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

Wawan Hafid Syaifudin; Endah R. M. Putri

doi:10.3934/jimo.2021119

Article Contents

2022, Volume 18, Issue 5: 3433-3443. 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

Wawan Hafid Syaifudin^1, , and
Endah R. M. Putri^2,

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
^* Corresponding author: Wawan Hafid Syaifudin

Received: July 31, 2020

Revised: February 28, 2021

Early access: July 2021

Published: November 2022

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

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Abstract

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.

Keywords:

Portfolio management,
Geometric Brownian Motion-Kalman Filter (GBM-KF),
stock price prediction,
Model Predictive Control (MPC),
stock portfolio optimization.

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

Citation:

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Figure 1. Daily Stock Price of Each Company

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Figure 2. Daily Stock Return of Each Company

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Figure 3. Stock's Forecasting Results

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Figure 4. Control Variables in Portfolio Optimization

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

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

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

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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}^- $

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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

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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 $

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