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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  November 2020 Early access  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 and 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. 

[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.

[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.

[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. 

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[15]

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

[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.

[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.

[18]

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

[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.

[20]

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

[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.

[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.

[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.

[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.

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. 

[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.

[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.

[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. 

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[15]

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

[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.

[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.

[18]

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

[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.

[20]

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

[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.

[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.

[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.

[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.

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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