本研究嘗試以常態混合GARCH模型( Normal Mixture GARCH Model)及NIG混合GARCH模型(Normal-inverse Gaissian Mixture GARCH Model)捕捉商品期貨市場之波動度行為。常態混合GARCH模型(以下簡稱為NM-GARCH模型)為一由二至數個常態分配以特定權重(mixing law)混合而成之模型，其變異數符合GARCH過程。NM-GARCH對於大部分具有高峰、厚尾現象之財務資料捕捉能力應較一般常態GARCH模型和Student’s t GARCH模型為佳。另外，NM-GARCH中權重較低之組成成分之變異數通常較高，而權重較高之組成成分波動度較小，說明了現實經濟情況中，大波動(shock)發生機率較小，小波動發生機率較高的現象，即在一般情況中不斷發生之波動幅度較平緩，較大的衝擊雖然幅度大，但較少發生。
NIG混合分配為一由兩個或以上組成成份加權平均而成的混合分配，而其任一個組成成份均符合NIG分配。相較於常態混合分配，NIG混合分配將NIG各項優點納入考量，不但解釋了高峰、資料之離散程度，又因NIG分配之雙尾下降速度較常態分配緩慢，NIG混合分配應能更完整的解釋資料之厚尾現象。
本研究將常態混合模型及NIG混合模型應用於原油期貨市場報酬率波動性預測，並經由參數估計、預測、損失函數及統計顯著性檢定，得以推論對原油期貨市場報酬率¬而言，此二種模型相較於其它波動度模型配適度及預測能力均顯著較佳。

This study attempts to capture the behavior of volatility in the commodity futures market by importing the normal mixture GARCH Model and the NIG mixture GARCH model (Normal-inverse Gaussian Mixture GARCH Model). Normal mixture GARCH Model (what follows called NM-GARCH Model) is a model mixed by two to several normal distributions with a specific weight portfolio, and its variance abide by GAECH process. The ability of capturing the financial data with leptokurtosis and fat-tail of NM-GARCH Model is better than Normal GARCH Model and Student’s t GARCH Model.。Also，The Variance of the factor with lower weight in NM-GARCH Model usually higher, and the volatility of the factor with higher weight is lower, which explains the situation happens in the real market that the probability of large fluctuations (shocks) is small, and the probability of small fluctuations are higher. Generally, the volatilities which keeping occurring in common cases are respectively flat, and the shocks usually bring large impacts but less frequent.
NIG Mixture Distribution is a distribution mixed by two to several weighted distributions, and the distribution of every factor abides by NIG Distribution. Compare to Normal Mixture Distribution, NIG Mixture Distribution takes the advantages of NIG Distribution into account, which can not only explain leptokurtosis and the deviation of data, but describe the fat-tail phenomenon more complete as well, because of the both tails of NIG Distribution decreasing slowly.
This study will apply the NM GARCH Model and NIG GARCH Model to the Volatility forecasting of the return rates in the crude oil futures market, and infer the predictive abilities of this two kinds of models are significantly better than other volatility model by implementing parameter estimation, forecasting, loss function and statistic significant test.

Contents
Abstract ii
Chapter 1　Introduction 1
1.1　Background 1
1.2　Purpose of Research, Process and Structure 4
Chapter 2　Literature Review 7
2.1　Volatility Models and The Empirical Results 7
Chapter 3　Methodology 15
3.1 The Characteristics and Definitions of Each Probability Density Function 15
3.2 Data Analysis 23
3.3 Model Selection 34
Chapter 4 Empirical Research 38
4.1　Research Process and Structure 38
4.2 Parameters Estimation 44
4.3 The Size of Volatility and Probability 47
4.4 Forecasting and Performance Evaluation 50
Chapter 5 Conclusion and Recommendations 54
Reference 57

Reference
Alexander, C. and E. Lazar (2004). "The equity index skew, market crashes and asymmetric normal mixture GARCH." ICMA Centre Discussion Papers in Finance.

Alexander, C. and E. Lazar (2005). "Asymmetries and volatility regimes in the european equity market." ICMA Centre Discussion Papers in Finance 14.

Alexander, C. and E. Lazar (2006). "Normal mixture garch (1, 1): Applications to exchange rate modelling." Journal of Applied Econometrics 21(3): 307-336.

Alexander, C. and E. Lazar (2006). "Symmetric normal mixture GARCH." Journal of Applied Econometrics 21: 307-336.

Bachelier, L. (1900). "Theory of speculation." The random character of stock market prices 1018: 17-78.

Bai, J. and P. Perron (2003). "Computation and analysis of multiple structural change models." Journal of Applied Econometrics 18(1): 1-22.

Ball, C. A. and W. N. Torous (1983). "A simplified jump process for common stock returns." Journal of Financial and Quantitative Analysis 18(01): 53-65.

Bauwens, L., M. Lubrano., & Richard, J. F.(1999). Bayesian inference in dynamic econometric models, Oxford University Press, USA.

Bollerslev, T. (1987). "A conditionally heteroskedastic time series model for speculative prices and rates of return." The review of economics and statistics 69(3): 542-547.

Boothe, P. and D. Glassman (1987). "The statistical distribution of exchange rates: empirical evidence and economic implications." Journal of International Economics 22(3-4): 297-319.

Ding, Z., C. W. J. Granger, & University of California, San Diego. (1993). "A long memory property of stock market returns and a new model." Journal of empirical finance 1(1): 83-106.

Drakos, A. A., G. P. Kouretas, & Zarangas, L. P. "Forecasting financial volatility of the Athens stock exchange daily returns: an application of the asymmetric normal mixture GARCH model." International Journal of Finance & Economics 15(4): 331-350.

Fama, E. F. (1965). "The behavior of stock-market prices." The Journal of Business 38(1): 34-105.

Forsberg, L. and T. Bollerslev (2002). "Bridging the gap between the distribution of realized (ECU) volatility and ARCH modelling (of the Euro): the GARCH NIG model." Journal of Applied Econometrics 17(5): 535-548.

Glosten, L. R., R. Jagannathan, Runkle, D. E., & Federal Reserve Bank of Minneapolis (1993). "On the relation between the expected value and the volatility of the nominal excess return on stocks." The Journal of Finance 48(5): 1779-1801.

Haas, M., S. Mittnik, & PAOLELLA, M. S. (2004). "Mixed normal conditional heteroskedasticity." Journal of Financial Econometrics 2(2): 211.

Hsieh, D. A. (1989). "Testing for nonlinear dependence in daily foreign exchange rates." The Journal of Business 62(3): 339-368.

Madan, D. B. and E. Seneta (1990). "The variance gamma (VG) model for share market returns." The Journal of Business 63(4): 511-524.

Mandelbrot, B. (1963). "New methods in statistical economics." The Journal of Political Economy 71(5): 421-440.

Nelson, D. B. (1991). "Conditional heteroskedasticity in asset returns: A new approach." Econometrica 59(2): 347-370.

Vlaar, P. J. G. and F. C. Palm (1993). "The message in weekly exchange rates in the European Monetary System: mean reversion, conditional heteroscedasticity, and jumps." Journal of Business & Economic Statistics: 351-360.

Xu, D. and T. S. Wirjanto (2010). "An Empirical Characteristic Function Approach to VaR Under a Mixture-of-Normal Distribution with Time-Varying Volatility." The Journal of Derivatives 18(1): 39-58.