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

## Pairs trading with illiquidity and position limits

 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.

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
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##### References:
Optimal investment boundaries for the illiquid asset without position limits. Parameter values: $\beta_1 = 0.1$, $\beta_2 = 0.15$, $\sigma_1 = 0.2$, $\sigma_2 = 0.25$, $\delta_1 = 1$, $\delta_2 = 0.4$, $\lambda = 1$, $\rho = 0.8$, $\alpha = 0.01$, $\theta = 0.01$, $\gamma = 0.5$, $r = 0.01$, $\nu = 0.02$
Optimal investment boundaries for the illiquid asset with position limits. Parameter values: $\beta_1 = 0.1$, $\beta_2 = 0.15$, $\sigma_1 = 0.2$, $\sigma_2 = 0.25$, $\delta_1 = 1$, $\delta_2 = 0.4$, $\lambda = 1$, $\rho = 0.8$, $\alpha = 0.01$, $\theta = 0.01$, $\gamma = 0.5$, $r = 0.01$, $\nu = 0.02$, $\underset{\bar{}}{l} = -0.5$, $\bar{l} = 0.7$
Optimal investment boundaries for the illiquid asset at $x = 15$. Parameter values: $\beta_1 = 0.1$, $\beta_2 = 0.15$, $\sigma_1 = 0.2$, $\sigma_2 = 0.25$, $\delta_1 = 1$, $\delta_2 = 0.4$, $\lambda = 1$, $\rho = 0.8$, $\alpha = 0.01$, $\theta = 0.01$, $\gamma = 0.5$, $r = 0.01$, $\nu = 0.02$, $\underset{\bar{}}{l} = -0.5$, $\bar{l} = 0.7$
Optimal investment boundaries for the illiquid asset at $x = 15$ with various transaction cost rates $\alpha$. Parameter values: $\beta_1 = 0.1$, $\beta_2 = 0.15$, $\sigma_1 = 0.2$, $\sigma_2 = 0.25$, $\delta_1 = 1$, $\delta_2 = 0.4$, $\lambda = 1$, $\rho = 0.8$, $\gamma = 0.5$, $r = 0.01$, $\nu = 0.02$
Optimal investment boundaries for the illiquid asset at $x = 15$ with various correlation coefficients $\rho$. Parameter values: $\beta_1 = 0.1$, $\beta_2 = 0.15$, $\sigma_1 = 0.2$, $\sigma_2 = 0.25$, $\delta_1 = 1$, $\delta_2 = 0.4$, $\lambda = 1$, $\alpha = 0.01$, $\theta = 0.01$, $\gamma = 0.5$, $r = 0.01$, $\nu = 0.02$
Optimal investment boundaries for creating positions with the illiquid asset at $x = 15$ with various lower bound $\underset{\bar{}}{l}$. Parameter values: $\beta_1 = 0.1$, $\beta_2 = 0.15$, $\sigma_1 = 0.2$, $\sigma_2 = 0.25$, $\delta_1 = 1$, $\delta_2 = 0.4$, $\lambda = 1$, $\alpha = 0.01$, $\theta = 0.01$, $\gamma = 0.5$, $r = 0.01$, $\nu = 0.02$
Optimal investment boundaries for the illiquid asset at $x = 15$ with various $\kappa$. Parameter values: $\beta_1 = 0.1$, $\beta_2 = 0.15$, $\sigma_1 = 0.2$, $\sigma_2 = 0.25$, $\delta_1 = 1$, $\delta_2 = 0.4$, $\lambda = 1$, $\alpha = 0.01$, $\theta = 0.01$, $\gamma = 0.5$, $r = 0.01$, $\nu = 0.02$, $\underset{\bar{}}{l} = -0.5$, $\bar{l} = 0.7$
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
 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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