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Recently, the volatility of financial markets has contributed a necessary part to risk management. Volatility risk is characterized as the standard deviation of the constantly compound return per day. This paper presents forecasting of volatility for the Jordanian industry sector after the crisis in 2009. ARIMA and ARIMA-Wavelet Transform (WT) have been conducted in order to select the best forecasting model in the content of industry stock market data collected from Amman Stock Exchange (ASE). As a result, the researcher found that ARIMA-WT has more accuracy than ARIMA directly. The results have been introduced using MATLAB 2010a and R programming.

Recently, several financial categories of research have been concerned with the forecasting volatility modeling since it plays an important role in risk management and financial asset pricing such as bonds and stocks. The forecasting accuracy helps financial market participants to estimate future risks. After that the regulators can make decisions regarding the financial instruments (Bollersleva et al. 2014).

The experience of market risk was formally predicted with the accurate volatility forecast. The high fluctuation of stock prices highlights the importance of volatility forecast. Recently, a list of volatility models has been suggested in the educational works of literature for testing the fundamental trade-off between risks and return of financial assets, and for investigating the causes and consequences of the volatility dynamic in the economy [

One of the famous volatility models is GARCH (Generalized Autoregressive Conditional Heteroskedasticity) model which is commonly used for estimating and forecasting financial market volatility. Based on Engle’s ARCH (Autoregressive Conditional Heteroskedasticity) in 1982 and Bolllerselv’s GARCH models in 1986, plenty of GARCH models, such as NAGARCH, GJR-GARCH, FIGARCH, and EGARCH, are created in options markets, exchange, and bonds. A huge percentage of studies focused on the stock market volatility. Trück and Liang examined the performance of different models (GARCH, TARCH, TGARCH, and ARMA) in 2012 [

In this study, the volatility of the stock market prices will be modeled using WT. Also, this article improves the forecasting accuracy in the content of volatility model using industry data from ASE by combining ARIMA model with WT; then finally the results of ARIMA model directly are compared with WT + ARIMA. Generally, the volatility should be based on ARIMA model which has great concentration in finance and economic fields. George Box and Gwilym Jenkins (ARIMA) is a forecasting model which was popularized in the 1970s. ARIMA model is defined as: ARIMA

This paper is organized as follows: Section

Recently, forecasting the financial data got high attention such as [

In the WT field, there are some results in the forecasting accuracy in the stock market data that have been introduced such as [

The definition of volatility, WT, and ARIMA models will be discussed. After that, the several approaches to accuracy will be listed.

In risk management area, volatility is defined as the standard deviation of the continuously compounded return per day. Define

A variable’s volatility,

WT is defined by [

Generally, WT were calculated by using dilation equations and were defined as

The HWT was improved and developed the frequency–domain characteristics by Daubechies WT (DWT). However, there is no specific formula for DWT. Therefore, the square gain function of their scaling filter is used; it is defined as (Gencay et al., [

The auto-regressive moving average (ARMA) models are used in stock market to illustrate stationary time series. The ARMA model is a combination of an autoregressive (AR) model and a moving average (MA) model. A time series

A time series

An extension of the ordinary ARMA model is the auto-regressive integrated moving-average model (ARIMA

When

ARIMA model is the majority famous way of forecasting since there is no need for any assumptions and it is not limited to specific type of pattern. These models can be fitted to any set of time series data (stationary or non-stationary) by estimating the parameters

The author used some criteria in order to make fair comparison between ARIMA and ARIMA-WT that can be presented in this section. Some types of accuracy criteria have used root means squared error (RMSE), percentage root mean absolute percentage error (MAPE), and mean absolute error (MAE). For the mathematical formulas, refer to [

In order to show forecasting volatility risk for the industry sector in the stock market, daily close price is used from industry after crises 2009 for the time period 2009–2015 selected from ASE.

The DWT (Discrete wavelet transform) converts the data into two sets: approximation series (CA1 (

Firstly, decompose through dWT the available historical return data.

Secondly, develop the fitted ARIMA model directly.

Thirdly, use specific ARIMA model fitted to each one of the approximation series to make the forecasting, which means make forecasting using dWT with ARIMA model.

Finally, the results in the second and third points are compared.

Figure

Referring to Figure

ARIMA model has been used since it is suitable with linear and stationary data, and WT is a suitable model for nonstationary and nonlinear data. Therefore, the combination between ARIMA and WT will give a strong model to improve the forecasting accuracy.

In this article, the decomposition for level 4 has been used. However, the author can use any other level since the smoothed data only is used forecasting accuracy. Therefore, the level of decomposition is unjustified.

dWT (Daubechies WT) is used in this article since it is well known that dWT is the best function in the WT field.

The forecasting volatility of ARIMA-WT is presented in Figure

The forecasting results.

ARIMA directly | ARIMA + WT | |
---|---|---|

RMSE | 1.56 | 0.6 |

MAPE | 520 | 286 |

MAE | 1.34 | 0.4 |

dWT decomposition for industry data form ASE.

Regarding Figure

Therefore,

Signing a memorandum of understanding between Amman Stock Exchange and the Egyptian Exchange.

ASE has finished the stage of preparing the new website.

New version of the Electronic Trading System.

ASE launches the Internet Trading Service.

The meetings of the Working Committee of the Federation of Euro-Asian Stock Exchanges (FEAS).

S&P Indices and Arab Federation of Exchanges create S&P AFE 40 Index.

ASE receives an economic delegation from the French Embassy.

ASE receives a delegation from European Bank For Reconstruction and Development.

The national economy was able to achieve a real growth in GDP during the three first quarters of 2014.

Amman Stock Exchange receives a delegation from Libyan Stock Market.

Regarding the second target of this paper, Table

For the sake of fair comparison the same number of data set is selected. The suitable forecasted model for forecasting the sample data is the fitted dWT-

ARIMA model is the most general way of forecasting since there is no need for any assumptions and it is not limited to a specific type of pattern. These models can be fitted to any set of time series data (stationary or non-stationary) by estimating the parameters

The authors declare that they have no conflicts of interest.