臺灣博碩士論文加值系統

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

BIBLIOGRAPHY

Ball, C. A., & Torous, W. N. (1983). A simplified jump process for common stock returns. Journal of Financial and Quantitative Analysis, 18, 53-65.
Ball, C. A., & Torous, W. N. (1985). On jumps in common stock prices and their impact on call option pricing. Journal of Finance, 40(1), 155-173.
Bates, D. S. (1991). The crash of ’87: was it expected? the evidence from the options markets. Journal of Finance, 46, 1009-1044.
Billio, M. & Pelizon, L. (2000). Value-at-Risk: a multivariate switching regime approach. Journal of Empirical Finance, 7, 531-554.
Bollerslev, T. (1986). Generalized autoregressive conditional heteroskedasiticity. Journal of Econometrics, 31(3), 307–327.
Bollerslev, T. (1987). A conditionally heteroskedastic time series model for speculative prices and rates of return. Review of Economics and Statistics, 69, 542–547.
Chan, W. H. and J. M. Maheu (2002). Conditional jump dynamics in stock market return. Journal of Business & Economic Statistics, 20(3), 377-389.
Ding, Z. C., Granger, W. J. & Engle, R. F. (1993). A long memory property of stock market returns and a new model. Journal of Empirical Finance 1, 83–106.
Duffie, D., & Pan, J. (2001). Analytical Value-at-Risk with jumps and credit risk. Finance and Stochastics, 5, 155-180.
Engle, R. (1982). Autoregressive conditional heterosedasticity with estimates of the variance of United Kingdom inflation. Econometrica, 50, 987–1007.
Engle, J., & Gizycki, M. (1999). Conservatism, accuracy and efficiency: comparing Value-at-Risk models. Working Paper Series Number wp0002. Australian Prudential Regulation Authority.
Gibson, M. S. (2001). Incorporating event risk into Value-at-Risk. Trading Risk Analysis Section, Division of Research and Statistics, Federal Reserve Board, Working paper.
Giot, P., & Laurent, S. (2003a). Market risk in commodity markets: a VaR approach. Energy Economics, 25, 435-457.
Giot, P., & Laurent, S. (2003b). Value-at-Risk for long and short trading positions. Journal of Applied Econometrics, 18, 641-664.
Giot, P., & Laurent, S. (2004). Modelling daily Value-at-Risk using realized volatility and ARCH type models. Journal of Empirical Finance, 11, 379-398.
Hendricks, D. (1996). Evaluation of Value-at-Risk models using historical data Federal Reserve Bank of New York. Economic Policy Review, April, 39-69.
Huang, Y. C., & Lin, B. J. (2004). Value-at-Risk analysis for Taiwan stock index futures: fat tails and conditional asymmetries. Review of Quantitative Finance and Accounting, 22, 79-95.
Jorion, P. (1988). On jump processes in the foreignexchange and stock markets. Review of Financial Studies, 1, 427-445.
Jorion, P. (2000). Value at Risk: The New Benchmark for controlling market risk. McGraw-Hill.
Kupiec, P. (1995). Techniques for verifying the accuracy of risk management models. Journal of Derivatives, 3, 73–84.
Lopez, J. (1998). Methods for evaluating Value-at-Risk estimates. Federal Reserve Bank of New York, Research Paper No. 9802.
Maheu, J. M., & McCurdy, T. H. (2004). News arrival, jump dynamics and volatility components for individual stock returns. Journal of Finance, 59(2), 755-793.

BIBLIOGRAPHY

Adams, J., & Montesi, C. J. (1995). Major issues related to hedge accounting. Newark, CT: Financial Accounting Standard Board.
Baillie, R. T., & Myers, R. J. (1991). Bivariate GARCH estimation of the optimal commodity futures hedge. Journal of Applied Econometrics, 6(2), 109-124.
Baillie, R. T., & Bollerslev, T. (1989). Common stochastic trends in a system of exchange rates, Journal of Finance, 44, 167-181.
Bawa, V. S. (1975). Optimal rules for ordering uncertain prospects. Journal of Financial Economics, 2, 95-121.
Bawa, V. S. (1978). Safety-first, stochastic dominance and optimal portfolio choice. Journal of Financial and Quantitative Analysis, 33(2), 255-271.
Bollerslev, T. (1990). Modelling the coherence in short-run nominal exchange rates: a multivariate generalized ARCH approach. Review of Economics and Statistics, 72, 498-505.
Cecchetti, S. G.., Cumby, R. E., & Figlewski, S. (1988). Estimation of the optimal futures hedge. Review of Economics and Statistics, 70, 623-630.
Chen, S. S., Lee, C. F., & Shrestha, K. (2001). On a mean-generalized semi-variance approach to determining the hedge ratio. Journal of Futures Markets, 21, 581-598.
Chen, S. S., Lee, C. F., & Shrestha, K. (2003). Futures hedge ratios: a review. Quarterly Review of Economics and Finance, 43, 433-465.
De Jong, A., De Roon, F., & Veld, C. (1997). Out-of sample hedging effectiveness of currency futures for alternative models and hedging strategies. Journal of Futures Markets, 17, 817-837.
Eftekhari, B. (1998). Lower partial moment hedge ratio. Applied Financial Economics, 8(6), 645-652.
Engle, R. F., & Granger, C. W. J. (1987). Cointegration and error correction: representation, estimation and testing. Econometrica, 55, 251-276.
Fishburn, P. J. (1977). Mean-risk analysis with risk associated with below-target returns. American Economic Review, 67(2), 116-126.
Gregory-Allen, R. B., & Shalit, H. (1999). The estimation of systematic risk under differentiated risk-aversion: a mean-extended Gini approach. Review of Quantitative Finance and Accounting, 12, 135-157.
Hakkio, K. S., & Rush, M. (1989). Market efficiency and cointegration: an application to the Sterling and Deutschemark exchange rates. Journal of International Money and Finance, 8, 75-88.
Hsin, C. W., Kuo, J., & Lee, C. F. (1994). A new measure to compare the hedging effectiveness of foreign currency futures versus options. Journal of Futures Markets, 14(6), 685-707.
Howard, C. T., & D’Antonio, L. J. (1984). A risk-return measurement of hedging effectiveness. Journal of Financial and Quantitative Analysis, 19, 101-112.
Ingersoll, J. (1987). The theory of financial decision making. Lanham, MA: Rowman and Littlefield.
Jarque, C. M., & Bera, A. K. (1987). A test for normality of observations and regression residuals. International Statistics Review, 55, 163-172.
Johnson, L. L. (1960). The theory of hedging and speculation in commodity futures. Review of Economic Studies, 27, 139-151.
Jorion, P. (2000), Value-at-Risk2nd: the new benchmark for controlling market risk. University of California, Irvine.
Kolb, R. W., & Okunev, J. (1992). An empirical evaluation of the extended mean-Gini coefficient for futures hedging. Journal of Futures Markets, 12, 177-186.
Kolb, R. W., & Okunev, J. (1993). Utility maximizing hedge ratios in the extended mean Gini framework. Journal of Futures Markets, 13, 597-609.
Kroner, K. F., & Claessens, S. (1991). Optimal dynamic hedging portfolios and the currency composition of external debt. Journal of International Money and Finance, 10, 131-148.
Lien, D., & Luo, X. (1993). Estimating the extended mean-Gini coefficient for futures hedging. Journal of Futures Markets, 13, 665-676.
Lien, D., & Tse, Y. K. (1998). Hedging time-varying downside risk. Journal of Futures Markets, 18, 705-277.
Lien, D., & Tse, Y. K. (2000). Hedging downside risk with futures contracts. Applied Financial Economics, 10(2), 163-170.
Lien, D., Tse, Y. K., & Tsui, A. K. C. (2002). Evaluating the hedging performance of the constant-correlation GARCH model. Applied Financial Economics, 12(11), 791-798.
Kroner K. F., & Sultan, J. (1993). Time-varying distributions and dynamic hedging with foreign currency futures. Journal of Financial and Quantitative Analysis, 28(4), 535-551.
Park, T. H., & Switzer, L. N. (1995). Bivariate GARCH estimation of the optimal hedge ratios for stock index future: a note. Journal of Futures Markets, 15(1), 61-67.
Phillips, P. C. B., & Perron, P. (1988). Testing for a unit root in time series regression. Biometrika, 75, 335-346.
Harris, R. D. F., & Shen, J. (2006). Hedging and Value at Risk. Journal of Futures Markets, 26(4), 369-390.
Shalit, H. (1995). Mean-Gini hedging in futures markets. Journal of Futures Markets, 15, 617–635.

BIBLIOGRAPHY
Alizadeh, A., & Nomikos, N. (2004). A Markov regime switching approach for hedging stock indices. Journal of Futures Markets, 24(7), 649-647.
Bollen, N. P. B., Gray, S. F., & Robert, E. W. (2000). Regime switching in foreign exchange rates: evidence from currency option prices. Journal of Econometrics, 94, 239-276.
Brooks, C., & Chong, J. (2001). The cross-currency hedging performance of implied versus statistical forecasting models. Journal of Futures Markets, 21, 1043-1069.
Chen, S. S., Lee, C. F., & Shrestha, K. (2003). Futures hedge ratios: a review. Quarterly Review of Economics and Finance, 43, 433-465.
Chiu, C. L., Wu, P. S., Chen, C. D., & Cheng, W. H. (2005). Hedging with Floor-traded and E-mini stock futures. Quarterly Journal of Business and Economics, 44, 49-68.
Ederington, L. H. (1979). The hedging performance of the new futures markets. Journal of Finance, 34, 157-170.
Garcia, R. (1998). Asymptotic null distribution of the likelihood ratio test in Markov switching models. International Economic Review, 39(3), 763-788.
Gray, S. F. (1996). Modeling the conditional distribution of interest rates as a regime-switching process. Journal of Financial Economics, 42, 27-62.
Hamilton, J. D. (1988). Rational expectations econometric analysis of changes in regime: an investigation of the term structure of interest rates. Journal of Economic Dynamics and Control, 12, 385-423.
Hamilton, J. D. (1989). A new approach to the economic analysis of nonstationary time series and the business cycle. Econometrica, 57, 357-384.
Hamilton, J. D. (1994). Autoregressive conditional heteroskedasticity and changes in regime. Journal of Econometrics, 64, 307-333.
Hansen, B. E. (1992). The likelihood ratio test under nonstandard conditions: testing the Markov switching model of GNP. Journal of Applied Econometrics, 7, S61-S82.
Hansen, B. E. (1996). Erratum: the likelihood ratio test under nonstandard conditions: testing the Markov switching model of GNP. Journal of Applied Econometrics, 11, 195-198.
Howard, C. T., & D’Antonio, L. J. (1991). Multiperiod hedging using futures: A risk minimization approach in the presence of autocorrelation. Journal of Futures Markets, 11, 697-710.
Hung, J. C., Chiu, C. L., & Lee, M. C. (2006). Hedging with zero-Value at Risk hedge ratio. Applied Financial Economics, 16(3), 259-269.
Jarque, C. M., & Bera, A. K. (1987). A test for normality of observations and regression residuals. International Statistical Review, 55, 163-172.
Jorion, P. (2000). Value-at-Risk2nd: The New Benchmark for Managing Financial Risk. McGraw-Hill International Edition.
Lien, D., & Tse, Y. K. (2002). Some recent developments in futures hedging. Journal of Economic Surveys, 16, 357-396.
Lee, H. T. & Yoder, J. (2006) A bivariate markov regime switching GARCH spproach to estimate time varying minimum variance hedge ratios. Applied Economics, forthcoming.
Lien, D., & Luo, C. (1993). Estimating extended mean-Gini coefficient for futures hedging. Journal of Futures Markets, 13, 665-676.
Lien, D., & Luo, C. (1994). Multiperiod hedging in the presence of conditional heteroskedasticity. Journal of Futures Markets, 14, 927-955.
Phillips, P., & Perron, P. (1988). Testing for a unit root in time series regressions. Biometrica, 75, 335-346.