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The application of model predictive control on stock portfolio optimization with prediction based on Geometric Brownian Motion-Kalman Filter

  • * Corresponding author: Wawan Hafid Syaifudin

    * Corresponding author: Wawan Hafid Syaifudin 

The first author is supported by ITS grant 1191/PKS/ITS/2019

Abstract Full Text(HTML) Figure(7) / Table(3) Related Papers Cited by
  • A stock portfolio is a collection of assets owned by investors, such as companies or individuals. The determination of the optimal stock portfolio is an important issue for the investors. Management of investors' capital in a portfolio can be regarded as a dynamic optimal control problem. At the same time, the investors should also consider about the prediction of stock prices in the future time. Therefore, in this research, we propose Geometric Brownian Motion-Kalman Filter (GBM-KF) method to predict the future stock prices. Subsequently, the stock returns will be calculated based on the forecasting results of stock prices. Furthermore, Model Predictive Control (MPC) will be used to solve the portfolio optimization problem. It is noticeable that the management strategy of stock portfolio in this research considers the constraints on assets in the portfolio and the cost of transactions. Finally, a practical application of the solution is implemented on 3 company's stocks. The simulation results show that the performance of the proposed controller satisfies the state's and the control's constraints. In addition, the amount of capital owned by the investor as the output of system shows a significant increase.

    Mathematics Subject Classification: Primary: 93C95; Secondary: 91G10.

    Citation:

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  • Figure 1.  Daily Stock Price of Each Company

    Figure 2.  Daily Stock Return of Each Company

    Figure 3.  Stock's Forecasting Results

    Figure 4.  Control Variables in Portfolio Optimization

    Figure 5.  The Dynamic of Total Invested Capital in Each Stock

    Figure 6.  The Dynamic of Total Invested Capital in Risk Free Asset

    Figure 7.  The Dynamic of Total Invested Capital in the Portfolio

    Table 1.  Kalman Filter Algorithm

    System Model and System model : $ x_{k+1}=f(x_k,u_k,k)+Gw_k $
    Measurement Model Measurement model : $ z_k=h(x_k,k)+v_k $
    Assumption : $ x(0)\sim X({\tilde x}_0,P_0);\; \; w(k)\sim N(0,Q_k); $
    Assumption : $ v_k\sim N(0,R) $
    Initialization $ {\tilde x}(0)={\tilde x}_0;\; \; P(0)=P_0 $
    Time Predict Estimation : $ \hat{x}_{k+1}^-=f(\hat{x}_{k}^-,u_k) $
    Covariance : $ P_{k+1}^-=AP_kA^T+G_kQ_kG_k^T $
    Measurement Update Kalman gain :
    $ K_{k+1}=P_{k+1}^-H^T(H_{k+1}P_{k+1}^-H^T+R_{k+1})^{-1} $
    Estimation : $ \hat{x}_{k+1}=\hat{x}_{k+1}^-+K_{k+1}(z_{k+1}-H\hat{x}_{k+1}^-) $
    Error covariance : $ P_{k+1}=(I-K_{k+1}H)P_{k+1}^- $
     | Show Table
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    Table 2.  MAPE (%) GBM vs GBM-KF

    Stock GBM GBM-KF
    Stock 1 (Canon) 1.01 0.16
    Stock 2 (Starbucks) 0.59 0.097
    Stock 3 (Microsoft) 1.23 0.1
     | Show Table
    DownLoad: CSV

    Table 3.  Parameters of Stock Portfolio

    Variable $ \alpha $ $ \beta $ $ r_1 $ $ r_2 $ $ \boldsymbol{x}(0) $ $ N_p $
    Value $ 0.0002 $ $ 0.0002 $ $ 0.00003 $ $ 0.00031 $ $ [ 0,0,0,1\times10^5]^T $ $ 10 $
    Variable $ Q $ $ R $ $ r(k) $ $ p_i\max $ $ q_i\max $
    Value $ 1 $ $ 0,1 $ $ {10}^6 $ $ {10}^5 $ $ {10}^5 $
     | Show Table
    DownLoad: CSV
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    [8] J. GuoW. Huang and B. M. Williams, Adaptive Kalman filter approach forstochastic short-term traffic flow rate prediction and uncertainty quantification, Transportation Research Part C: Emerging Technologies, 43 (2014), 50-64.  doi: 10.1016/j.trc.2014.02.006.
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    [13] L. Wang, Model Predictive Control System Design and Implementation Using MATLABⓇ, Springer Science & Business Media, 2009.
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