Perubahan dinamika pasar sebagai akibat liberalisasi perdagangan memberikan dampak terhadap fluktuasi harga komoditas pertanian. Pergerakan harga kakao yang sangat volatil dan fluktuatif mencerminkan adanya risiko harga dan pasar. Ketidakpastian harga dan pasar dapat menyulitkan pelaku ekonomi untuk mengambil keputusan dan menentukan arah pengembangan usaha. Penelitian ini bertujuan untuk 1) mengetahui karakteristik pola pergerakan harga kakao pada pasar berjangka komoditas, dan 2) menganalisis volatilitas harga kakao menggunakan model alternatif **ARCH** dan **GARCH**. Penelitian dilakukan dengan mengamati pola pergerakan harga pada pasar berjangka dan menganalisis volatilitas dengan data sekunder periode 2008 - 2013. Sumber data berasal dari Interconti- nental Exchange (ICE) Futures U.S. Reports. Hasil analisis menunjukkan bahwa model **GARCH** (1,1) merupakan model terbaik untuk mengestimasi nilai volatilitas **return** harga rerata kakao, karena memenuhi kriteria tiga uji diagnostik, yaitu uji efek **ARCH**, uji korelasi serial residual dan uji normalitas residual. Hasil uji efek **ARCH** memperlihatkan bahwa model **GARCH** tidak mengandung unsur

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Akgiray’s [1] work proceeds further which not only investigates the statistical properties but also presents the evidence on the forecasting ability of **ARCH** and **GARCH** **models** vis-à-vis EWMA. Expanding further, Tse [2] and Tse and Tung [3] found that EWMA **models** provide better forecasts than **GARCH** **models**. Philip and Dick [4] studied the performance of the **GARCH** **models** and two of its non-linear modifications to forecast weekly stock market **volatility**. This study concludes that the QGARCH model is best and that the GJR model cannot be recommended for forecasting.

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Al-Najjar [13] examined the **volatility** characteristics on Jordan’s capital market using **ARCH**, **GARCH** and EGARCH **models** to investigate the behaviour of stock **return** **volatility** for Amman Stock Exchange (ASE) covering the period from January 2005 to December 2014. The symmetric **ARCH**/**GARCH** **models** showed evidence for both **volatility** clustering and leptokurtosis, while the asymmetric EGARCH model showed no evidence for the existence of leverage effect in the stock returns at Amman Stock Exchange. Harrison and Paton [14] employed stock markets data from Romania and the Czech Republic to identify the correct **GARCH** specification under varying distributional assumptions and found that when log returns are characterized by heavy tails or kurtosis the use of **GARCH**-type model with student-t innovation specification is appropriate. Tudor [15] conducted a study on the Romanian stock market to investigate the Risk-**Return** Tradeoff using basic **GARCH**, **GARCH**-in-Mean and EGARCH **models**. He found that EGARCH model outperformed the other **GARCH** **models** in the Bucharest Stock Exchange composite index **volatility** in terms of sample-fit. In studying the **volatility** of Chinese stock returns during the crisis and pre-crisis period from 2000 to 2010, [16] found that EGARCH model fits the sample data better than basic **GARCH** model. He also found long term **volatility** to be more volatile during the crisis period and that leverage effect was present in the Chinese stock market during the crisis.

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Abstract: Inflation and its **volatility** is one of the serious macro-economic problems in every countries economy. Inflation in Ethiopia is not immune from **volatility** problem nowadays and it was vital to model and forecast it. Therefore, this study aimed at modelling and forecasting **price** inflation **volatility** in Ethiopia using **GARCH** family **models** for the general inflation data, which spans from 1995 to2016. The Lagrangian Multiplier (**ARCH** -LM) test statistic was used for testing the existence of **ARCH** effect in the residuals of conditional mean or ARMA (1,1) model and it confirmed the existence of **ARCH** effect in the log-**return** series of the **price** inflation in Ethiopia. This indicates that **price** inflation in Ethiopia is suffered from **volatility** problems and applying **GARCH** family **models** is relevant and necessary. To model and forecast the **price** inflation **volatility**, ARMA (1,1)-**GARCH** (1,1) was selected as an appropriate model among EGARCH (1,1) and **GARCH** (1,1) **models** with GED, normal and t-distributional assumption for residuals. To select an appropriate model, forecasting error measure statistics such as: MAE (Mean Absolute Error), RMSE(Root Mean Square Error)and Uthail’s inequality coefficient were used in addition to well-known information criteria’s such as: AIC (Akeike Information Criteria) and BIC (Byesian Information Criteria).Moreover, macro-economic variables such as: Broad Money Supply, Exchange Rate and Lending Interest Rate have direct contribution for the **price** inflation **volatility** in Ethiopia except Deposit Interest Rate and GDP(Gross Domestic Product). The finding of this study also clearly showed that last shock and **volatility** had significant contribution to **price** inflation **volatility**. Finally, the **price** inflation **volatility** was forecasted using ARMA (1,1)-**GARCH** (1,1) model with GED distributional assumption. The forecast showed the existence of fluctuation of variance which is declining at the end of the study period. This study suggested that, to come up with stable **price** inflation **volatility** in Ethiopia, the government as well as concerning bodies must pay great effort to control macro-economic factors of inflation **volatility**.

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Table 1 presents the summary statistics of the data. The sample spans from January 4, 1990 to December 31, 2009, and a total of 5,043 daily observations (5,042 for returns) are obtained from Yahoo finance. DJIA denotes the closing **price** of the Dow Jones Industrial Average, Returns are the log differences of DJIA, and Volume is the trading volume at the end of day. We use two sets of volume specifications for robustness check. They are the square root of the natural logarithm of volume V1* and the square root of the absolute value of the log difference of volume V2*, where taking square root of volume is required due to the specifications of stable GACH and TGACH **models** given in equation (3). From Table 1, the average daily **return** is 0.026% (annualized **return** of 9.45%), while the largest daily gain is 10.5% and the largest loss is 8.2%, resulting in substantial variations in the daily returns. According to the Augmented Dickey and Fuller (ADF) tests, V1 * is stationary with intercept and trend at one percent significance level, but nonstationary without trend. V2 * is stationary regardless of its specifications. We also used other unit root tests and the results are basically the same. 5 DJIA and Returns show negative skewness, but Volume series are positively skewed. Returns and V2* show kurtosis, and according to the Jarque-Bera statistics, all variables are non-normally distributed. Further, the residuals from equation (1) indicate strong **ARCH** effect (not presented).

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Asset **return** **volatility** is an imperative factor in pricing of derivatives and portfolio allocation in the financial world. As a result, many conventional methods for measuring the risk associated with financial assets are done through studies on the variance (**volatility**) of the asset **price** [1]. **Volatility** is considered as a measure of risk which is used by investors as a premium for investing in risky assets and therefore an efficient model for fore- casting an asset’s **price** **volatility** is a crucial ingredient in financial decision making.

Regarding the occurrence of **volatility** clustering it is when large changes in stock prices are followed by large changes in **price** of both signs, and vice versa, i.e. the small **price** changes are followed by periods of small changes in prices. On the other hand, the case of non-normal distribution of financial **return** which tends to be fat tailed is called leptokurtosis. Furthermore, studies such as Kosapattarapim et.al (2011), Rousan and AL Khouri (2005), Liu and Huang (2010), Freedi et.al (2012), and Gokbulut and Pekkaya (2014); observed new insights encountered in time series studies that is leverage effect (Asymmetric) that was first detected by Black (1976). It occurs when stock **return** tends to have a negative correlation with changes in **volatility**, i.e. **volatility** is expected to rise in response to bad news and fall in response to good news.

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Komoditi kokoa memegang peranan yang penting dalam menghasilkan devisa negara, mengingat kokoa merupakan salah satu komoditi andalan ekspor Indonesia. Dilain pihak, sebagai salah satu jenis tanaman perkebunan, harga komoditi kokoa cenderung mengalami volatilitas yang tinggi sepanjang waktu. Studi ini memiliki dua tujuan, yaitu untuk menguji kemampuan model-model **GARCH** (**ARCH**, **GARCH**, **GARCH**-M, EGARCH, dan TGARCH) dalam memprediksi volatilitas tingkat pengembalian (**return**) komoditi kokoa dan menentukan model terbaik diantara model-model tersebut. Dua variabel independen yang digunakan dalam studi ini adalah nilai residual dari persamaan rata-rata dan volatilitas varians kesalahan pada periode-periode sebelumnya. Harga komoditi kokoa yang digunakan adalah harga spot komoditi tersebut selama periode Januari 2005 sampai dengan Juni 2011 yang diperoleh dari BAPPEBTI (Badan Pengawas Perdagangan Berjangka Komoditi). Hasil-hasil penelitian menunjukkan bahwa model **GARCH**-M dan model EGARCH memberikan prediksi terbaik dalam mengestimasi volatilitas harga komoditi kokoa.

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where r t is the Bitcoin **price** **return** on day t , u t is the error term, z t is a white noise process, and h t is the conditional standard deviation. Table A.1 (Appendices) presents the different **GARCH**-type **models** used in this research, namely **GARCH**, EGARCH, TGARCH, Asymmetric Power **ARCH** (APARCH), Component **GARCH** (CGARCH) and Asymmetric Component **GARCH** (ACGARCH).

This paper provides a method to forecast day-ahead electricity prices based on autoregressive integrated moving average (ARIMA) and generalized autoregressive conditional heteroskedastic (**GARCH**) **models**. In the competitive power market environment, electricity **price** forecasting is an essential task for market participants. However, time series of electricity **price** has complex behavior such as nonlinearity, nonstationarity, and high **volatility**. ARIMA is suitable in forecasting, but it is not able to handle nonlinearity and **volatility** are existent in time series. Therefore, **GARCH** **models** are used to handle **volatility** in the in time series forecasting. The proposed method is computed using the daily electricity **price** data of Iran market for a five-year period from March 2013 to February 2018. The results reported in this paper illustrate the potential of the proposed ARMA-**GARCH** model and this combined model has been successfully applied to real prices in the Iranian power market .

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The initial discovery of asymmetry in the relationship between returns and **volatility** is usu- ally accredited to Black (1976) and Christie (1982) with their observation that current returns and future **volatility** are negatively correlated, commonly referred to in the literature as Black’s leverage effect. The historic explanations of such market behavior are grounded in the firms’ dept–equity ratio that changes with movements in the **return** and, thus, alters the stock’s risk- iness. However, an increasing number of studies challenge this fundamental reasoning. For example, Hasanhodzic and Lo (2011) find the leverage effect also present in all–equity financed companies and report its effect even stronger than for leveraged firms. Similarly, Hens and Steude (2009) find the effect in the laboratory environment absent of any leverage implying that the inverse relationship between **price** and **volatility** is not driven by financial leverage. In addition, Figlewski and Wang (2000) present evidence that the leverage effect is largely independent of a change in the firms’ capital structure. More evidence for the leverage effect in assets for which the traditional explanation cannot hold is provided in Park (2011), who conjectures a herding type of behavior to explain it.

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Perhaps the most useful result concerns forecast combination for this problem. Al- though many methods are available, a simple average is often competitive with more sophisticated forecast combination devices, and we use here the sample mean of two forecasts to illustrate the potential for forecast combination. The QML and LAD- **ARCH** estimators are based on different information sets, and so a priori we might expect that a combined estimator could perform better than either alone, at least in regions of the parameter space where neither is dominated. This is what we observe: the combined estimator produces the lowest loss in a wide variety of cases, a result which is robust to sample period, to choice of loss function and to parameter choices for estimation methods. Cross-validated combination weights also perform well, but tend overall to be outperformed by an equally weighted combination. Statistical infer- ence tends to confirm that the advantage of forecast combination over either constituent method, whether by equal weighting or cross-validated weights, is genuine.

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This paper provides a method to forecast day-ahead electricity prices based on autoregressive integrated moving average (ARIMA) and generalized autoregressive conditional heteroskedastic (**GARCH**) **models**. In the competitive power market environment, electricity **price** forecasting is an essential task for market participants. However, time series of electricity **price** has complex behavior such as nonlinearity, nonstationarity, and high **volatility**. ARIMA is suitable in forecasting, but it is not able to handle nonlinearity and **volatility** are existent in time series. Therefore, **GARCH** **models** are used to handle **volatility** in the in time series forecasting. The proposed method is computed using the daily electricity **price** data of Iran market for a five-year period from March 2013 to February 2018. The results reported in this paper illustrate the potential of the proposed ARMA-**GARCH** model and this combined model has been successfully applied to real prices in the Iranian power market .

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DOI: 10.4236/jmf.2019.94030 594 Journal of Mathematical Finance However, there have also been several studies on modelling **volatility** dynam- ics of the cryptocurrency market recently, for instance, Dyhrberg [15] estimated the **volatility** of the Bitcoin, Gold and the US Dollar using the **GARCH** and asymmetric EGARCH **models** and concludes that they have similarities and re- spond the same way to variables in the **GARCH** model, arguing that it can be used for hedging. Katsiampa [16] analyzed the Bitcoin **volatility** using a range of **GARCH**-type **models** assuming normally distributed errors and concludes that AR (1)-CGARCH (1, 1) is the best model to estimate Bitcoin returns **volatility**. Charles and Darn [17] replicate the study of Katsiampa considering the presence of extreme observations and using jump-filtered returns and the AR (1)-**GARCH** (1, 1) model is selected as the optimal model. Pichl and Kaizoji [18] study the time-varying realized **volatility** of Bitcoin and conclude that it is sig- nificantly bigger compared to that of fiat currencies. Bariviera [19] investigate the time-varying **volatility** the behaviour of long memory on Bitcoin returns us- ing the Hurst exponent analysis. Urquhart and Zhang [20] model a range of **GARCH** **volatility** **models** and analysis the hedging ability of the crypto-coin against other currencies. In terms of different innovations distributions, Liu and Tsyvinski [21] compare the performance of the normal reciprocal inverse Gaus- sian (NRIG) with the normal distribution and the Student’s t error distributions under the **GARCH** framework and concludes that the **GARCH**-type model with Student’s, t distributed innovations outperform the new heavy-tailed distribu- tion in modelling the Bitcoin returns. Chu et al. [22] estimated the **volatility** of seven cryptocurrencies using **GARCH**-type **models** with different innovations distributions and conclude that the IGARCH (1, 1) model is the most appropri- ate in **estimating** Bitcoin **volatility**.

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This paper presents a comparison of the performance of the **GARCH** family of econometric **models** and neural networks in **estimating** the returns and conditional variance of the Ibex-35 Spanish Stock Exchange Index. As a fairly long period (11) of daily closing prices was analysed, the sample contained a signiﬁcant number of observations (2,658) with stages of both rising and falling stock prices, as well as high and low **volatility**.

This study aims to know on the variance **volatility** to exchange rate in Sudan at the period (2007 – 2018) and estimate the variance of exchange rate in Sudan also, and how to forecasting by exchange rate in Sudan by using **ARCH** and **GARCH** model, and also we found that the model is not suffering from the **ARCH** effect.The approach using in analysis is **ARCH** and **GARCH** **models** (**Volatility** **models** or time-varying dynamic time series).

Page | 305 **models** and based on the results it was concluded that there was high persistent **volatility** for the NSE **return** series and no asymmetric shock phenomena observed in the series (Adesina, 2013). Nairobi Securities Stock Market was analysed on the basis of **GARCH** **models** and results supported the positive relationship between the **volatility** and expected stock returns (Maqsood, et al. 2017).A study aimed at identifying the relationship between returns and **volatility** among South Africa and China using **GARCH** **models** concluded that there existed enough evidence of high **volatility** in both the markets (Cheteni, 2017). Similar kind of study made in Sudan and Egypt stock markets evidenced explosive and persistent process of conditional variance among **return** series respectively (Zakaria, et.al., 2012). Jordan Sock Market **volatility** study based on the family of **GARCH** **models** showed that there existed **volatility** clustering in **return** series but EGARCH output did not support the presence of leverage effect (Najjar, 2016). The literature review provided the required theoretical ground for analysing the **volatility** of the BSE Sensex **return** series in India with the help of **GARCH** **models**. The following section presents the theoretical framework adopted in the present analysis.

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In this thesis, we examine **models** from both categories. We use three differ- ent **volatility** **models**; a **GARCH**(1,1), a Standard stochastic **volatility** model (SV- model), and a bivariate-BEKK(1,1). The **GARCH** model first proposed by Bollerslev (1986) has become a workhorse for **volatility** forecasting. The literature covering this model is extensive, and it has been applied to a variety of financial assets. See for example Andersen and Bollerslev (1998), Hansen and Lunde (2005), and Wang and Wu (2012). Andersen and Bollerslev’s findings suggest that both **GARCH** **models** and stochastic **volatility** **models** provide **volatility** estimates that are closely corre- lated with the future **volatility**. Hansen and Lunde do an extensive study using over 300 **models** from the **ARCH** model universe; their findings show no evidence of the **GARCH**(1,1) being outperformed by any of the more sophisticated **models** on exchange rates. Moreover, Wang and Wu compare forecast performance between univariate and multivariate **GARCH** type **models**. They use, amongst other **models**, a **GARCH**(1,1) and a full BEKK(1,1). To evaluate the **models** they use six different loss functions. Two which we use in our analysis, the Mean squared error (MSE) and Mean absolute error (MAE). While the majority of the loss criteria prefer the multi- variate over the univariate **models**, the MSE and MAE prefer the **GARCH**(1,1) over the full BEKK(1,1). Moreover, both Franses et al. (2008) and Boscher et al. (2000) compare the **GARCH** model and the SV-model on stock **return** data and interest rate data respectively. They both find evidence that the SV-model performs better than the **GARCH** model. However, this evidence is more distinct the in-sample than in the out-of-sample evaluation.

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up; 5000 observations is not a very large sample in terms of accuracy with which parameters are estimated, but it is a reasonable length with which it can be worked with). **GARCH** **models** require several years of daily data in order to be trustworthy. Among further shortcomings to be mentioned we find that the model takes into account only the size of the movement of the returns (magnitude), not the direction as well. Investors behave and plan their actions differently depending on whether a share moves up or down, which explains why the **volatility** is not symmetric in the stance of the directional movements. Market declines forecast higher **volatility** than comparable market increases. This represents the leverage effect described by Gourieroux and Jasiak (2002). Both **GARCH** and **ARCH** have this limitation that impedes them from very accurate forecasts.

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Bollerslev (1986) showed that the **GARCH** model outperformed the **ARCH** model. However, Baillie and Bollerslev (1991) used the **GARCH** model to examine patterns of **volatility** in the US forex market and results were generally poor. In the two decades after the arrival of **ARCH** and **GARCH**, several approaches building on **GARCH** has been created. EGARCH was introduced by Nelson (1991), NGARCH by Higgins and Bera (1992), GJR-**GARCH** by Glosten, Jagannathan and Runkle (1993), TGARCH by Zakoian (1994), QGARCH by Sentana (1995), and many more are available see for example Bollerslev (2008). In an interesting study, Hansen and Lunde (2005) finds that none of the **models** in the **GARCH** family outperforms the simple **GARCH** (1,1) which might be surprising since the **GARCH** (1,1) does not rely upon a leverage effect. While Nelson`s EGarch has several advantages over the linear **GARCH** model authors such as Brownlees and Gallo (2010) find that while at some horizons EGARCH produces the most accurate forecast, but at other horizons EGARCH is outperformed by the linear **GARCH** model. Donaldson and Kamstra (1997) used GJR-**GARCH** (1,1) to forecast international stock **return** **volatility**, and found that this model yielded better forecasts than the **GARCH**(1,1) and EGARCH(1,1). However using **ARCH**, **GARCH**, GJR-**GARCH** and EGARCH, Balaban (2004) found that the standard **GARCH** **models** was overall the most accurate forecast for monthly U.S. dollar-Deutsche mark exchange rate **volatility**.

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