Switching between a pair of stocks: An optimal trading rule

Jingzhi Tie; Qing Zhang

doi:10.3934/mcrf.2018042

Article Contents

2018, Volume 8, Issue 3&4: 965-999. Doi: 10.3934/mcrf.2018042

This issue Previous Article Controllability under positivity constraints of semilinear heat equations Next Article Admissible controls and controllable sets for a linear time-varying ordinary differential equation

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

Abstract Full Text(HTML) Figure(6) / Table(8) Related Papers Cited by

Abstract

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:

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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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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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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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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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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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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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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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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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