# American Institute of Mathematical Sciences

doi: 10.3934/naco.2022014
Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

## Optimal pairs trading of mean-reverting processes over multiple assets

 1 School of Mathematical Sciences, Nankai University, Tianjin, 300071, China 2 School of Finance, Nankai University, Tianjin, 300071, China

*Corresponding author: Lihua Bai

Received  November 2021 Revised  April 2022 Early access May 2022

This paper studies multi-asset pairs trading strategies of maximizing the expected exponential utility of terminal wealth. We model the log-relationship between each pair of stock prices as an Ornstein-Uhlenbeck (O-U) process, and formulate a portfolio optimization problem. Using the classical stochastic control approach based on the Hamilton-Jacobi-Bellman (HJB) equation, we characterize the optimal strategies and provide a verification result for the value function. Finally, we give some numerical results to show the characteristics of pairs trading.

Citation: Pengxu Xie, Lihua Bai, Huayue Zhang. Optimal pairs trading of mean-reverting processes over multiple assets. Numerical Algebra, Control and Optimization, doi: 10.3934/naco.2022014
##### References:
 [1] B. Angoshtari and T. Leung, Optimal dynamic basis trading, Annals of Finance, 15 (2019), 307-335.  doi: 10.1007/s10436-019-00348-x. [2] F. E. Benth and K. H. Karlsen, A note on Merton's portfolio selection problem for the Schwartz mean-reversion model, Stochastic Analysis and Applications, 23 (2005), 687-704.  doi: 10.1081/SAP-200064457. [3] B. Do, R. Faff and K. Hamza, A new approach to modeling and estimation for pairs trading, In Proceedings of 2006 Financial Man agement Association European Conference, Stockholm, June 2006. [4] R. J. Elliott, J. van der Hoek and W. P. Malcolm, Pairs trading, Quantitative Finance, 5 (2005), 271-276.  doi: 10.1080/14697680500149370. [5] E. Gatev, W. N. Goetzmann and K. G. Rouwenhorst, Pairs trading: performance of a relative value arbitrage rule, Social Science Electronic Publishing, 19 (2006), 797-827. [6] J. W. Jurek and H. Yang, Dynamic Portfolio Selection in Arbitrage, SSRN Electronic Journal, 2007. [7] S. Mudchanatongsuk, J. A. Primbs and W. Wong, Optimal pairs trading: A stochastic control approach, American Control Conference IEEE, 2008. [8] A. Tourin and R. Yan, Dynamic pairs trading using the stochastic control approach, Journal of Economic Dynamics and Control, 37 (2013), 1972-1981.  doi: 10.1016/j.jedc.2013.05.010. [9] G. Vidyamurthy, Pairs Trading: Quantitative Methods and Analysis, John Wiley Sons, 2004.

show all references

##### References:
 [1] B. Angoshtari and T. Leung, Optimal dynamic basis trading, Annals of Finance, 15 (2019), 307-335.  doi: 10.1007/s10436-019-00348-x. [2] F. E. Benth and K. H. Karlsen, A note on Merton's portfolio selection problem for the Schwartz mean-reversion model, Stochastic Analysis and Applications, 23 (2005), 687-704.  doi: 10.1081/SAP-200064457. [3] B. Do, R. Faff and K. Hamza, A new approach to modeling and estimation for pairs trading, In Proceedings of 2006 Financial Man agement Association European Conference, Stockholm, June 2006. [4] R. J. Elliott, J. van der Hoek and W. P. Malcolm, Pairs trading, Quantitative Finance, 5 (2005), 271-276.  doi: 10.1080/14697680500149370. [5] E. Gatev, W. N. Goetzmann and K. G. Rouwenhorst, Pairs trading: performance of a relative value arbitrage rule, Social Science Electronic Publishing, 19 (2006), 797-827. [6] J. W. Jurek and H. Yang, Dynamic Portfolio Selection in Arbitrage, SSRN Electronic Journal, 2007. [7] S. Mudchanatongsuk, J. A. Primbs and W. Wong, Optimal pairs trading: A stochastic control approach, American Control Conference IEEE, 2008. [8] A. Tourin and R. Yan, Dynamic pairs trading using the stochastic control approach, Journal of Economic Dynamics and Control, 37 (2013), 1972-1981.  doi: 10.1016/j.jedc.2013.05.010. [9] G. Vidyamurthy, Pairs Trading: Quantitative Methods and Analysis, John Wiley Sons, 2004.
The stock price for $A^1$, $B^1$, $A^2$ and $B^2$
The optimal wealth process $X^*$
 [1] Jean-Claude Zambrini. On the geometry of the Hamilton-Jacobi-Bellman equation. Journal of Geometric Mechanics, 2009, 1 (3) : 369-387. doi: 10.3934/jgm.2009.1.369 [2] Bian-Xia Yang, Shanshan Gu, Guowei Dai. Existence and multiplicity for Hamilton-Jacobi-Bellman equation. Communications on Pure and Applied Analysis, 2021, 20 (11) : 3767-3793. doi: 10.3934/cpaa.2021130 [3] Daniele Castorina, Annalisa Cesaroni, Luca Rossi. On a parabolic Hamilton-Jacobi-Bellman equation degenerating at the boundary. Communications on Pure and Applied Analysis, 2016, 15 (4) : 1251-1263. doi: 10.3934/cpaa.2016.15.1251 [4] Thi Tuyen Nguyen. Large time behavior of solutions of local and nonlocal nondegenerate Hamilton-Jacobi equations with Ornstein-Uhlenbeck operator. Communications on Pure and Applied Analysis, 2019, 18 (3) : 999-1021. doi: 10.3934/cpaa.2019049 [5] Steven Richardson, Song Wang. The viscosity approximation to the Hamilton-Jacobi-Bellman equation in optimal feedback control: Upper bounds for extended domains. Journal of Industrial and Management Optimization, 2010, 6 (1) : 161-175. doi: 10.3934/jimo.2010.6.161 [6] Xuhui Wang, Lei Hu. A new method to solve the Hamilton-Jacobi-Bellman equation for a stochastic portfolio optimization model with boundary memory. Journal of Industrial and Management Optimization, 2021  doi: 10.3934/jimo.2021137 [7] Zhen-Zhen Tao, Bing Sun. A feedback design for numerical solution to optimal control problems based on Hamilton-Jacobi-Bellman equation. Electronic Research Archive, 2021, 29 (5) : 3429-3447. doi: 10.3934/era.2021046 [8] Pengxu Xie, Lihua Bai, Huayue Zhang. Optimal proportional reinsurance and pairs trading under exponential utility criterion for the insurer. Journal of Industrial and Management Optimization, 2022  doi: 10.3934/jimo.2022020 [9] Filomena Feo, Pablo Raúl Stinga, Bruno Volzone. The fractional nonlocal Ornstein-Uhlenbeck equation, Gaussian symmetrization and regularity. Discrete and Continuous Dynamical Systems, 2018, 38 (7) : 3269-3298. doi: 10.3934/dcds.2018142 [10] Tomasz Komorowski, Lenya Ryzhik. Fluctuations of solutions to Wigner equation with an Ornstein-Uhlenbeck potential. Discrete and Continuous Dynamical Systems - B, 2012, 17 (3) : 871-914. doi: 10.3934/dcdsb.2012.17.871 [11] Virginia Giorno, Serena Spina. On the return process with refractoriness for a non-homogeneous Ornstein-Uhlenbeck neuronal model. Mathematical Biosciences & Engineering, 2014, 11 (2) : 285-302. doi: 10.3934/mbe.2014.11.285 [12] Yin Li, Xuerong Mao, Yazhi Song, Jian Tao. Optimal investment and proportional reinsurance strategy under the mean-reverting Ornstein-Uhlenbeck process and net profit condition. Journal of Industrial and Management Optimization, 2022, 18 (1) : 75-93. doi: 10.3934/jimo.2020143 [13] Mohamed Assellaou, Olivier Bokanowski, Hasnaa Zidani. Error estimates for second order Hamilton-Jacobi-Bellman equations. Approximation of probabilistic reachable sets. Discrete and Continuous Dynamical Systems, 2015, 35 (9) : 3933-3964. doi: 10.3934/dcds.2015.35.3933 [14] Annalisa Cesaroni, Matteo Novaga, Enrico Valdinoci. A symmetry result for the Ornstein-Uhlenbeck operator. Discrete and Continuous Dynamical Systems, 2014, 34 (6) : 2451-2467. doi: 10.3934/dcds.2014.34.2451 [15] Samuel Herrmann, Nicolas Massin. Exit problem for Ornstein-Uhlenbeck processes: A random walk approach. Discrete and Continuous Dynamical Systems - B, 2020, 25 (8) : 3199-3215. doi: 10.3934/dcdsb.2020058 [16] Mondher Damak, Brice Franke, Nejib Yaakoubi. Accelerating planar Ornstein-Uhlenbeck diffusion with suitable drift. Discrete and Continuous Dynamical Systems, 2020, 40 (7) : 4093-4112. doi: 10.3934/dcds.2020173 [17] Kai Liu. Quadratic control problem of neutral Ornstein-Uhlenbeck processes with control delays. Discrete and Continuous Dynamical Systems - B, 2013, 18 (6) : 1651-1661. doi: 10.3934/dcdsb.2013.18.1651 [18] Tomasz Komorowski, Łukasz Stȩpień. Kinetic limit for a harmonic chain with a conservative Ornstein-Uhlenbeck stochastic perturbation. Kinetic and Related Models, 2018, 11 (2) : 239-278. doi: 10.3934/krm.2018013 [19] Antonio Avantaggiati, Paola Loreti. Hypercontractivity, Hopf-Lax type formulas, Ornstein-Uhlenbeck operators (II). Discrete and Continuous Dynamical Systems - S, 2009, 2 (3) : 525-545. doi: 10.3934/dcdss.2009.2.525 [20] Tiziana Durante, Abdelaziz Rhandi. On the essential self-adjointness of Ornstein-Uhlenbeck operators perturbed by inverse-square potentials. Discrete and Continuous Dynamical Systems - S, 2013, 6 (3) : 649-655. doi: 10.3934/dcdss.2013.6.649

Impact Factor: