本論文以台灣股價指數期貨及美國SPDRs自2001年至2008年之日資料為實證標的，全球金融海嘯(2008)為預測期間，進行波動性預測能力比較及風險值績效評估。若預測模型可以在金融危機期間具有良好表現，實務上應該具有相當的重要性。有別於傳統文獻大多使用報酬率的平方作為市場真實波動的代理變數，本論文改以PK變幅、GK變幅、RS變幅及已實現波動度(RV)，並同時採用對稱與不對稱損失函數評估模型的波動性預測績效。更進一步加入已實現波動為基礎的風險值模型(RV-VaR)，利用Kupiec(1995)提出之非條件涵蓋率檢定，比較RV-VaR與GARCH族為基礎的風險值模型之風險管理績效。
實證結果皆指出，以不對稱GARCH模型的波動性預測能力較佳，顯示不對稱的變異數方程式設定能提升波動預測績效，其中以EGARCH模型最佳，而GARCH模型表現最差。在風險值評估部份，台灣股價指數期貨的實證結果，發現RV-VaR模型有低估風險值之虞，以致未能通過回溯測試；反之，GARCH族模型卻能提供準確的風險值預測績效。而美國SPDR指數型股票基金的結果則顯示各模型皆通過回溯測試，其中以RV-VaR模型較能準確估算真實風險值。整體來說，EGARCH與RV-VaR 模型分別為TAIFEX與SPDRs的最佳模型，此結果可提供機構法人、執政當局、風險管理者、投資大眾在面對未來極端事件時的參考依據，並提升風險控管績效。

This thesis applies alternative GARCH-type models to daily volatility forecasting with Value-at-Risk (VaR) application to the Taiwanese stock index futures and Standard & Poor’s Depositary Receipts (SPDRs) that suffered the global financial tsunami that occurred during 2008. Instead of using squared returns as a proxy for true volatility, this thesis adopts four volatility proxy measures, the PK-range, GK-range, RS-range, and RV, for use in the empirical exercise. The volatility forecast evaluation is conducted with a variety of volatility proxies according to both symmetric and asymmetric types of loss functions regarding forecasting accuracy. These models are also evaluated in terms of their ability to provide adequate VaR estimates with the inclusion of realized-volatility-based VaR model. Moreover, the predictive performance of the RV-based VaR model is compared with various GARCH-based VaR models according to both unconditional coverage test (Kupiec,1995) and utility-based loss functions with respect to risk management practice.
Empirical results indicate that the EGARCH model provides the most accurate daily volatility forecasts, whereas the performances of the standard GARCH model are relatively poor. Such evidence suggests that asymmetry in volatility dynamics should be taken into account for forecasting financial markets volatility. Moreover, I find a consistent result that the forecasting performance of models remains constant across various volatility proxies for both empirical data in most cases. In the area of risk management,the RV-VaR model tends to underestimate VaR and has been rejected for lacking correct unconditional coverage for the TAIFEX returns data, while the GARCH genre of models is capable of providing satisfactory and reliable daily VaR forecasts. In particular, the asymmetric EGARCH model is the most preferred. For SPDRs case, while all models have passed the back-test, the RV-VaR is considered the optimal VaR model both for a regulator and for a firm at alternative confidence levels during the whole year of 2008. The empirical findings presented here provide crucial implications for market practitioners, such as, policy makers, institutional risk managers, and common investors in risk management.

TABLE OF CONTENTS

ACKNOWLEDGEMENTS i 
ABSTRACT IN CHINESE ii 
ABSTRACT IN ENGLISH iii 
LIST OF TABLES     viii 
LIST OF FIGURES	  ix 

CHAPTER
1.Introduction 1
1.1 Research Background 1
1.1.1 Lessons from Long-Term Capital Management Crisis 1
1.1.2 The Subprime Mortgage Storm in America 4
1.1.3 The Expatiation of ETFs and Futures 7
1.2 Motivations 9
1.3 Objectives 12
1.4 Flow Chart 14
2.Literature Review 15
2.1 Volatility Forecasting 15
2.1.1 Volatility Proxy Measures 15
2.1.2 Related Literature Review 18
2.2 Applications to Risk Management: Value-at-Risk 22
2.2.1 The definition of Value-at-Risk 22
2.2.2 Related Literature Review 24
2.3 Contribution of the Thesis 34
3. Data Description and Preliminary Analysis 36
3.1 Data Description 36
3.1.1 Taiwanese Stock Index Futures 37
3.1.2 Standard and Poor's Depositary Receipts 37
3.2 Preliminary Analysis of Empirical Data 39
3.2.1 Descriptive Statistics 39
3.2.2 Descriptive Graphs 44
4. Econometric Frameworks 48
4.1 Various Adaptations of GARCH Models 48
4.1.1 GARCH (1,1) Model 48
4.1.2 GJR-GARCH Model 49
4.1.3 QGARCH Model 50
4.1.4 EGARCH Model 50
4.2 The Skewed Generalized T (SGT) Distribution 51
4.3 Alternative Volatility Proxies 53
4.4 Performance Evaluation for Volatility Forecasting Models 54
4.4.1 R-squared from the Mincer and Zarnowitz (1969) Regression 54
4.4.2 Symmetric Loss Functions 55
4.4.3 Asymmetric Loss Function 55
4.5 Model-Based VaRs and their Evaluations 56
4.5.1 The LR Test for Unconditional Coverage 56
4.5.2 The Utility-Based Loss Functions 57
4.5.2.1 Regulatory Loss Function (RLF) 57
4.5.2.2 Firm Loss Function (FLF) 58
5.Empirical Results and Analysis 59
5.1 Model estimates and diagnostic test 59
5.2 Analyzing Volatility forecasting performance of models62
5.2.1 R-squared from Mincer and Zarnowitz (1969) regression 62
5.2.2 Forecasting performance based on symmetric loss errors 63
5.2.3 Forecasting performance based on asymmetric loss errors 66
5.3 Analyzing predictive accuracy of the model-based VaR 75
5.3.1 VaR performance for various models at the 99% confidence level 75
5.3.2 VaR performance for various models at the 99.5% confidence level 81
6.Conclusions and Suggestions 87
BIBLIOGRAPHY 91


LIST OF TABLES

Table 3.1 The Contract Specifications of the Taiwanese Stock Index Futures36
Table 3.2 The Specifications of the SPDRs 39
Table 3.3 Descriptive Statistics of Daily TAIFEX Futures and SPDRs Returns 43
Table 3.4 Descriptive Statistics of alternative volatility proxies for TAIFEX Futures and SPDRs 44
Table 5.1 Estimation results of various GARCH models 61
Table 5.2 The Mincer and Zarnowitz (1969) regression test results 62
Table 5.3 Out-of-sample forecasting performance based on symmetric loss errors:
The case for TAIFEX 64
Table 5.4 Out-of-sample forecasting performance based on symmetric loss errors:
The case for SPDRs 65
Table 5.5 Out-of-sample forecasting performance based on asymmetric loss errors:
The case for TAIFEX 68
Table 5.6 Out-of-sample forecasting performance based on asymmetric loss errors:
The case for SPDRs 69
Table 5.7 Value-at-Risk forecasting results for TAIFEX at the 99% confidence level78
Table 5.8 Value-at-Risk forecasting results for SPDRs at the 99% confidence level 78
Table 5.9 Value-at-Risk forecasting results for TAIFEX at the 99.5% confidence level 83
Table 5.10 Value-at-Risk forecasting results for SPDRs at the 99.5% confidence level 83


LIST OF FIGURES

Figure 3.1 Daily TAIFEX futures prices in level 41
Figure 3.2 Daily SPDRs prices in level 41
Figure 3.3 Daily TAIFEX futures returns 45
Figure 3.4 Daily SPDRs returns 45
Figure 3.5 Daily TAIFEX futures QQ-plot 46
Figure 3.6 Daily SPDRs QQ-plot 46
Figure 3.7 Daily TAIFEX futures returns density 47
Figure 3.8 Daily SPDRs returns density 47
Figure 5.1 PK versus various model-based volatility forecasts for TAIFEX 70
Figure 5.2 GK versus various model-based volatility forecasts for TAIFEX 71
Figure 5.3 RS versus various model-based volatility forecasts for TAIFEX 71
Figure 5.4 RV versus various model-based volatility forecasts for TAIFEX 72
Figure 5.5 PK versus various model-based volatility forecasts for SPDRs 72
Figure 5.6 GK versus various model-based volatility forecasts for SPDRs 73
Figure 5.7 RS versus various model-based volatility forecasts for SPDRs 73
Figure 5.8 RV versus various model-based volatility forecasts for SPDRs 74
Figure 5.9 Daily returns versus the EGARCH-based VaR forecasts at the 99% confidence level:The case of TAIFEX 80
Figure 5.10 Daily returns versus the RV-based VaR forecasts at the 99% confidence level:The case of SPDRs 80
Figure 5.11 Daily returns versus the EGARCH-based VaR forecasts at the 99.5% confidence level:The case of TAIFEX 85
Figure 5.12 Daily returns versus the RV-based VaR forecasts at the 99.5% confidence level:The case of SPDRs 85

1.Alexander, C. and A. Barbosa (2008), ‘Hedging Index Exchange Traded Funds’, Journal of Banking and Finance, 32: 326-337.
2.Alizadeh, S., M. Brandt and F. Diebold (2002), ‘Range-based Estimation of Stochastic Volatility Models’, Journal of Finance, 57: 1047-1091.
3.Andersen, T. G. and T. Bollerslev (1998), ‘Answering the Skeptics: Yes, Standard Volatility Models Do Provide Accurate Forecasts’, International Economic Review, 39: 885-905.
4.Andersen, T. G., T. Bollerslev and S. Lange (1999), ‘Forecasting Financial Market Volatility: Sample Frequency vis-à-vis Forecast Horizon’, Journal of Empirical Finance, 6: 457-477.
5.Angelidis, T. and S. Degiannakis (2005), ‘Modeling Risk for Long and Short Trading Positions’, Journal of Risk Finance, 6(3): 226-238. 
6.Angelidis, T. and S. Degiannakis (2008), ‘Forecasting One-day-ahead VaR and Intra-day Realized Volatility in the Athens Stock Exchange Market’, Managerial Finance, 34(7): 489-497.
7.Awartani, B. M. A. and V. Corradi (2005), ‘Predicting the Volatility of the S&P-500 Stock Index via GARCH Models: The Role of Asymmetries’, International Journal of Forecasting, 21(1): 167-183.
8.Balaban, E. (2004), ‘Comparative Forecasting Performance of Symmetric and Asymmetric Conditional Volatility Models of An Exchange Rate’, Economics Letters, 83: 99-105.
9.Bali, T. G. and D. Weinbaum (2005), ‘A Comparative Study of Alternative Extreme Value Volatility Estimators’, Journal of Futures Markets, 25: 873-892.
10.Bams, D., T. Lehnert and C. C. P. Wolff (2005), ‘An Evaluation Framework for Alternative VaR-models’, Journal of International Money and Finance, 24: 944-958.
11.Bollen, B. and B. Inder (2002), ‘Estimating Daily Volatility in Financial Markets Utilizing Intraday Data’, Journal of Empirical Finance, 9: 551-562.
12.Bollerslev, T. (1986), ‘Generalized Autoregressive Heteroskedasticity’, Journal of Econometrics, 31: 307-327.
13.Brailsford, T. J. and R. W. Faff (1996), ‘An Evaluation of Volatility Forecasting Techniques’, Journal of Banking and Finance, 20: 419-438.
14.Brandt, M. and C. Jones (2006), ‘Volatility Forecasting with Range-based EGARCH Models’, Journal of Business and Economic Statistics, 24: 470-486.
15.Chou, R. Y. (2005), ‘Forecasting Financial Volatilities with Extreme Values: The Conditional Autoregressive Range (CARR) Model’, Journal of Money Credit and Banking, 37: 561-582.
16.Christensen, K. and M. Podolskij (2007), ‘Realized Range-based Estimation of Integrated Variance’, Journal of Econometrics, 141: 323-349.
17.Chu, Q. C. and W. L. Hsieh (2002), ‘Pricing Efficiency of the S&P 500 Index Market: Evidence from the Standard & Poor’s Depositary Receipts’, Journal of Futures Markets, 22: 877-900.
18.Chuang, Y., J. R. Lu and P. H. Lee (2007), ‘Forecasting Volatility in the Financial Markets: A Comparison of Alternative Distributional Assumptions’, Applied Financial Economics, 17: 1051-1060.
19.Dowd, K. (1998), Beyond Value at Risk: The New Science of Risk Management, John Wiley & Sons Ltd.
20.Engle, R. F. (1982), ‘Autoregressive Conditional Heteroscedasticity with Estimates of the Variance of U.K. Inflation’, Econometrica, 50: 987-1007.
21.Engle, R. F. (1990), ‘Discussion: Stock Market Volatility and the Crash of 87’, Review of Financial Studies, 3: 103-106.
22.Engle, R. F. and V. Ng (1993), ‘Measuring and Testing the Impact of News on Volatility’, Journal of Finance, 48: 1749-1778.
23.Evans, T. and D. G. McMillan (2007), ‘Volatility Forecasts: The Role of Asymmetric and Long-memory Dynamics and Regional Evidence’, Applied Financial Economics, 17: 1421-1430.
24.Franses, P. H. and D. V. Dijk (1996), ‘Forecasting Stock Market Volatility Using (Non-linear) GARCH Models’, Journal of Forecasting, 15: 229-235.
25.Garman, M. and M. Klass (1980), ‘On the Estimation of Security Price Volatilities from Historical Data’, Journal of Business, 53: 67-78.
26.Giot, P. and S. Laurent (2003a), ‘Market Risk in Commodity Markets: A VaR Approach’, Energy Economics, 25: 435-457.
27.Giot, P. and S. Laurent (2003b), ‘Value-at-Risk for Long and Short Trading Positions’, Journal of Applied Econometrics, 18: 641-664.
28.Giot, P. and S. Laurent (2004), ‘Modelling Daily Value-at-Risk Using Realized Volatility and ARCH Type Models’, Journal of Empirical Finance, 11: 379-398.
29.Glosten, L., R. Jagannathan and D. Runkle (1993), ‘On the Relation Between the Expected Value and the Volatility Nominal Excess Return on Stocks’, Journal of Finance, 46: 1779-1801. 
30.Hansen, B. E. (1994), ‘Autoregressive Conditional Density Estimation’, International Economic Review, 35: 705-730.
31.Hansen, P. and A. Lunde (2006), ‘Consistent Ranking of Volatility Models’, Journal of Econometrics, 131: 97-121.
32.Harper, J. T., J. Mdura and O. Schnusenberg (2006), ‘Performance Comparison between Exchange-traded Funds and Closed-end Country Funds’, International Financial Markets, Institutions & Money, 16: 104-122.
33.Hung, J. C., M. C. Lee and H. C. Liu (2008), ‘Estimation of Value-at-Risk for Energy Commodities via Fat-tailed GARCH Models’, Energy Economics, 30: 1173-1191.
34.Huang, Y. C. and B. J. Lin (2004), ‘Value-at-Risk Analysis for Taiwan Stock Index Futures: Fat Tails and Conditional Asymmetries in Return Innovations’, Review of Quantitative Finance and Accounting, 22: 79-95.
35.Jacob, J. and Vipul (2008), ‘Estimation and Forecasting of Stock Volatility with Range-based Estimators’, Journal of Futures Markets, 28(6): 561-581.
36.Jondeau, E. and M. Rockinger (2003), ‘Conditional Volatility, Skewness, and Kurtosis: Existence, Persistence, and Comovements’, Journal of Economic Dynamics and Control, 27(10): 1699-1737.
37.Jorion, P., (2001), Value at Risk — The New Benchmark for Managing Financial Risk’, The McGraw-Hill Co.
38.Jorion, P., (2007), Value at Risk — The New Benchmark for Managing Financial Risk (3rd ed.), The McGraw-Hill Co.
39.Kupiec, P. (1995), ‘Techniques for Verifying the Accuracy of Risk Management Models’, The Journal of Derivatives, 3: 73-84.
40.Kwiatkowski, D., P. C. B. Phillips P. Schmidt and Y. Shin (1992), ‘Testing the Null Hypothesis of Stationarity against the Alternative of a Unit Root: How Sure are We that Economic Time Series Have a Unit Root?’, Journal of Econometrics, 54: 159-178.
41.Lee, M. C., J. B. Su and H. C. Liu (2008), ‘Value-at-Risk in US Stock Indices with Skewed Generalized Error Distribution’, Applied Financial Economics Letters, 4: 425-431.
42.Liu, H. C., Y. H. Lee and M. C. Lee (2009), ‘Forecasting China Stock Markets Volatility via GARCH Models under Skewed-GED Distribution’, Journal of Money, Investment and Banking, 7: 5-15. 
43.Liu, H. C., M. C. Lee and C. C. Chang (2009), ‘The Role of SGT Distribution in Value-at-Risk Estimation: Evidence from the WTI Crude Oil Market’, Investment Management and Financial Innovations, 6: 86-95.
44.Liu, H. C., Y. J. Cheng and Y. P. Tzou (2009), ‘Value-at-Risk-based Risk Management on Exchange Traded Funds: The Taiwanese Experience’, Banks and Bank Systems, 4: 20-28.
45.Lopez, J. A. (1999), ‘Methods for Evaluating Value-at-Risk Estimates’, Federal Reserve Bank of San Francisco Economic Review, 2: 3-17.
46.Marcucci, J. (2005), ‘Forecasting Stock Market Volatility with Regime-switching GARCH Models’, Studies in Nonlinear Dynamics & Econometrics, 9: 1-53.
47.Marsh, T. A. and E. R. Rosenfeld (1986), ‘Non-trading, Market Making, and Estimates of Stock Price Volatility’, Journal of Financial Economics, 15: 359-372.
48.Martens, M. and D. Dijk (2007), ‘Measuring Volatility with the Realized Range’, Journal of Econometrics, 138: 181-207.
49.Mazin, A. M. and Janabi Al (2006), ‘Foreign-exchange Trading Risk Management with Value-at-Risk Case Analysis of the Moroccan Market’, Journal of Risk Finance, 7: 273-291.
50.McDonald, J. B. and W. K. Newey (1988), ‘Partially Adaptive Estimation of Regression Models via the Generalized t Distribution’, Econometric Theory, 4: 428-457.
51.McMillan, D. G., A. Speight and O. Apgwilym (2000), ‘Forecasting U.K. Stock Market Volatility’, Applied Financial Economics, 10(4): 435-448. 
52.McMillan, D. G. and A. E. H. Speight (2004), ‘Daily Volatility Forecasts: Reassessing the Performance of GARCH Models’, Journal of Forecasting, 23: 449-460.
53.McMillan, D. G. and A. E. H. Speight (2007), ‘Value-at-Risk in Emerging Equity Markets: Comparative Evidence for Symmetric, Asymmetric, and Long Memory GARCH Models’, International Review of Finance, 7: 1-19.
54.Mincer, J. and V. Zarnowitz (1969), ‘The Evaluation of Economic Forecasts. In: Mincer, J. (Ed.), Economic Forecasts and Expectations’, National Bureau of Economic Research, New York.
55.Nelson, D. B. (1991), ‘Conditional Heteroskedasticity in Asset Returns: A New Approach’, Econometrica, 59: 347-370.
56.Olienyk, J. P., R. G. Schwebach and J. K. Zumwalt (1999), ‘WEBS, SPDRs and Country Funds: Analysis of International Cointegration’, Journal of Multinational Financial Management, 9: 217-232.
57.Parkinson, M. (1980), ‘The Extreme Value Method for Estimating the Variance of the Rate of Return’, Journal of Business, 53: 61-65.
58.Phillips, P. C. B. and P. Perron (1988), ‘Testing for a Unit Root in Time Series Regression’, Biometrika, 75: 335-346.
59.Poon, S. H. and C. W. J. Granger (2003), ‘Forecasting Volatility in Financial Markets: A Review’, Journal of Economic Literature, 41: 478-539.
60.Rivera, G. G., T. H. Lee and S. Mishra (2004), ‘Forecasting Volatility: A Reality Check Based on Option Pricing, Utility Function, Value-at-Risk, and Predictive Likelihood’, International Journal of Forecasting, 20: 629-645.
61.Rogers, L. C. G. and S. E. Satchell (1991), ‘Estimating Variance from High, Low and Closing Prices’, Annals of Applied Probability, 1(4): 504-512.
62.Sarma, M., S. Thomas and A. Shah (2003), ‘Selection of Value-at-Risk Models’, Journal of Forecasting, 22: 337-358.
63.Sentana, E. (1995), ‘Quadratic ARCH Models’, Review of Economic Studies, 62: 639-661.
64.Shao, X. D., Y. J. Lian and L. Q. Yin (2009), ‘Forecasting Value-at-Risk Using High Frequency Data: The Realized Range Model’, Global Finance Journal, 20: 128-136.
65.Shu, J. and J. E. Zhang (2006), ‘Testing Range Estimators of Historical Volatility’, Journal of Futures Markets, 26: 297-313.
66.So, M. K. P. and P. L. H. Yu (2006), ‘Empirical Analysis of GARCH Models in Value at Risk Estimation’, International Financial Markets, Institutions & Money, 16: 180-197.
67.Subbotin, M. T. H. (1923), ‘On the Law of Frequency of Error’, Matematicheskii Sbornik, 31: 296-301.
68.Theodossiou, P. (1998), ‘Financial Data and the Skewed Generalized t Distribution’, Management Science, 44: 1650-1661.
69.Tse, Y. and V. Martinez (2007), ‘Price Discovery and Informational Efficiency of International iShares Funds’, Global Finance Journal, 18: 1-15.
70.Vipul, and J. Jacob (2007), ‘Forecasting Performance of Extreme-Value Volatility Estimators’, Journal of Futures Markets, 27(11): 1085-1105.
71.Wiggins, J. B. (1991), ‘Empirical Tests of the Bias and Efficiency of the Extreme-Value Variance Estimator for Common Stocks’, Journal of Business, 64: 417-432.
72.Wiggins, J. B. (1992), ‘Estimating the Volatility of S&P 500 Futures Prices Using the Extreme-Value Method’, Journal of Futures Markets, 12: 265-273.
73.Wilhelmsson, A. (2006), ‘GARCH Forecasting Performance under Different Distribution Assumptions’, Journal of Forecasting, 25: 561-578.