||The estimation and forecasting of Value-at-Risk
for financial commodities
||Department of Banking and Finance
Composite Simpson’s rule
第三部份本文假設金融資產之對數價格服從一般化偏態誤差分配(SGED)下推導歐式選擇權買權之定價公式，然後再使用第二部份中所用複合辛普森數值積分法求SGED分配及其特例分配在不同偏態與峰態組合下選擇權買權之價格，再分別探討偏態與峰態對Black-Scholes偏誤之影響。由數值分析結果，我們可得以下結果：在定價過程存在非對稱現象；對任意 ( )值，過度定價或不足定價程度會隨 ( )絕對值增加（減少）而增加；有關峰態對Black-Scholes偏誤之影響，當 =2時，Black-Scholes在齊質偏誤點左側會對具有負（正）偏態產生過度定價（不足定價），在齊質偏誤點右側會對具有負（正）偏態產生不足定價（過度定價）。當 =1.5 及 1.0時，過度區域會隨偏態係數 由 增加至 0.25而往右移動。過度定價或不足定價之程度也會隨 由2減少至1而增加。有關偏態對Black-Scholes偏誤之影響，當 = 0時，Black-Scholes在左邊的齊質偏誤點左側及右邊的齊質偏誤點右側會產生不足定價，在這兩個齊質偏誤點間之區域會產生過度定價情形。當 = 時，過度區域會隨 由2減少至1而往右（左）移動致使過度區域範圍變大。以上的發現將有助於解釋各種已知Black-Scholes的偏誤現象。
||This study focuses on VaR measurement and Option pricing, and it contains
three parts. The first part is titled “Value-at-Risk Forecasts in Gold Market under Oil Shocks”, the second part is named “Value-at-Risk Forecasts in U.S. Crude Oil Market with Skewed Generalized Error Distributions.”, and the last one is “Option Pricing with Skewed Generalized Error Distributions.”
A brief introduction of these three parts is described as follow: The first part investigates the value-at-risk in gold markets by considering both oil volatilities and the flexible model construction. The oil volatility is estimated using the dynamic jump model, and the volatility is distinguished further into stochastic and jump volatility. The flexible models include the BHK and PGARCH models. Finally, by combining the data with the rolling window approach, the appropriate out-of-sample VaR estimates are clearly obtained in this paper. The empirical results demonstrate that the BHK-PGARCH-HV-type model, which distinguish both the crude oil volatility and focus on the high volatilities, perform best in this paper. That is to say, the high volatility and jump volatility cannot be ignored in forecasting gold VaR.
In the second part, we propose a composite Simpson’s rule, a numerical integral method, for estimating quantiles on the skewed generalized error distribution (SGED). Daily spot prices of Brent and WTI crude oil are used to examine the one-day-ahead VaR forecasting performance of the ARJI-N and ARJI-SGED models. Empirical results show that Brent crude oil exhibits slightly skewed to the left while WTI exhibits slightly skewed to the right. Therefore the ARJI-N model may overestimate the true VaR for Brent crude oil and underestimate the true VaR for WTI crude oil. These findings demonstrate that the use of SGED distribution, which explicitly accommodates both skewness and kurtosis, is essential for out-of-sample VaR forecasting in U.S. oil markets.
The last part presents a novel option-pricing model based on the Skewed Generalized Error Distribution (SGED). A composite Simpson’s rule is used to acquire numerical results under the SGED and its degenerative distributions with varying degrees of skewness and kurtosis. The impact of skewness and kurtosis on Black-Scholes biases is investigated. The following analytical results are based on numerical analyses. Some asymmetrical phenomena exist. For any ( ), the extent of overpricing or underpricing increases when the absolute value of ( ) increases (decreases). For the impact of skewness, when =2, the Black-Scholes model overprices (underprices) the options price for a negative (positive) on the left of the homo-bias point, whereas the model underprices (overprices) for a negative (positive) on the right of the homo-bias point. For = 1.5 and 1.0, the overpricing areas shift to the right when the value of increases from to 0.25. The degree of underpricing or overpricing increases when decreases from 2.0 to 1.0. For the impact of kurtosis, when = 0, the Black-Scholes model underprices the options price on the left of the left homo-bias point and on the right of the right homo-bias point, and overprices between these two points. For = (0.2), the overpriced areas shift to the right (left) and then increase in size when decreases from 2.0 to 1.0. This survey will help explain the various known Black-Scholes biases.
||TABLE OF CONTENTS
ABSTRACT IN CHINESE ii
ABSTRACT IN ENGLISH iv
LIST OF TABLES ix
LIST OF FIGURES x
PART I Value-at-Risk Forecasts in Gold Market under Oil Shocks 1
1. Introduction 3
1.1 Motivations and Objectives 3
1.2 Flow Chart 4
2. Literature Review 5
2.1 Literature review of the gold market 5
2.2 Volatility measurement in the oil market 6
3. Econometric Methodology 8
3.1 Autoregressive Conditional Jump Intensity (ARJI) model for crude oil Volatility 8
3.2 The variety of BHK model 10
4. Model-based VaR Estimates and Evaluation Methods 13
4.1 Value-at-Risk (VaR) definition 13
4.2 Evaluation Methods 13
4.2.1 General Loss Functions 13
4.2.2 Binary Loss Function 14
4.2.3 Quadratic Loss Function 14
4.2.4 LR test of unconditional coverage 14
5. Empirical Results and Analysis 16
5.1 Data and descriptive statistics 16
5.2 Empirical Results 17
5.2.1 The empirical results of ARJI model for crude oil volatility 17
5.2.2 The empirical results of six different models for gold returns 18
5.2.3 The measurements of out-of-sample VaR forecasting 19
6. Conclusions 26
PART II Value-at-Risk Forecasts in U.S. Crude Oil Market with Skewed Generalized Error Distributions 29
1. Introduction 31
1.1 Motivations and Objectives 31
1.2 Flow Chart 32
2. Literature Review 33
3. Econometric Methodology 35
3.1 ARJI-N model 35
3.2 ARJI-SGED model 37
4. Model-based VaR Estimates and Evaluation Methods 39
4.1 Value-at-Risk (VaR) definition 39
4.2 Evaluation Methods 39
4.2.1 Binary Loss Function 40
4.2.2 Quadratic Loss Function 40
4.2.3 LR test of unconditional coverage 40
5. Empirical Results. And Analysis 42
5.1 Data and descriptive statistics 42
5.2 Empirical Results 43
5.2.1 The empirical results of ARJI-N and ARJI-SGED models for Brent and WTI crude oil price 43
5.2.2 The measurements of out-of-sample VaR forecasting 55
6. Conclusions 62
Appendix A 65
PART III Options Pricing with Skewed Generalized Error Distributions 66
1. Introduction 68
1.1 Motivations and Objectives 68
1.2 Flow Chart 69
2. Literature Review 70
3. SGED Distribution and its Degenerative Distributions 73
3.1 Skewed Generalized Error Distribution (SGED) 73
3.2 Skew Laplace Distribution (SLD) 74
3.3 Skew Normal Distribution (SND) 75
3.4 Generalized Error Distribution (GED) 75
4. Options Pricing under SGED Distribution 77
4.1 Geometric Brownian motion 77
4.2 Options Pricing under SGED Distribution 78
5. Numerical Results and Analysis 81
5.1 The impact of kurtosis on Black-Scholes biases 82
5.1.1 kappa=2.0 (SND) 82
5.1.2 kappa=1.5 86
5.1.3 kappa=1.0 (SLD) 89
5.2 The impact of skewness on Black-Scholes biases 93
5.2.1 lambda=0.0 (GED) 93
5.2.2 lambda=-0.2 97
5.2.3 lambda=0.2 101
6. Conclusions 104
LIST OF TABLES
Table 1-1. Descriptive statistics of daily return 16
Table 1-2. Empirical results of the ARJI model for crude oil 18
Table 1-3. Empirical results of various models 19
Table 1-4. The out-of-sample forecasting of VaR 21
Table 2-1. Descriptive statistics of daily return(in sample period) 42
Table 2-2. Estimation results of ARJI-N and ARJI-SGED models for Brent and
WTI crude oil 44
Table 2-3. Descriptive statistics of model parameters during rolling period for
Brent crude oil 46
Table 2-4. Descriptive statistics of model parameters during rolling period for
WTI crude oil 47
Table 2-5. Out-of-sample VaR performance 57
Table 2-A1. Quantiles of SGED distribution with various combinations (kappa,lambda)
at alternate levels 65
Table 3-1A. Option Price with % in-the-Money and Skewness(Skew Normal
Table 3-1B Black-Scholes Bias in $ due to Skew Normal Distribution 83
Table 3-1C Black-Scholes Bias in % due to Skew Normal Distribution 84
Table 3-2A. Option Price with % in-the-Money and Skewness(Skew Laplace
Table 3-2B Black-Scholes Bias in $ due to Skew Laplace Distribution 90
Table 3-2C Black-Scholes Bias in % due to Skew Laplace Distribution 90
Table 3-3A. Option Price with % in-the-Money and Kappa(GED Distribution) 94
Table 3-3B Black-Scholes Bias in $ due to GED Distribution 94
Table 3-3C Black-Scholes Bias in % due to GED Distribution 95
Table 3-4A Option Price with % in-the-Money and Kappa (SGED Distribution) 98
Table 3-4B. Black-Scholes Bias in $ due to SGED Distribution (lambda=-0.2) 98
Table 3-4C. Black-Scholes Bias in % due to SGED Distribution (lambda=-0.2) 98
LIST OF FIGURES
Figure 1-1.The time series plot of gold and crude oil prices 17
Figure 1-2. Conditional Variance Components, Crude Oil 18
Figure 1-3. Returns and VaR forecasts at different confidence levels with a sequence of GARCH, GARCH-HV, GARCH-HV-A, PGARCH, PGARCH-HV, and PGARCH-HV-A model 23
Figure 1-4. Returns and VaR forecasts under each model at a sequence of 90% , 95%, 99%, and 99.5%confidence levels 25
Figure 2-1. The time series plot of Brent and WTI crude oil price 43
Figure 2-2. The trend of parameters estimates of ARJI-N and ARJI-SGED models during rolling period for Brent crude oil 51
Figure 2-3. The trend of parameters estimates of ARJI-N and ARJI-SGED models during rolling period for WTI crude oil 55
Figure 2-4. Returns and VaR forecasts under ARJI-N and ARJI-SGED models at a sequence of 90%, 95%, and 99% confidence levels for Brent Crude oil 58
Figure 2-5. Returns and VaR forecasts under ARJI-N and ARJI-SGED models at a sequence of 90%, 95%, and 99% confidence levels for WTI Crude oil 59
Figure 2-6. Returns and VaR forecasts at different confidence levels with a sequence of ARJI-N and ARJI-SGED model for Brent Crude oil 60
Figure 2-7. Returns and VaR forecasts at different confidence levels with a sequence of ARJI-N and ARJI-SGED model for WTI Crude oil 61
Figure 3-1. Black-Scholes Bias in $ due to kappa=2.0 85
Figure 3-2. Black-Scholes Bias in % due to kappa=2.0 86
Figure 3-3. Black-Scholes Bias in $ due to kappa=1.5 88
Figure 3-4. Black-Scholes Bias in % due to kappa=1.5 89
Figure 3-5. Black-Scholes Bias in $ due to kappa=1.0 92
Figure 3-6. Black-Scholes Bias in % due to kappa=1.0 93
Figure 3-7. Black-Scholes Bias in $ due to lambda=0.0 96
Figure 3-8. Black-Scholes Bias in % due to lambda=0.0 97
Figure 3-9. Black-Scholes Bias in $ due to lambda=-0.2 100
Figure 3-10. Black-Scholes Bias in % due to lambda=-0.2 101
Figure 3-11. Black-Scholes Bias in $ due to lambda=0.2 103
Figure 3-12. Black-Scholes Bias in % due to lambda=0.2 103
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