# Comparison of Financial Models for Stock Price Prediction

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Literature Review

## 3. Methodology

#### 3.1. Data

#### 3.2. Autoregressive Integrated Moving Average Process

#### 3.2.1. ARIMA(p,d,q) Models

#### 3.2.2. ARIMA(p,d,q) Model for S & P 500 Index

#### 3.3. Stochastic Model Geometric Brownian Motion

- $W\left(0\right)=0$
- It has independent increments. That is, for any ${t}_{1},{t}_{2},\cdots ,{t}_{n}$, $W\left({t}_{2}\right)-W\left({t}_{1}\right),W\left({t}_{3}\right)-W\left({t}_{4}\right)\cdots ,W\left({t}_{n}\right)-W\left({t}_{n-}\right)$ are independent random variables.
- For every $0\le s<t\le T,W\left(t\right)-W\left(s\right)\sim \mathbb{N}(0,t-s).$

#### 3.3.1. Geometric Brownian Motion (GBM) Model

#### 3.3.2. Geometric Brownian Motion Model for S & P 500 Index: GBM($\mu ,{\sigma}^{2}$) Simulation

#### 3.4. Artificial Neural Network

#### 3.4.1. Model Descriptions

#### 3.4.2. Artificial Neural Network for S & P 500 Index

## 4. Results

#### 4.1. Autoregressive Integrated Moving Average

#### 4.1.1. ARIMA Model Result

#### 4.1.2. ARIMA Model Diagnostics

#### 4.2. Stochastic Model

#### 4.2.1. Stochastic Model Result

#### 4.2.2. Stochastic Model Diagnostics

#### 4.3. Artificial Neural Network

#### 4.3.1. ANN(7-15-1) Results

#### 4.3.2. ANN(7-15-1) Model Diagnostics

#### 4.4. Comparison

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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Model | AIC | BIC | AICc |
---|---|---|---|

ARIMA(0,2,0) | $-7144.11$ | $-7139.03$ | $-7144.11$ |

ARIMA(0,2,1) | $-7981.48$ | $-7971.31$ | $-7981.47$ |

ARIMA(0,2,2) | $-7979.84$ | $-7964.59$ | $-7979.82$ |

ARIMA(0,2,3) | $-7981.77$ | $-7961.43$ | $-7981.73$ |

ARIMA(0,2,4) | $-7980.02$ | $-7954.61$ | $-7979.97$ |

ARIMA(1,2,0) | $-7455.29$ | $-7445.12$ | $-7455.28$ |

ARIMA(1,2,1) | $-7979.81$ | $-7964.56$ | $-7979.79$ |

ARIMA(1,2,2) | $-7982.16$ | $-7961.83$ | $-7982.13$ |

ARIMA(1,2,3) | $-7983.04$ | $-7957.62$ | $-7982.98$ |

ARIMA(1,2,4) | $-7981.53$ | $-7951.03$ | $-7981.45$ |

ARIMA(2,2,0) | $-7631.73$ | $-7616.48$ | $-7631.71$ |

ARIMA(2,2,1) | $-7981.50$ | $-7961.16$ | $-7981.46$ |

ARIMA(2,2,2) | $-7982.88$ | $-7957.46$ | $-7982.83$ |

ARIMA(2,2,3) | $-7982.62$ | $-7952.12$ | $-7982.55$ |

ARIMA(2,2,4) | $-7979.91$ | $-7944.32$ | $-7979.81$ |

ARIMA(3,2,0) | $-7692.63$ | $-7672.30$ | $-7692.60$ |

ARIMA(3,2,1) | $-7979.84$ | $-7954.42$ | $-7979.79$ |

ARIMA(3,2,2) | $-7981.50$ | $-7951.00$ | $-7981.43$ |

ARIMA(3,2,3) | $-7977.84$ | $-7942.25$ | $-7977.74$ |

ARIMA(3,2,4) | $-7978.07$ | $-7937.40$ | $-7977.95$ |

ARIMA(4,2,0) | $-7738.80$ | $-7713.39$ | $-7738.75$ |

ARIMA(4,2,1) | $-7980.13$ | $-7949.63$ | $-7980.06$ |

ARIMA(4,2,2) | $-7978.69$ | $-7943.11$ | $-7978.60$ |

ARIMA(4,2,3) | $-7978.28$ | $-7937.61$ | $-7978.15$ |

ARIMA(4,2,4) | $-7984.84$ | $-7939.09$ | $-7984.69$ |

Model | Arima(x = tr.stock, order = c(0, 2, 1)) | ||||||
---|---|---|---|---|---|---|---|

MA(1) Coefficient | −1.00 | ||||||

Standard Error | 0.0027 | ||||||

Sigma-squared estimated as | 0.0000718 | ||||||

Log likelihood | 3992.74 | ||||||

AIC | −7981.48 | ||||||

AICc | −7981.47 | ||||||

BIC | −7971.31 | ||||||

Training set error measures | ME | RMSE | MAE | MPE | MAPE | MASE | ACF1 |

0.00013 | 0.00846 | 0.00573 | 0.00220 | 0.10561 | 0.99516 | −0.01682 |

MODEL | ${\mathit{R}}^{2}$ | APE | AAE | ARPE | RMSE |
---|---|---|---|---|---|

7-2-1 | 0.3137 | 0.0231 | 7.0435 | 0.1118 | 0.3344 |

7-3-1 | 0.4158 | 0.0211 | 6.412 | 0.1018 | 0.319 |

7-4-1 | 0.0422 | 0.028 | 8.5128 | 0.1351 | 0.3676 |

7-5-1 | 0.0368 | 0.0292 | 8.8796 | 0.1409 | 0.3754 |

7-6-1 | 0.3716 | 0.0215 | 6.5496 | 0.104 | 0.3224 |

7-7-1 | 0.3431 | 0.0226 | 6.8687 | 0.109 | 0.3302 |

7-8-1 | 0.3907 | 0.0219 | 6.6781 | 0.106 | 0.3256 |

7-9-1 | 0.4036 | 0.0212 | 6.4713 | 0.1027 | 0.3205 |

7-10-1 | 0.4108 | 0.0214 | 6.5237 | 0.1036 | 0.3218 |

7-11-1 | 0.5155 | 0.0195 | 5.9284 | 0.0941 | 0.3068 |

7-12-1 | 0.5392 | 0.0187 | 5.6802 | 0.0902 | 0.3003 |

7-13-1 | 0.4777 | 0.0194 | 5.8945 | 0.0936 | 0.3059 |

7-14-1 | 0.4676 | 0.0196 | 5.9702 | 0.0948 | 0.3078 |

7-15-1 | 0.6216 | 0.0167 | 5.0928 | 0.0808 | 0.2843 |

7-16-1 | 0.5701 | 0.0182 | 5.5382 | 0.0879 | 0.2965 |

Date | Actual | Predicted | Error |
---|---|---|---|

11-12-2019 | 305.69 | 305.17 | 0.17 |

11-13-2019 | 305.79 | 305.81 | 0.01 |

11-14-2019 | 306.23 | 305.91 | 0.10 |

11-15-2019 | 308.45 | 306.35 | 0.68 |

11-18-2019 | 308.68 | 308.57 | 0.04 |

11-19-2019 | 308.59 | 308.80 | 0.07 |

11-20-2019 | 307.44 | 308.71 | 0.41 |

11-21-2019 | 306.95 | 307.56 | 0.20 |

11-22-2019 | 307.63 | 307.07 | 0.18 |

11-25-2019 | 310.01 | 307.75 | 0.73 |

11-26-2019 | 310.72 | 310.14 | 0.19 |

11-27-2019 | 312.10 | 310.85 | 0.40 |

11-29-2019 | 310.94 | 312.23 | 0.41 |

12-2-2019 | 308.30 | 311.07 | 0.90 |

12-3-2019 | 306.23 | 308.43 | 0.72 |

12-4-2019 | 308.12 | 306.36 | 0.57 |

12-5-2019 | 308.68 | 308.25 | 0.14 |

12-6-2019 | 311.50 | 308.81 | 0.86 |

12-9-2019 | 310.52 | 311.64 | 0.36 |

12-10-2019 | 310.17 | 310.65 | 0.15 |

12-11-2019 | 311.05 | 310.30 | 0.24 |

12-12-2019 | 313.73 | 311.18 | 0.81 |

12-13-2019 | 313.92 | 313.86 | 0.02 |

12-16-2019 | 316.08 | 314.05 | 0.64 |

12-17-2019 | 316.15 | 316.21 | 0.02 |

12-18-2019 | 316.17 | 316.29 | 0.04 |

12-19-2019 | 317.46 | 316.31 | 0.36 |

12-20-2019 | 318.86 | 317.60 | 0.40 |

12-23-2019 | 319.34 | 319.00 | 0.11 |

12-24-2019 | 319.35 | 319.48 | 0.04 |

12-26-2019 | 321.05 | 319.49 | 0.49 |

12-27-2019 | 320.97 | 321.20 | 0.07 |

12-30-2019 | 319.20 | 321.12 | 0.60 |

MODEL | APE | AAE | ARPE | RMSE |
---|---|---|---|---|

ARIMA(0,2,1) | 0.0044 | 1.3651 | 0.0217 | 0.1472 |

Date | Actual | Predicted | Error |
---|---|---|---|

11-12-2019 | 305.69 | 305.17 | 0.17 |

11-13-2019 | 305.79 | 305.81 | 0.01 |

11-14-2019 | 306.23 | 305.91 | 0.10 |

11-15-2019 | 308.45 | 306.35 | 0.68 |

11-18-2019 | 308.68 | 308.59 | 0.03 |

11-19-2019 | 308.59 | 308.80 | 0.07 |

11-20-2019 | 307.44 | 308.71 | 0.41 |

11-21-2019 | 306.95 | 307.58 | 0.21 |

11-22-2019 | 307.63 | 307.06 | 0.19 |

11-25-2019 | 310.01 | 307.74 | 0.73 |

11-26-2019 | 310.72 | 310.15 | 0.18 |

11-27-2019 | 312.10 | 310.86 | 0.40 |

11-29-2019 | 310.94 | 312.23 | 0.41 |

12-2-2019 | 308.30 | 311.09 | 0.90 |

12-3-2019 | 306.23 | 308.43 | 0.72 |

12-4-2019 | 308.12 | 306.36 | 0.57 |

12-5-2019 | 308.68 | 308.25 | 0.14 |

12-6-2019 | 311.50 | 308.82 | 0.86 |

12-9-2019 | 310.52 | 311.63 | 0.36 |

12-10-2019 | 310.17 | 310.64 | 0.15 |

12-11-2019 | 311.05 | 310.30 | 0.24 |

12-12-2019 | 313.73 | 311.21 | 0.80 |

12-13-2019 | 313.92 | 313.88 | 0.01 |

12-16-2019 | 316.08 | 314.06 | 0.64 |

12-17-2019 | 316.15 | 316.21 | 0.02 |

12-18-2019 | 316.17 | 316.27 | 0.03 |

12-19-2019 | 317.46 | 316.31 | 0.36 |

12-20-2019 | 318.86 | 317.59 | 0.40 |

12-23-2019 | 319.34 | 319.00 | 0.11 |

12-24-2019 | 319.35 | 319.47 | 0.04 |

12-26-2019 | 321.05 | 319.51 | 0.48 |

12-27-2019 | 320.97 | 321.20 | 0.07 |

12-30-2019 | 319.20 | 321.11 | 0.60 |

MODEL | APE | AAE | ARPE | RMSE |
---|---|---|---|---|

GBM | 0.0044 | 1.3341 | 0.0212 | 0.1455 |

Date | Actual | Predicted | Error |
---|---|---|---|

11-12-2019 | 305.69 | 302.13 | 1.17 |

11-13-2019 | 305.79 | 302.52 | 1.07 |

11-14-2019 | 306.23 | 302.26 | 1.30 |

11-15-2019 | 308.45 | 302.39 | 1.97 |

11-18-2019 | 308.68 | 303.56 | 1.66 |

11-19-2019 | 308.59 | 304.19 | 1.43 |

11-20-2019 | 307.44 | 303.54 | 1.27 |

11-21-2019 | 306.95 | 302.52 | 1.44 |

11-22-2019 | 307.63 | 302.86 | 1.55 |

11-25-2019 | 310.72 | 305.06 | 1.82 |

11-27-2019 | 312.10 | 305.81 | 2.02 |

11-29-2019 | 310.94 | 306.38 | 1.47 |

12-2-2019 | 308.30 | 306.02 | 0.74 |

12-3-2019 | 306.23 | 303.59 | 0.86 |

12-4-2019 | 308.12 | 301.63 | 2.11 |

12-5-2019 | 308.68 | 304.03 | 1.51 |

12-6-2019 | 311.50 | 304.28 | 2.32 |

12-9-2019 | 310.52 | 306.05 | 1.44 |

12-10-2019 | 310.17 | 306.07 | 1.33 |

12-11-2019 | 311.05 | 305.23 | 1.87 |

12-12-2019 | 313.73 | 305.55 | 2.61 |

12-13-2019 | 313.92 | 306.48 | 2.37 |

12-16-2019 | 316.08 | 306.73 | 2.96 |

12-17-2019 | 316.15 | 307.56 | 2.72 |

12-18-2019 | 316.17 | 308.05 | 2.57 |

12-19-2019 | 317.46 | 308.40 | 2.85 |

12-20-2019 | 318.86 | 308.04 | 3.39 |

12-23-2019 | 319.34 | 306.20 | 4.11 |

12-24-2019 | 319.35 | 308.94 | 3.26 |

12-26-2019 | 321.05 | 309.71 | 3.53 |

12-27-2019 | 320.97 | 310.31 | 3.32 |

12-30-2019 | 319.20 | 309.93 | 2.90 |

MODEL | APE | AAE | ARPE | RMSE |
---|---|---|---|---|

ANN(7-15-1) | 0.0167 | 5.09279 | 0.08084 | 0.28432 |

Date | Actual | ARIMA | GBM | ANN |
---|---|---|---|---|

11-12-2019 | 305.69 | 305.17 | 305.17 | 302.13 |

11-13-2019 | 305.79 | 305.81 | 305.81 | 302.52 |

11-14-2019 | 306.23 | 305.91 | 305.91 | 302.26 |

11-15-2019 | 308.45 | 306.35 | 306.35 | 302.39 |

11-18-2019 | 308.68 | 308.57 | 308.59 | 303.56 |

11-19-2019 | 308.59 | 308.80 | 308.80 | 304.19 |

11-20-2019 | 307.44 | 308.71 | 308.71 | 303.54 |

11-21-2019 | 306.95 | 307.56 | 307.58 | 302.52 |

11-22-2019 | 307.63 | 307.07 | 307.06 | 302.86 |

11-25-2019 | 310.01 | 307.75 | 307.74 | 303.50 |

11-26-2019 | 310.72 | 310.14 | 310.15 | 305.06 |

11-27-2019 | 312.10 | 310.85 | 310.86 | 305.81 |

11-29-2019 | 310.94 | 312.23 | 312.23 | 306.38 |

12-2-2019 | 308.30 | 311.07 | 311.09 | 306.02 |

12-3-2019 | 306.23 | 308.43 | 308.43 | 303.59 |

12-4-2019 | 308.12 | 306.36 | 306.36 | 301.63 |

12-5-2019 | 308.68 | 308.25 | 308.25 | 304.03 |

12-6-2019 | 311.50 | 308.81 | 308.82 | 304.28 |

12-9-2019 | 310.52 | 311.64 | 311.63 | 306.05 |

12-10-2019 | 310.17 | 310.65 | 310.64 | 306.04 |

12-11-2019 | 311.05 | 310.30 | 310.30 | 305.23 |

12-12-2019 | 313.73 | 311.18 | 311.21 | 305.55 |

12-13-2019 | 313.92 | 313.86 | 313.88 | 306.48 |

12-16-2019 | 316.08 | 314.05 | 314.06 | 306.73 |

12-17-2019 | 316.15 | 316.21 | 316.21 | 307.56 |

12-18-2019 | 316.17 | 316.29 | 316.27 | 308.05 |

12-19-2019 | 317.46 | 316.31 | 316.31 | 308.40 |

12-20-2019 | 318.86 | 317.60 | 317.59 | 308.04 |

12-23-2019 | 319.34 | 319.00 | 319.00 | 306.20 |

12-24-2019 | 319.35 | 319.48 | 319.47 | 308.94 |

12-26-2019 | 321.05 | 319.49 | 319.51 | 309.71 |

12-27-2019 | 320.97 | 321.20 | 321.20 | 310.31 |

12-30-2019 | 319.20 | 321.12 | 321.11 | 309.93 |

MODEL | APE | AAE | ARPE | RMSE |
---|---|---|---|---|

ARIMA(0,2,1) | 0.00438 | 1.33476 | 0.02119 | 0.14556 |

GBM | 0.00438 | 1.33426 | 0.02118 | 0.14553 |

ANN(7-15-1) | 0.01672 | 5.09279 | 0.08084 | 0.28432 |

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**MDPI and ACS Style**

Islam, M.R.; Nguyen, N.
Comparison of Financial Models for Stock Price Prediction. *J. Risk Financial Manag.* **2020**, *13*, 181.
https://doi.org/10.3390/jrfm13080181

**AMA Style**

Islam MR, Nguyen N.
Comparison of Financial Models for Stock Price Prediction. *Journal of Risk and Financial Management*. 2020; 13(8):181.
https://doi.org/10.3390/jrfm13080181

**Chicago/Turabian Style**

Islam, Mohammad Rafiqul, and Nguyet Nguyen.
2020. "Comparison of Financial Models for Stock Price Prediction" *Journal of Risk and Financial Management* 13, no. 8: 181.
https://doi.org/10.3390/jrfm13080181