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

• 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.

Mathematics Subject Classification: Primary: 93E20, 91G80; Secondary: 49L20.

 Citation: • • Figure 1.  Switching Regions ($\alpha _t = 1$)

Figure 2.  Value Functions

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

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

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

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

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 2.562 $b_1$ 1.8936 2.0566 2.2203 2.3849 2.5503 $s_2$ 1.6435 1.7812 1.9193 2.0576 2.1964

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 $s_1$ 3.493 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

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 2.23 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

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 2.414 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

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 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 7.  $(b_1,b_2,s_1,s_2)$ 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

Table 8.  $(b_1,b_2,s_1,s_2)$ 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
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Tables(8)

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