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Pairs trading with illiquidity and position limits

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  • 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.

    Mathematics Subject Classification: 91G10, 91G80.

    Citation:

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  • 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 $

    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 $

    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 $

    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 $

    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 $

    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 $

    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 $

    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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