American Institute of Mathematical Sciences

Advanced
September  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 & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

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

[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.  Google Scholar

[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.  Google Scholar

[12]

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

[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.  Google Scholar

[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.  Google Scholar

[15]

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

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

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

[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.  Google Scholar

[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.  Google Scholar

[12]

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

[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.  Google Scholar

[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.  Google Scholar

[15]

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

Figure 1.  Switching Regions ($\alpha _t = 1$)
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 2.  Value Functions
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 3.  ${\bf{S}}^1$ = QQQ, ${\bf{S}}^2$ = SPY: The threshold levels ${b_1},{b_2}$ and the corresponding equity curve
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 4.  ${\bf{S}}^1$ = QQQ, ${\bf{S}}^2$ = SPY: The threshold levels ${b_1},{b_2}$ and the corresponding equity curve
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 5.  log(Daily closing prices) of KO and PEP
Figure Options

Download full-size image

Download as PowerPoint slide

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
Figure Options

Download full-size image

Download as PowerPoint slide

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
Table Options

Download as excel

$\mu_1$ 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
Table Options

Download as excel

$\mu_2$ 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
Table Options

Download as excel

$\sigma_{11}$ 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
Table Options

Download as excel

$\sigma_{22}$ 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
Table Options

Download as excel

$\sigma_{12}(=\sigma_{21})$ 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
Table Options

Download as excel

$\rho$ 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
Table Options

Download as excel

$\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
Table Options

Download as excel

$K$ 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
[1]

Lori Badea. Multigrid methods for some quasi-variational inequalities. Discrete & Continuous Dynamical Systems - S, 2013, 6 (6) : 1457-1471. doi: 10.3934/dcdss.2013.6.1457
[2]

Yusuke Murase, Risei Kano, Nobuyuki Kenmochi. Elliptic Quasi-variational inequalities and applications. Conference Publications, 2009, 2009 (Special) : 583-591. doi: 10.3934/proc.2009.2009.583
[3]

Yurii Nesterov, Laura Scrimali. Solving strongly monotone variational and quasi-variational inequalities. Discrete & Continuous Dynamical Systems - A, 2011, 31 (4) : 1383-1396. doi: 10.3934/dcds.2011.31.1383
[4]

Laura Scrimali. Mixed behavior network equilibria and quasi-variational inequalities. Journal of Industrial & Management Optimization, 2009, 5 (2) : 363-379. doi: 10.3934/jimo.2009.5.363
[5]

Yusuke Murase, Atsushi Kadoya, Nobuyuki Kenmochi. Optimal control problems for quasi-variational inequalities and its numerical approximation. Conference Publications, 2011, 2011 (Special) : 1101-1110. doi: 10.3934/proc.2011.2011.1101
[6]

Jie Yu, Qing Zhang. Optimal trend-following trading rules under a three-state regime switching model. Mathematical Control & Related Fields, 2012, 2 (1) : 81-100. doi: 10.3934/mcrf.2012.2.81
[7]

Haisen Zhang. Clarke directional derivatives of regularized gap functions for nonsmooth quasi-variational inequalities. Mathematical Control & Related Fields, 2014, 4 (3) : 365-379. doi: 10.3934/mcrf.2014.4.365
[8]

Samir Adly, Tahar Haddad. On evolution quasi-variational inequalities and implicit state-dependent sweeping processes. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 0-0. doi: 10.3934/dcdss.2020105
[9]

Lin Xu, Rongming Wang, Dingjun Yao. Optimal stochastic investment games under Markov regime switching market. Journal of Industrial & Management Optimization, 2014, 10 (3) : 795-815. doi: 10.3934/jimo.2014.10.795
[10]

Fuke Wu, George Yin, Zhuo Jin. Kolmogorov-type systems with regime-switching jump diffusion perturbations. Discrete & Continuous Dynamical Systems - B, 2016, 21 (7) : 2293-2319. doi: 10.3934/dcdsb.2016048
[11]

Nguyen Huu Du, Nguyen Thanh Dieu, Tran Dinh Tuong. Dynamic behavior of a stochastic predator-prey system under regime switching. Discrete & Continuous Dynamical Systems - B, 2017, 22 (9) : 3483-3498. doi: 10.3934/dcdsb.2017176
[12]

Kun Fan, Yang Shen, Tak Kuen Siu, Rongming Wang. On a Markov chain approximation method for option pricing with regime switching. Journal of Industrial & Management Optimization, 2016, 12 (2) : 529-541. doi: 10.3934/jimo.2016.12.529
[13]

Hongfu Yang, Xiaoyue Li, George Yin. Permanence and ergodicity of stochastic Gilpin-Ayala population model with regime switching. Discrete & Continuous Dynamical Systems - B, 2016, 21 (10) : 3743-3766. doi: 10.3934/dcdsb.2016119
[14]

Rui Wang, Xiaoyue Li, Denis S. Mukama. On stochastic multi-group Lotka-Volterra ecosystems with regime switching. Discrete & Continuous Dynamical Systems - B, 2017, 22 (9) : 3499-3528. doi: 10.3934/dcdsb.2017177
[15]

Jiaqin Wei. Time-inconsistent optimal control problems with regime-switching. Mathematical Control & Related Fields, 2017, 7 (4) : 585-622. doi: 10.3934/mcrf.2017022
[16]

Zhuo Jin, Linyi Qian. Lookback option pricing for regime-switching jump diffusion models. Mathematical Control & Related Fields, 2015, 5 (2) : 237-258. doi: 10.3934/mcrf.2015.5.237
[17]

Mourad Bellassoued, Raymond Brummelhuis, Michel Cristofol, Éric Soccorsi. Stable reconstruction of the volatility in a regime-switching local-volatility model. Mathematical Control & Related Fields, 2020, 10 (1) : 189-215. doi: 10.3934/mcrf.2019036
[18]

Christoforidou Amalia, Christian-Oliver Ewald. A lattice method for option evaluation with regime-switching asset correlation structure. Journal of Industrial & Management Optimization, 2017, 13 (5) : 0-0. doi: 10.3934/jimo.2020042
[19]

Masao Fukushima. A class of gap functions for quasi-variational inequality problems. Journal of Industrial & Management Optimization, 2007, 3 (2) : 165-171. doi: 10.3934/jimo.2007.3.165
[20]

Nobuyuki Kenmochi. Parabolic quasi-variational diffusion problems with gradient constraints. Discrete & Continuous Dynamical Systems - S, 2013, 6 (2) : 423-438. doi: 10.3934/dcdss.2013.6.423

2018 Impact Factor: 1.292

Tools

Metrics

  • PDF downloads (72)
  • HTML views (352)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences