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November  2020, 16(6): 2991-3009. doi: 10.3934/jimo.2019090

Pairs trading with illiquidity and position limits

Menglu Feng 1, , Mei Choi Chiu 2, and  Hoi Ying Wong 3,,

1. 

Societe Generale Hong Kong, Three Pacific Place, 1 Queen's Road East, Hong Kong
2. 

Department of Mathematics and Information Technology, Education University of Hong Kong, Hong Kong SAR, China
3. 

Department of Statistics, The Chinese University of Hong Kong, Hong Kong SAR, China

*Corresponding author

Received  August 2018 Revised  February 2019 Published  July 2019

We investigate the optimal investment among the money market account, a liquid risky asset (e.g. stock index) and an illiquid risky asset (e.g. individual stock), where the two risky assets are cointegrated. The illiquid risky asset is subject to a proportional transaction cost and the portfolio of the three assets faces certain position limits. We develop the optimal investment strategy to maximize the gain function, which is realized through an expected sum of discounted utilities given transaction costs and position limits. The problem formulation uses a singular control framework with cointegration that determines optimal trading boundaries among holding, selling and no-trading regions. We conduct comprehensive numerical analysis on the optimal investment strategy and features of the optimal trading boundaries given various levels of position limits.

Keywords: Cointegration, liquidity, pairs trading, position limits, singular control problem.
Mathematics Subject Classification: 91G10, 91G80.
Citation: Menglu Feng, Mei Choi Chiu, Hoi Ying Wong. Pairs trading with illiquidity and position limits. Journal of Industrial & Management Optimization, 2020, 16 (6) : 2991-3009. doi: 10.3934/jimo.2019090
References:
[1]

A. Almazan, K. C. Brown, M. Carlson and D. Chapman, Why constrain your mutual fund manager?, J. of Financial Econ., 73 (2004), 289-321.   Google Scholar

[2]

M. Akian, J. L. Menaldi and and A. Sulem, On an investment-consumption model with transaction costs, SIAM J. on Control and Optimization, 34 (1996), 329-364.  doi: 10.1137/S0363012993247159.  Google Scholar

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K. Chen, M. C. Chiu and H. Y. Wong, Time-consistent mean-variance pairs-trading under regime-switching cointegration, SIAM J. on Finan. Math., 10 (2019), 632-665.  doi: 10.2139/ssrn.3250340.  Google Scholar

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K. Chen and H. Y. Wong, Time-consistent mean-variance hedging of an illiquid asset with a cointegrated liquid asset, Finan. Research Letters, 29 (2019), 184-192.  doi: 10.1016/j.frl.2018.07.004.  Google Scholar

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M. C. Chiu and H. Y. Wong, Mean-variance portfolio selection of cointegrated assets, J. of Econ. Dynamics and Control, 35 (2011), 1369-1385.  doi: 10.1016/j.jedc.2011.04.003.  Google Scholar

[9]

M. C. Chiu and H. Y. Wong, Mean-variance asset-liability management: Cointegrated assets and insurance liabilities, European J. of Oper. Research, 223 (2012), 785-793.  doi: 10.1016/j.ejor.2012.07.009.  Google Scholar

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M. C. Chiu and H. Y. Wong, Optimal investment for an insurer with cointegrated assets: CRRA utility, Insurance: Math. and Econ., 52 (2013), 52-64.  doi: 10.1016/j.insmatheco.2012.11.004.  Google Scholar

[11]

M. C. Chiu and H. Y. Wong, Dynamic cointegrated pairs trading: Mean-variance time-consistent strategies, J. of Computational and Applied Math., 290 (2015), 516-534.  doi: 10.1016/j.cam.2015.06.004.  Google Scholar

[12]

M. Dai, H. Jin and H. Liu, Illiquidity, position limits, and optimal investment for mutual funds, J. of Econ. Theory, 146 (2011), 1598-1630.  doi: 10.1016/j.jet.2011.03.014.  Google Scholar

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J. Duan and S. R. Pliska, Option valuation with co-integrated asset prices, J. of Econ. Dynamics and Control, 28 (2004), 727-754.  doi: 10.1016/S0165-1889(03)00042-3.  Google Scholar

[15]

R. Engle and C. Granger, Co-integration and error correction: Representation, estimation, and testing, Econometrica, 55 (1987), 251-276.  doi: 10.2307/1913236.  Google Scholar

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P. A. Forsyth and K. R. Vetzal, Quadratic convergence for valuing American options using a penalty method, SIAM J. on Scientific Computing, 23 (2002), 2095-2122.  doi: 10.1137/S1064827500382324.  Google Scholar

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E. Gatev, W. N. Goetzmann and K. G. Rouwenhorst, Pairs trading: Performance of a relative-value arbitrage rule, The Review of Finan. Studies, 19 (2006), 797-827.  doi: 10.1093/rfs/hhj020.  Google Scholar

[18]

J. Karceski, M. Livingston and E. S. O'Neal, Portfolio transactions costs at US equity mutual funds, Working Paper, (2004). Google Scholar

[19]

Y. Lei and J. Xu, Costly arbitrage through pairs trading, J. of Econ. Dynamics and Control, 56 (2015), 1-19.  doi: 10.1016/j.jedc.2015.04.006.  Google Scholar

[20]

J. Liu and A. Timmermann, Optimal convergence trade strategies, The Review of Finan. Studies, 26 (2013), 1048-1086.  doi: 10.1093/rfs/hhs130.  Google Scholar

[21]

R. Merton, Optimum consumption and portfolio rules in a continuous-time model, Journal of Econ. Theory, 3 (1971), 373-413.  doi: 10.1016/0022-0531(71)90038-X.  Google Scholar

[22]

A. Tourin and R. Yan, Dynamic pairs trading using the stochastic control approach, J. of Econ. Dynamics and Control, 37 (2013), 1972-1981.  doi: 10.1016/j.jedc.2013.05.010.  Google Scholar

[23]

N. Touzi, Optimal Stochastic Control, Stochastic Target Problems, and Backward SDE, Fields Institute Monographs, 29, Springer, New York, 2013. doi: 10.1007/978-1-4614-4286-8.  Google Scholar

[24]

R. Wermers, Mutual fund performance: An empirical decomposition into stock-picking talent, style, transactions costs, and expenses, The J. of Finance, 55 (2000), 1655-1695.  doi: 10.1111/0022-1082.00263.  Google Scholar

show all references

References:
[1]

A. Almazan, K. C. Brown, M. Carlson and D. Chapman, Why constrain your mutual fund manager?, J. of Financial Econ., 73 (2004), 289-321.   Google Scholar

[2]

M. Akian, J. L. Menaldi and and A. Sulem, On an investment-consumption model with transaction costs, SIAM J. on Control and Optimization, 34 (1996), 329-364.  doi: 10.1137/S0363012993247159.  Google Scholar

[3]

R. Baillie and T. Bollerslev, Common stochastic trends in a system of exchange rates, The J. of Finance, 44 (1989), 167-181.  doi: 10.1111/j.1540-6261.1989.tb02410.x.  Google Scholar

[4]

S. Basak, A. Pavlova and A. Shapiro, Optimal asset allocation and risk shifting in money management, The Review of Financial Studies, 20 (2007), 1583-1621.   Google Scholar

[5]

M. Cerchi and A. Havenner, Cointegration and stock prices: The random walk on Wall Street revisited, J. of Econ. Dynamics and Control, 12 (1988), 333-346.  doi: 10.1016/0165-1889(88)90044-9.  Google Scholar

[6]

K. Chen, M. C. Chiu and H. Y. Wong, Time-consistent mean-variance pairs-trading under regime-switching cointegration, SIAM J. on Finan. Math., 10 (2019), 632-665.  doi: 10.2139/ssrn.3250340.  Google Scholar

[7]

K. Chen and H. Y. Wong, Time-consistent mean-variance hedging of an illiquid asset with a cointegrated liquid asset, Finan. Research Letters, 29 (2019), 184-192.  doi: 10.1016/j.frl.2018.07.004.  Google Scholar

[8]

M. C. Chiu and H. Y. Wong, Mean-variance portfolio selection of cointegrated assets, J. of Econ. Dynamics and Control, 35 (2011), 1369-1385.  doi: 10.1016/j.jedc.2011.04.003.  Google Scholar

[9]

M. C. Chiu and H. Y. Wong, Mean-variance asset-liability management: Cointegrated assets and insurance liabilities, European J. of Oper. Research, 223 (2012), 785-793.  doi: 10.1016/j.ejor.2012.07.009.  Google Scholar

[10]

M. C. Chiu and H. Y. Wong, Optimal investment for an insurer with cointegrated assets: CRRA utility, Insurance: Math. and Econ., 52 (2013), 52-64.  doi: 10.1016/j.insmatheco.2012.11.004.  Google Scholar

[11]

M. C. Chiu and H. Y. Wong, Dynamic cointegrated pairs trading: Mean-variance time-consistent strategies, J. of Computational and Applied Math., 290 (2015), 516-534.  doi: 10.1016/j.cam.2015.06.004.  Google Scholar

[12]

M. Dai, H. Jin and H. Liu, Illiquidity, position limits, and optimal investment for mutual funds, J. of Econ. Theory, 146 (2011), 1598-1630.  doi: 10.1016/j.jet.2011.03.014.  Google Scholar

[13]

M. Davis and A. Norman, Portfolio selection with transaction costs, Math. of Ops. Research, 15 (1990), 676-713.  doi: 10.1287/moor.15.4.676.  Google Scholar

[14]

J. Duan and S. R. Pliska, Option valuation with co-integrated asset prices, J. of Econ. Dynamics and Control, 28 (2004), 727-754.  doi: 10.1016/S0165-1889(03)00042-3.  Google Scholar

[15]

R. Engle and C. Granger, Co-integration and error correction: Representation, estimation, and testing, Econometrica, 55 (1987), 251-276.  doi: 10.2307/1913236.  Google Scholar

[16]

P. A. Forsyth and K. R. Vetzal, Quadratic convergence for valuing American options using a penalty method, SIAM J. on Scientific Computing, 23 (2002), 2095-2122.  doi: 10.1137/S1064827500382324.  Google Scholar

[17]

E. Gatev, W. N. Goetzmann and K. G. Rouwenhorst, Pairs trading: Performance of a relative-value arbitrage rule, The Review of Finan. Studies, 19 (2006), 797-827.  doi: 10.1093/rfs/hhj020.  Google Scholar

[18]

J. Karceski, M. Livingston and E. S. O'Neal, Portfolio transactions costs at US equity mutual funds, Working Paper, (2004). Google Scholar

[19]

Y. Lei and J. Xu, Costly arbitrage through pairs trading, J. of Econ. Dynamics and Control, 56 (2015), 1-19.  doi: 10.1016/j.jedc.2015.04.006.  Google Scholar

[20]

J. Liu and A. Timmermann, Optimal convergence trade strategies, The Review of Finan. Studies, 26 (2013), 1048-1086.  doi: 10.1093/rfs/hhs130.  Google Scholar

[21]

R. Merton, Optimum consumption and portfolio rules in a continuous-time model, Journal of Econ. Theory, 3 (1971), 373-413.  doi: 10.1016/0022-0531(71)90038-X.  Google Scholar

[22]

A. Tourin and R. Yan, Dynamic pairs trading using the stochastic control approach, J. of Econ. Dynamics and Control, 37 (2013), 1972-1981.  doi: 10.1016/j.jedc.2013.05.010.  Google Scholar

[23]

N. Touzi, Optimal Stochastic Control, Stochastic Target Problems, and Backward SDE, Fields Institute Monographs, 29, Springer, New York, 2013. doi: 10.1007/978-1-4614-4286-8.  Google Scholar

[24]

R. Wermers, Mutual fund performance: An empirical decomposition into stock-picking talent, style, transactions costs, and expenses, The J. of Finance, 55 (2000), 1655-1695.  doi: 10.1111/0022-1082.00263.  Google Scholar

Figure 1.  Optimal investment boundaries for the illiquid asset without position limits. Parameter values: $ \beta_1 = 0.1 $, $ \beta_2 = 0.15 $, $ \sigma_1 = 0.2 $, $ \sigma_2 = 0.25 $, $ \delta_1 = 1 $, $ \delta_2 = 0.4 $, $ \lambda = 1 $, $ \rho = 0.8 $, $ \alpha = 0.01 $, $ \theta = 0.01 $, $ \gamma = 0.5 $, $ r = 0.01 $, $ \nu = 0.02 $
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Figure 2.  Optimal investment boundaries for the illiquid asset with position limits. Parameter values: $ \beta_1 = 0.1 $, $ \beta_2 = 0.15 $, $ \sigma_1 = 0.2 $, $ \sigma_2 = 0.25 $, $ \delta_1 = 1 $, $ \delta_2 = 0.4 $, $ \lambda = 1 $, $ \rho = 0.8 $, $ \alpha = 0.01 $, $ \theta = 0.01 $, $ \gamma = 0.5 $, $ r = 0.01 $, $ \nu = 0.02 $, $ \underset{\bar{}}{l} = -0.5 $, $ \bar{l} = 0.7 $
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Figure 3.  Optimal investment boundaries for the illiquid asset at $ x = 15 $. Parameter values: $ \beta_1 = 0.1 $, $ \beta_2 = 0.15 $, $ \sigma_1 = 0.2 $, $ \sigma_2 = 0.25 $, $ \delta_1 = 1 $, $ \delta_2 = 0.4 $, $ \lambda = 1 $, $ \rho = 0.8 $, $ \alpha = 0.01 $, $ \theta = 0.01 $, $ \gamma = 0.5 $, $ r = 0.01 $, $ \nu = 0.02 $, $ \underset{\bar{}}{l} = -0.5 $, $ \bar{l} = 0.7 $
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Figure 4.  Optimal investment boundaries for the illiquid asset at $ x = 15 $ with various transaction cost rates $ \alpha $. Parameter values: $ \beta_1 = 0.1 $, $ \beta_2 = 0.15 $, $ \sigma_1 = 0.2 $, $ \sigma_2 = 0.25 $, $ \delta_1 = 1 $, $ \delta_2 = 0.4 $, $ \lambda = 1 $, $ \rho = 0.8 $, $ \gamma = 0.5 $, $ r = 0.01 $, $ \nu = 0.02 $
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Figure 5.  Optimal investment boundaries for the illiquid asset at $ x = 15 $ with various correlation coefficients $ \rho $. Parameter values: $ \beta_1 = 0.1 $, $ \beta_2 = 0.15 $, $ \sigma_1 = 0.2 $, $ \sigma_2 = 0.25 $, $ \delta_1 = 1 $, $ \delta_2 = 0.4 $, $ \lambda = 1 $, $ \alpha = 0.01 $, $ \theta = 0.01 $, $ \gamma = 0.5 $, $ r = 0.01 $, $ \nu = 0.02 $
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Figure 6.  Optimal investment boundaries for creating positions with the illiquid asset at $ x = 15 $ with various lower bound $ \underset{\bar{}}{l} $. Parameter values: $ \beta_1 = 0.1 $, $ \beta_2 = 0.15 $, $ \sigma_1 = 0.2 $, $ \sigma_2 = 0.25 $, $ \delta_1 = 1 $, $ \delta_2 = 0.4 $, $ \lambda = 1 $, $ \alpha = 0.01 $, $ \theta = 0.01 $, $ \gamma = 0.5 $, $ r = 0.01 $, $ \nu = 0.02 $
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Figure 7.  Optimal investment boundaries for the illiquid asset at $ x = 15 $ with various $ \kappa $. Parameter values: $ \beta_1 = 0.1 $, $ \beta_2 = 0.15 $, $ \sigma_1 = 0.2 $, $ \sigma_2 = 0.25 $, $ \delta_1 = 1 $, $ \delta_2 = 0.4 $, $ \lambda = 1 $, $ \alpha = 0.01 $, $ \theta = 0.01 $, $ \gamma = 0.5 $, $ r = 0.01 $, $ \nu = 0.02 $, $ \underset{\bar{}}{l} = -0.5 $, $ \bar{l} = 0.7 $
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Table 1.  Summary of Default Parameters for Numerical Analysis
Parameter Value Parameter Value
$ \beta_1\; \; $ 0.1 $ \rho\; \; $ 0.8
$ \beta_2\; \; $ 0.15 $ \alpha\; \; $ 0.01
$ \delta_1\; \; $ 1 $ \theta\; \; $ 0.01
$ \delta_2\; \; $ 0.4 $ \gamma\; \; $ 0.5
$ \sigma_1\; \; $ 0.2 $ r\; \; $ 0.01
$ \sigma_2\; \; $ 0.25 $ \nu\; \; $ 0.02
$ \lambda\; \; $ 1
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Parameter Value Parameter Value
$ \beta_1\; \; $ 0.1 $ \rho\; \; $ 0.8
$ \beta_2\; \; $ 0.15 $ \alpha\; \; $ 0.01
$ \delta_1\; \; $ 1 $ \theta\; \; $ 0.01
$ \delta_2\; \; $ 0.4 $ \gamma\; \; $ 0.5
$ \sigma_1\; \; $ 0.2 $ r\; \; $ 0.01
$ \sigma_2\; \; $ 0.25 $ \nu\; \; $ 0.02
$ \lambda\; \; $ 1
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