 2.2298 2.2288
$b_1$ 2.2211 2.2207 2.2203 2.2201 2.2199
$s_2$ 1.9011 1.9102 1.9193 1.9283 1.9374
Table 6.  $(b_1,b_2,s_1,s_2)$ with varying $\rho$
$\rho$ 0.01 0.02 0.03 0.04 0.05
$s_1$ 2.1006 2.2373 2.4152 2.6561 3.0009
$b_2$ 1.9439 2.0685 2.2309 2.4515 2.7681
$b_1$ 1.9356 2.0592 2.2203 2.4391 2.7529
$s_2$ 1.7008 1.7961 1.9193 2.0844 2.3174
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$\rho$ 0.01 0.02 0.03 0.04 0.05
$s_1$ 2.1006 2.2373 2.4152 2.6561 3.0009
$b_2$ 1.9439 2.0685 2.2309 2.4515 2.7681
$b_1$ 1.9356 2.0592 2.2203 2.4391 2.7529
$s_2$ 1.7008 1.7961 1.9193 2.0844 2.3174
Table 7.  $(b_1,b_2,s_1,s_2)$ with varying $\lambda $
$\lambda $ 0.8 0.9 1.0 1.1 1.2
$s_1$ 2.4070 2.4111 2.4152 2.4193 2.4201
$b_2$ 2.2266 2.2288 2.2309 2.2332 2.2336
$b_1$ 2.2158 2.2181 2.2203 2.2226 2.2231
$s_2$ 1.9208 1.9200 1.9193 1.9184 1.9183
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$\lambda $ 0.8 0.9 1.0 1.1 1.2
$s_1$ 2.4070 2.4111 2.4152 2.4193 2.4201
$b_2$ 2.2266 2.2288 2.2309 2.2332 2.2336
$b_1$ 2.2158 2.2181 2.2203 2.2226 2.2231
$s_2$ 1.9208 1.9200 1.9193 1.9184 1.9183
Table 8.  $(b_1,b_2,s_1,s_2)$ with varying $K$
$K$ 0.00050 0.00075 0.00100 0.00500 0.01000
$s_1$ 2.3315 2.3762 2.4152 2.8814 3.4914
$b_2$ 2.1908 2.2119 2.2309 2.4752 2.7979
$b_1$ 2.1840 2.2031 2.2203 2.4459 2.7528
$s_2$ 1.9543 1.9347 1.9193 1.7878 1.6694
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$K$ 0.00050 0.00075 0.00100 0.00500 0.01000
$s_1$ 2.3315 2.3762 2.4152 2.8814 3.4914
$b_2$ 2.1908 2.2119 2.2309 2.4752 2.7979
$b_1$ 2.1840 2.2031 2.2203 2.4459 2.7528
$s_2$ 1.9543 1.9347 1.9193 1.7878 1.6694
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