# 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
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##### 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
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