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doi: 10.3934/jimo.2021119
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## The application of model predictive control on stock portfolio optimization with prediction based on Geometric Brownian Motion-Kalman Filter

 1 Department of Actuarial Science, Faculty of Science and Data Analytics, Institut Teknologi Sepuluh Nopember, Indonesia 2 Department of Mathematics, Faculty of Science and Data Analytics, Institut Teknologi Sepuluh Nopember, Indonesia

* Corresponding author: Wawan Hafid Syaifudin

Received  August 2020 Revised  March 2021 Early access July 2021

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

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.

Citation: Wawan Hafid Syaifudin, Endah R. M. Putri. The application of model predictive control on stock portfolio optimization with prediction based on Geometric Brownian Motion-Kalman Filter. Journal of Industrial and Management Optimization, doi: 10.3934/jimo.2021119
##### References:
 [1] T. R. Bielecki and S. R. Pliska, Risk-sensitive dynamic asset management, Appl. Math. Optim., 39 (1999), 337-360.  doi: 10.1007/s002459900110. [2] Z. Bodie, A. Kane and A. J. Marcus, Investments, NY : McGraw-Hill/Irwin, 2011. [3] E. F. Camacho and C. B. Alba, Model Predictive Control, Springer-VerlagLondon, 2007. [4] F. Cassola and M. Burlando, Wind speed and wind energy forecast through Kalman filtering of numerical weather prediction model output, Applied Energy, 99 (2012), 154-166.  doi: 10.1016/j.apenergy.2012.03.054. [5] A. Dmouj, Stock price modelling: Theory and practice, Masters Degree Thesis, Vrije Universiteit, 2006. [6] V. Dombrovsky and E. Lashenko, Dynamic model of active portfolio management with stochastic volatility in incomplete market, SICE 2003 Annual Conference (IEEE Cat. No. 03TH8734), IEEE, 1 (2003), 516-521. [7] G. Galanis, P. Louka, P. Katsafados, I. Pytharoulis and G. Kallos, Applications of Kalman filters based on non-linear functions to numerical weather predictions, Ann. Geophys., 24 (2006), 2451-2460.  doi: 10.5194/angeo-24-2451-2006. [8] J. Guo, W. 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. [9] H. Markowitz, Portfolio selection, The Journal of Finance, 7 (1952), 77-91. [10] J. A. Primbs, Portfolio optimization applications of stochastic receding horizon control, in 2007 American Control Conference, IEEE, (2007), 1811–1816. [11] K. Reddy and V. Clinton, Simulating stock prices using geometric brownian motion: Evidence from Australian companies, Australasian Accounting, Business and Finance Journal, 10 (2016), 23-47.  doi: 10.14453/aabfj.v10i3.3. [12] G. Welch and G. Bishop, An introduction to the Kalman filter, Proc. Siggraph Course, 8 (2006). [13] L. Wang, Model Predictive Control System Design and Implementation Using MATLABⓇ, Springer Science & Business Media, 2009.

show all references

##### References:
 [1] T. R. Bielecki and S. R. Pliska, Risk-sensitive dynamic asset management, Appl. Math. Optim., 39 (1999), 337-360.  doi: 10.1007/s002459900110. [2] Z. Bodie, A. Kane and A. J. Marcus, Investments, NY : McGraw-Hill/Irwin, 2011. [3] E. F. Camacho and C. B. Alba, Model Predictive Control, Springer-VerlagLondon, 2007. [4] F. Cassola and M. Burlando, Wind speed and wind energy forecast through Kalman filtering of numerical weather prediction model output, Applied Energy, 99 (2012), 154-166.  doi: 10.1016/j.apenergy.2012.03.054. [5] A. Dmouj, Stock price modelling: Theory and practice, Masters Degree Thesis, Vrije Universiteit, 2006. [6] V. Dombrovsky and E. Lashenko, Dynamic model of active portfolio management with stochastic volatility in incomplete market, SICE 2003 Annual Conference (IEEE Cat. No. 03TH8734), IEEE, 1 (2003), 516-521. [7] G. Galanis, P. Louka, P. Katsafados, I. Pytharoulis and G. Kallos, Applications of Kalman filters based on non-linear functions to numerical weather predictions, Ann. Geophys., 24 (2006), 2451-2460.  doi: 10.5194/angeo-24-2451-2006. [8] J. Guo, W. 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. [9] H. Markowitz, Portfolio selection, The Journal of Finance, 7 (1952), 77-91. [10] J. A. Primbs, Portfolio optimization applications of stochastic receding horizon control, in 2007 American Control Conference, IEEE, (2007), 1811–1816. [11] K. Reddy and V. Clinton, Simulating stock prices using geometric brownian motion: Evidence from Australian companies, Australasian Accounting, Business and Finance Journal, 10 (2016), 23-47.  doi: 10.14453/aabfj.v10i3.3. [12] G. Welch and G. Bishop, An introduction to the Kalman filter, Proc. Siggraph Course, 8 (2006). [13] L. Wang, Model Predictive Control System Design and Implementation Using MATLABⓇ, Springer Science & Business Media, 2009.
Daily Stock Price of Each Company
Daily Stock Return of Each Company
Stock's Forecasting Results
Control Variables in Portfolio Optimization
The Dynamic of Total Invested Capital in Each Stock
The Dynamic of Total Invested Capital in Risk Free Asset
The Dynamic of Total Invested Capital in the Portfolio
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 : $P_{k+1}=(I-K_{k+1}H)P_{k+1}^-$
 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 : $P_{k+1}=(I-K_{k+1}H)P_{k+1}^-$
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
 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
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 $1$ $0,1$ ${10}^6$ ${10}^5$ ${10}^5$
 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 $1$ $0,1$ ${10}^6$ ${10}^5$ ${10}^5$
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