本論文著重在風險值的衡量與以風險值為基礎的避險比率上，共包含三個部份，分別為「在動態跳躍與訊息不對稱下的風險值計算」、「絕對風險值避險比率」與「以雙變量馬可夫狀態轉換模型估計多期的絕對風險值避險比率與最小變異避險比率」。
    將此三部份的內容簡敘如下。第一部分使用GARJI、ARJI與不對稱GARCH等三個模型在估計一天的相對風險值之績效。本文使用此三個模型計算兩個股價指數(道瓊指數與S&P 500指數)與一個匯率(日圓)的多部位風險值，其用意在於探討價格跳躍與訊息不對稱效果對於衡量風險值績效的影響。實證結果發現，在資產報酬率具有隨時間變動的跳躍現象以及訊息不對稱的效果下，由GARJI與ARJI模型所估算出來的風險值無論信心水準的高低，均能提供令人信賴的準確度。另外，由MRSB顯示，GARJI所估算出的風險值最具效率性。
    第二部份本文以風險值為基礎，推出以絕對風險值為目標函數的絕對風險值避險比率。當期貨報酬率服從單純平賭過程或是風險趨避程度趨於無窮大的條件下，絕對風險值避險比率將縮減成為最小變異避險比率。於實證過程中，本文採用包含誤差修正項的雙變量固定相關係數GARCH(1,1)模型估計計算最小變異避險比率所需的參數，並且比較其與Hsin et al. (1994)所提出的以極大化效用函數為目標的避險比率。
    第三部份本文推廣單期的絕對風險值避險比率(Hung et al., 2006)成為多期的情況，並使用四狀態雙變量馬可夫轉換模型和雙變量對角化VECH GARCH(1,1)模型估計道瓊指數與S&P 500的絕對風險值避險比率與最小變異避險比率。與Bollen et al. (2000)不同之處在於，本文分別從樣本內與樣本外避險績效的角度，探討在雙變量的情況下，狀態轉換與GARCH這兩種方法何者對於樣本內的配適度以及樣本外的變異數預測較佳。實證結果顯示狀態轉換的方式提供較佳的樣本內配適度；然而，於大多數的情況下，GARCH在樣本外的變異數預測上較具有優勢。

This study focuses on VaR measurement and VaR-based hedge ratio, and it contains three parts. The first part is titled “Estimation of Value-at-Risk under Jump Dynamics and Asymmetric Information”, the second part is named “Hedging with Zero-Value at Risk Hedge Ratio”, and the last one is “Bivariate Markov Regime Switching Model for Estimating Multi-period zero-VaR Hedge Ratios and Minimum Variance Hedge Ratios”.
  A brief introduction of these three parts is described as follow: The first part employs GARJI, ARJI and asymmetric GARCH models to estimate the one-step-ahead relative VaR and compare their performances among these three models. Two stock indices (Dow Jones industry index and S&P 500 index) and one exchange rate (Japanese yen) are used to estimate the model-based VaR, and we investigate the influences of price jumps and asymmetric information on the performance of VaR measurement. The empirical results demonstrate that, while asset returns exhibited time-varying jump and the information asymmetric effect, the GARJI-based and ARJI-based VaR provide reliable accuracy at both low and high confidence levels. Moreover, as MRSB indicates, the GARJI model is more efficient than alternatives. 
  In the second part, a mean-risk hedge ratio is derived on the foundation of Value-at-Risk. The proposed zero-VaR hedge ratio converges to the MV hedge ratio under a pure martingale process or an infinite risk-averse level. In empirical section, a bivariate constant correlation GARCH(1,1) model with an error correction term is adopted to calculate zero-VaR hedge ratio, and we compare it with the one proposed by Hsin et al. (1994) which maximized the utility function as their objective.
The last part extends one period zero-VaR hedge ratio (Hung et al., 2006) to the multi-period case, and also employed a four-regime bivariate Markov regime switching model and diagonal VECH GARCH(1,1) model to estimate both zero-VaR and MV hedge ratios for Dow Jones and S&P 500 stock indices. Dissimilar with Bollen et al. (2000), the in-sample fitting abilities and out-of-sample variance forecasts between regime-switching and GARCH approaches are investigated in a bivariate case through in- and out-of-sample hedging performances. The empirical evidences show that the regime switching approach provides better in-sample fitting ability; however, GARCH approach has better out-of-sample variance forecast ability for most cases.

TABLE OF CONTENTS                                     
                       Page
ACKNOWLEDGEMENT        ii
ABSTRACT IN CHINESE    iii
ABSTRACT IN ENGLISH    iv
LIST OF TABLES         ix
LIST OF FIGURES        x

PART I	1
Estimation of Value-at-Risk under Jump Dynamics and Asymmetric Information
ABSTRACT	2
CHAPTER
1. Introduction	3
1.1 Motivations and Objectives	3
1.2 Flow Chart	5
2. Literature Review	6
3. Model-based VaR Estimates and Evaluation Methods	8
3.1 VaR Definition	8
3.2 Evaluation Methods	9
3.2.1 General Loss Functions	9
3.2.2 Binary Loss Function	9
3.2.3 Quadratic Loss Function	10
3.2.4 Likelihood Ratio Tests	10
3.2.5 Mean Relative Scaled Bias	11
4. Econometric Methodology	12
4.1 GARJI Model	12
4.2 Asymmetric GARCH Model	15
5. Empirical Results and Analysis	17
5.1 Data Description and Preliminary Analysis	17
5.2 Empirical Results and Analysis	17
5.2.1 Estimation and Analytical Computation of Model-based VaR	17
5.2.2 Prediction Performance of the Model-based VaR	22
6. Conclusions	27
BIBLIOGRAPHY	28

PART II	30
Hedging with Zero-Value at Risk Hedge Ratio
ABSTRACT	31
CHAPTER
1. Introduction	32
1.1 Motivations and Objectives	32
1.2 Flow Chart	34
2. Literature Review of The Hedge Ratios	35
3. The Derivation of Zero-VaR Hedge Ratio	38
4. Econometric Methodology	42
4.1 Bivariate Constant-Correlation GARCH Model	42
4.2 Evaluation of Hedging Effectiveness	44
5. Empirical Results	45
5.1 Data Description	45
5.2 Empirical Results	46
6. Conclusions	53
BIBLIOGRAPHY	54
Appendix A	57
Appendix B	58

PART III	59
Bivariate Markov Regime Switching Model for Estimating Multi-period zero-VaR Hedge Ratios and Minimum Variance Hedge Ratios
ABSTRACT	60
CHAPTER
1. Introduction	61
1.1 Motivations and Objectives	61
1.2 Flow Chart	64
2. Multi-period Zero-VaR Hedge Ratio	65
2.1 Derivation of Multi-period Zero-VaR hedge ratio	65
2.2 Relationship Between Multi-period Zero-VaR Hedge Ratios and Risk-
averse Level	66
2.3 Relationship Between Multi-period Zero-VaR Hedge Ratios and Hedging Horizon	66
3. Econometric Methodology	68
3.1 Four-regime Bivariate Markov Switching Model	68
3.2 The Diagonal VECH GARCH Model	71
3.3 Evaluation of Hedging Performance	72
4. Empirical Results	74
4.1 Data Description	74
4.2 Empirical Results	75
4.2.1 Estimation of Four-regime Bivariate Markov Switching Model	75
4.2.2 Estimation of Diagonal VECH GARCH(1,1) Model	78
4.2.3 In- and Out-of-sample Hedging Performance	79
5. Conclusions	82
BIBLIOGRAPHY	83


 
LIST OF TABLES
                                                      Page
PART I
Table 1. Descriptive statistics of daily return	17
Table 2. Estimation results of GARJI model	18
Table 3. Estimation results of ARJI model	19
Table 4. Estimation results of asymmetric GARCH model	20
Table 5. Forecasting performance summary for different VaR models at 95%
confidence level	23
Table 6. Forecasting performance summary for different VaR models at 99% 
confidence level	24
Table 7. Forecasting performance summary for different VaR models at 99.5% 
confidence level	25
Table 8. Forecasting performance summary for different VaR models at 99.9%  
confidence level	26

PART II
Table 1. Summary statistics of the spot and futures returns of S&P 500	45
Table 2. Bivariate constant correlation GARCH(1,1) model	47
Table 3. HKL and zero-VaR hedge ratios for different risk-averse levels	51
Table 4. Comparison of within-sample hedging effectiveness	52

PART III
Table 1. Summary statistics of the spot and futures returns for daily stock indices	74
Table 2. Estimation results of four-regime bivariate Markov switching model	76
Table 3. Estimation results of diagonal VECH GARCH(1,1) model	79
Table 4. Comparison of hedging performance	81




LIST OF FIGURES
                                                     Page
PART I

Figure 1. Conditional jump intensity for Dow Jones industry index	21
Figure 2. Conditional jump intensity for S&P 500	22
Figure 3. Conditional Jump Intensity for Japanese yen	22

PART II
Figure 1. MV hedge ratio and zero-VaR hedge ratio	48
Figure 2. Comparison of MV, HKL2.5, and zero-VaR75% hedge ratios	50

PART III
Figure 1. The relationship between multi-period zero-VaR hedge ratio and risk-averse 
level	67
Figure 2. The relationship between multi-period zero-VaR hedge ratio and hedging 
horizon	67
Figure 3. Ex-post probabilities of high variance regime for spot and futures of Dow Jones	77
Figure 4. Ex-post probabilities of high variance regime for spot and futures of S&P 500.	78

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