2015, 5(3): 489-499. doi: 10.3934/mcrf.2015.5.489

Pairs trading: An optimal selling rule

1. 

Citi, 2859 Paces Ferry Rd., Ste. 900, Atlanta, GA 30339, United States

2. 

Department of Mathematics, University of Georgia, Athens, GA 30602, United States, United States, United States

3. 

Department of Mathematics, Massachusetts College of Liberal Arts, 375 Church Street, North Adams, MA 01247, United States

Received  March 2014 Revised  June 2014 Published  July 2015

Pairs trading involves two cointegrated securities. When divergence is underway, i.e., one stock moves up while the other moves down, a pairs trade is entered consisting of a short position in the outperforming stock and a long position in the underperforming one. Such a strategy bets the ``spread'' between the two would eventually converge. This paper is concerned with an optimal pairs-trade selling rule. In this paper, a difference of the pair is governed by a mean-reverting model. The trade will be closed whenever the difference reaches a target level or a cutloss limit. Given a fixed cutloss level, the objective is to determine the optimal target so as to maximize an overall return. This optimization problem is related to an optimal stopping problem as the cutloss level vanishes. Expected holding time and profit probability are also obtained. Numerical examples are reported to demonstrate the results.
Citation: Kevin Kuo, Phong Luu, Duy Nguyen, Eric Perkerson, Katherine Thompson, Qing Zhang. Pairs trading: An optimal selling rule. Mathematical Control & Related Fields, 2015, 5 (3) : 489-499. doi: 10.3934/mcrf.2015.5.489
References:
[1]

C. Blanco and D. Soronow, Mean reverting processes - Energy price processes used for derivatives pricing and risk management,, Commodities Now, (2001), 68.

[2]

A. Cowles and H. Jones, Some posteriori probabilities in stock market action,, Econometrica, 5 (1937), 280. doi: 10.2307/1905515.

[3]

E. Fama and K. R. French, Permanent and temporary components of stock prices,, Journal of Political Economy, 96 (1988), 246. doi: 10.1086/261535.

[4]

L. A. Gallagher and M. P. Taylor, Permanent and temporary components of stock prices: Evidence from assessing macroeconomic shocks,, Southern Economic Journal, 69 (2002), 345. doi: 10.2307/1061676.

[5]

E. Gatev, W. N. Goetzmann and K. G. Rouwenhorst, Pairs trading: Performance of a relative-value arbitrage rule,, Review of Financial Studies, 19 (2006), 797.

[6]

X. Guo and Q. Zhang, Optimal selling rules in a regime switching model,, IEEE Transactions on Automatic Control, 50 (2005), 1450. doi: 10.1109/TAC.2005.854657.

[7]

C. M. Hafner and H. Herwartz, Option pricing under linear autoregressive dynamics, heteroskedasticity, and conditional leptokurtosis,, Journal of Empirical Finance, 8 (2001), 1. doi: 10.1016/S0927-5398(00)00024-4.

[8]

J. Liu and A. Timmermann, Optimal convergence trade strategies,, Review of Financial Studies, 26 (2013), 1048. doi: 10.1093/rfs/hhs130.

[9]

G. Vidyamurthy, Pairs Trading: Quantitative Methods and Analysis,, Wiley, (2004).

[10]

Q. Zhang, Stock trading: An optimal selling rule,, SIAM J. Contr. Optim., 40 (2001), 64. doi: 10.1137/S0363012999356325.

[11]

H. Zhang and Q. Zhang, Trading a mean-reverting asset: Buy low and sell high,, Automatica, 44 (2008), 1511. doi: 10.1016/j.automatica.2007.11.003.

show all references

References:
[1]

C. Blanco and D. Soronow, Mean reverting processes - Energy price processes used for derivatives pricing and risk management,, Commodities Now, (2001), 68.

[2]

A. Cowles and H. Jones, Some posteriori probabilities in stock market action,, Econometrica, 5 (1937), 280. doi: 10.2307/1905515.

[3]

E. Fama and K. R. French, Permanent and temporary components of stock prices,, Journal of Political Economy, 96 (1988), 246. doi: 10.1086/261535.

[4]

L. A. Gallagher and M. P. Taylor, Permanent and temporary components of stock prices: Evidence from assessing macroeconomic shocks,, Southern Economic Journal, 69 (2002), 345. doi: 10.2307/1061676.

[5]

E. Gatev, W. N. Goetzmann and K. G. Rouwenhorst, Pairs trading: Performance of a relative-value arbitrage rule,, Review of Financial Studies, 19 (2006), 797.

[6]

X. Guo and Q. Zhang, Optimal selling rules in a regime switching model,, IEEE Transactions on Automatic Control, 50 (2005), 1450. doi: 10.1109/TAC.2005.854657.

[7]

C. M. Hafner and H. Herwartz, Option pricing under linear autoregressive dynamics, heteroskedasticity, and conditional leptokurtosis,, Journal of Empirical Finance, 8 (2001), 1. doi: 10.1016/S0927-5398(00)00024-4.

[8]

J. Liu and A. Timmermann, Optimal convergence trade strategies,, Review of Financial Studies, 26 (2013), 1048. doi: 10.1093/rfs/hhs130.

[9]

G. Vidyamurthy, Pairs Trading: Quantitative Methods and Analysis,, Wiley, (2004).

[10]

Q. Zhang, Stock trading: An optimal selling rule,, SIAM J. Contr. Optim., 40 (2001), 64. doi: 10.1137/S0363012999356325.

[11]

H. Zhang and Q. Zhang, Trading a mean-reverting asset: Buy low and sell high,, Automatica, 44 (2008), 1511. doi: 10.1016/j.automatica.2007.11.003.

[1]

Hoi Tin Kong, Qing Zhang. An optimal trading rule of a mean-reverting asset. Discrete & Continuous Dynamical Systems - B, 2010, 14 (4) : 1403-1417. doi: 10.3934/dcdsb.2010.14.1403

[2]

Edward Allen. Environmental variability and mean-reverting processes. Discrete & Continuous Dynamical Systems - B, 2016, 21 (7) : 2073-2089. doi: 10.3934/dcdsb.2016037

[3]

Jingzhi Tie, Qing Zhang. An optimal mean-reversion trading rule under a Markov chain model. Mathematical Control & Related Fields, 2016, 6 (3) : 467-488. doi: 10.3934/mcrf.2016012

[4]

Jingzhen Liu, Ka Fai Cedric Yiu, Alain Bensoussan. Ergodic control for a mean reverting inventory model. Journal of Industrial & Management Optimization, 2017, 13 (4) : 1-20. doi: 10.3934/jimo.2017079

[5]

Huaying Guo, Jin Liang. An optimal control model of carbon reduction and trading. Mathematical Control & Related Fields, 2016, 6 (4) : 535-550. doi: 10.3934/mcrf.2016015

[6]

Shuren Liu, Qiying Hu, Yifan Xu. Optimal inventory control with fixed ordering cost for selling by internet auctions. Journal of Industrial & Management Optimization, 2012, 8 (1) : 19-40. doi: 10.3934/jimo.2012.8.19

[7]

Volker Rehbock, Iztok Livk. Optimal control of a batch crystallization process. Journal of Industrial & Management Optimization, 2007, 3 (3) : 585-596. doi: 10.3934/jimo.2007.3.585

[8]

Jingzhen Liu, Ka Fai Cedric Yiu, Alain Bensoussan. The optimal mean variance problem with inflation. Discrete & Continuous Dynamical Systems - B, 2016, 21 (1) : 185-203. doi: 10.3934/dcdsb.2016.21.185

[9]

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

[10]

Yongxia Zhao, Rongming Wang, Chuancun Yin. Optimal dividends and capital injections for a spectrally positive Lévy process. Journal of Industrial & Management Optimization, 2017, 13 (1) : 1-21. doi: 10.3934/jimo.2016001

[11]

Tan H. Cao, Boris S. Mordukhovich. Optimal control of a perturbed sweeping process via discrete approximations. Discrete & Continuous Dynamical Systems - B, 2016, 21 (10) : 3331-3358. doi: 10.3934/dcdsb.2016100

[12]

Jacques Henry. For which objective is birth process an optimal feedback in age structured population dynamics?. Discrete & Continuous Dynamical Systems - B, 2007, 8 (1) : 107-114. doi: 10.3934/dcdsb.2007.8.107

[13]

Lukáš Adam, Jiří Outrata. On optimal control of a sweeping process coupled with an ordinary differential equation. Discrete & Continuous Dynamical Systems - B, 2014, 19 (9) : 2709-2738. doi: 10.3934/dcdsb.2014.19.2709

[14]

Yvette Kosmann-Schwarzbach. Dirac pairs. Journal of Geometric Mechanics, 2012, 4 (2) : 165-180. doi: 10.3934/jgm.2012.4.165

[15]

Ying Li, Miyuan Shan, Michael Z.F. Li. Advance selling decisions with overconfident consumers. Journal of Industrial & Management Optimization, 2016, 12 (3) : 891-905. doi: 10.3934/jimo.2016.12.891

[16]

Yan Zeng, Zhongfei Li, Jingjun Liu. Optimal strategies of benchmark and mean-variance portfolio selection problems for insurers. Journal of Industrial & Management Optimization, 2010, 6 (3) : 483-496. doi: 10.3934/jimo.2010.6.483

[17]

Yuefen Chen, Yuanguo Zhu. Indefinite LQ optimal control with process state inequality constraints for discrete-time uncertain systems. Journal of Industrial & Management Optimization, 2017, 13 (4) : 1-18. doi: 10.3934/jimo.2017082

[18]

Darryl D. Holm, Cornelia Vizman. Dual pairs in resonances. Journal of Geometric Mechanics, 2012, 4 (3) : 297-311. doi: 10.3934/jgm.2012.4.297

[19]

V. Balaji, P. Barik, D. S. Nagaraj. On degenerations of moduli of Hitchin pairs. Electronic Research Announcements, 2013, 20: 103-108. doi: 10.3934/era.2013.20.105

[20]

Hongyong Deng, Wei Wei. Existence and stability analysis for nonlinear optimal control problems with $1$-mean equicontinuous controls. Journal of Industrial & Management Optimization, 2015, 11 (4) : 1409-1422. doi: 10.3934/jimo.2015.11.1409

2017 Impact Factor: 0.542

Metrics

  • PDF downloads (2)
  • HTML views (0)
  • Cited by (0)

[Back to Top]