# American Institute of Mathematical Sciences

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

## Switching between a pair of stocks: An optimal trading rule

 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.

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:

show all references

##### References:
Switching Regions ($\alpha _t = 1$)
Value Functions
${\bf{S}}^1$ = QQQ, ${\bf{S}}^2$ = SPY: The threshold levels ${b_1},{b_2}$ and the corresponding equity curve
${\bf{S}}^1$ = QQQ, ${\bf{S}}^2$ = SPY: The threshold levels ${b_1},{b_2}$ and the corresponding equity curve
log(Daily closing prices) of KO and PEP
${\bf{S}}^1$ = KO, ${\bf{S}}^2$ = PEP: The threshold levels $b_1,b_2,s_1,s_2$ and the corresponding equity curve
$(b_1,b_2,s_1,s_2)$ 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 2.562 $b_1$ 1.8936 2.0566 2.2203 2.3849 2.5503 $s_2$ 1.6435 1.7812 1.9193 2.0576 2.1964
 $\mu_1$ 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 2.562 $b_1$ 1.8936 2.0566 2.2203 2.3849 2.5503 $s_2$ 1.6435 1.7812 1.9193 2.0576 2.1964
$(b_1,b_2,s_1,s_2)$ with varying $\mu_2$
 $\mu_2$ 0.0787 0.0887 0.0987 0.1087 0.1187 $s_1$ 3.493 2.857 2.4152 2.0905 1.842 $b_2$ 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
 $\mu_2$ 0.0787 0.0887 0.0987 0.1087 0.1187 $s_1$ 3.493 2.857 2.4152 2.0905 1.842 $b_2$ 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
$(b_1,b_2,s_1,s_2)$ 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 2.23 2.2309 2.232 2.2331 $b_1$ 2.2199 2.2201 2.2203 2.2206 2.221 $s_2$ 1.935 1.927 1.9193 1.9116 1.9042
 $\sigma_{11}$ 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 2.23 2.2309 2.232 2.2331 $b_1$ 2.2199 2.2201 2.2203 2.2206 2.221 $s_2$ 1.935 1.927 1.9193 1.9116 1.9042
$(b_1,b_2,s_1,s_2)$ with varying $\sigma_{22}$
 $\sigma_{22}$ 0.1071 0.1171 0.1271 0.1371 0.1471 $s_1$ 2.4132 2.414 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
 $\sigma_{22}$ 0.1071 0.1171 0.1271 0.1371 0.1471 $s_1$ 2.4132 2.414 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
$(b_1,b_2,s_1,s_2)$ 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
 $\sigma_{12}(=\sigma_{21})$ 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
$(b_1,b_2,s_1,s_2)$ 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
 $\rho$ 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
$(b_1,b_2,s_1,s_2)$ with varying $\lambda$
 $\lambda$ 0.8 0.9 1 1.1 1.2 $s_1$ 2.407 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 1.92 1.9193 1.9184 1.9183
 $\lambda$ 0.8 0.9 1 1.1 1.2 $s_1$ 2.407 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 1.92 1.9193 1.9184 1.9183
$(b_1,b_2,s_1,s_2)$ with varying $K$
 $K$ 0.0005 0.00075 0.001 0.005 0.01 $s_1$ 2.3315 2.3762 2.4152 2.8814 3.4914 $b_2$ 2.1908 2.2119 2.2309 2.4752 2.7979 $b_1$ 2.184 2.2031 2.2203 2.4459 2.7528 $s_2$ 1.9543 1.9347 1.9193 1.7878 1.6694
 $K$ 0.0005 0.00075 0.001 0.005 0.01 $s_1$ 2.3315 2.3762 2.4152 2.8814 3.4914 $b_2$ 2.1908 2.2119 2.2309 2.4752 2.7979 $b_1$ 2.184 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] 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 [9] 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 [10] 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 [11] 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 [12] 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 [13] 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 [14] Jiaqin Wei. Time-inconsistent optimal control problems with regime-switching. Mathematical Control & Related Fields, 2017, 7 (4) : 585-622. doi: 10.3934/mcrf.2017022 [15] 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 [16] Mourad Bellassoued, Raymond Brummelhuis, Michel Cristofol, Éric Soccorsi. Stable reconstruction of the volatility in a regime-switching local-volatility model. Mathematical Control & Related Fields, 2019, 0 (0) : 0-0. doi: 10.3934/mcrf.2019036 [17] 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 [18] 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 [19] Zhuo Jin, George Yin, Hailiang Yang. Numerical methods for dividend optimization using regime-switching jump-diffusion models. Mathematical Control & Related Fields, 2011, 1 (1) : 21-40. doi: 10.3934/mcrf.2011.1.21 [20] Ka Chun Cheung, Hailiang Yang. Optimal investment-consumption strategy in a discrete-time model with regime switching. Discrete & Continuous Dynamical Systems - B, 2007, 8 (2) : 315-332. doi: 10.3934/dcdsb.2007.8.315

2018 Impact Factor: 1.292