金融商品風險值之估計與預測

The estimation and forecasting of Value-at-Risk for financial commodities

蘇榮斌(Jung-Bin Su)

指導教授 ： 李命志 ； 邱建良

淡江大學/商學院/財務金融學系博士班/博士(2008年)

https://doi.org/10.6846/TKU.2008.00667

摘要

本論文著重在風險值的衡量與選擇權定價，共包含三個部份，分別為「石油衝擊下預測黃金市場之風險值」、「一般化偏態誤差分配預測美國原油市場之風險值」與「一般化偏態誤差分配的選擇權定價」。將此三部份的內容簡述如下： 第一部分在考慮石油波動性與使用較彈性化模型如BHK-PGARCH-HV等一系列模型來探討黃金市場之風險值。石油波動性採用ARJI模型估計並將此分成隨機與跳躍波動。最後以滾動視窗法預測樣本外之風險值，由實證結果可知將波動分成隨機與跳躍並考慮高波動之BHK-PGARCH-HV模型預測能力最佳，也就是說高波動與跳躍波動在黃金市場之風險值預測時是不可以忽略。 第二部份中我們以複合辛普森數值積分法，來計算一般化偏態誤差分配(SGED)之尾部臨界值，並使用ARJI-N 和ARJI-SGED模型來對布倫特與西德州原油日現貨價預測樣本外之風險值。由實證結果可知布倫特原油呈現稍微左偏而西德州原油呈現稍微右偏，因此ARJI-N模型會高估布倫特原油真實的風險值而會低估西德州原油真實的風險值。此發現可說明一般化偏態誤差分配在美國原油市場樣本外風險值預測之重要性。 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。

關鍵字

風險值 ； 黃金 ； 油波動性 ； 自我迴歸跳躍強度模型 ； 一般化偏態誤差分配 ； 複合辛普森數值積分法 ； 選擇權定價 ； Black-Scholes偏誤 ； 齊質偏誤點

並列摘要

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.

並列關鍵字

Value-at-risk ； Gold ； Oil volatility ； ARJI ； PGARCH ； BHK ； SGED ； Composite Simpson’s rule ； Option Pricing ； Black-Scholes biases ； homo-bias point

參考文獻

1. Bahmani-Oskooee, M. and Brown, F. (2004) Kalman Filter Approach to Estimate the Demand for International Reserves, Applied Economics, 36, 1655-1668.

2. Baker, S. A. and Van-Tassel, R. C. (1985) Forecasting the Price of Gold: A Fundamental Approach, Journal of Atlantic Economics, 13, 43-52.

3. Bollerslev, T. (1986) Generalized Autoregressive Conditional Heteroskedasticity, Journal of Economics, 31, 307-327.

5. Cai, J., Cheung, Y. L. and Wong, M. C. S. (2001) What Moves the Gold Market? Journal of Futures Markets, 21, 257-278.

6. Chan, W. H. and Maheu, J. M. (2002) Conditional Jump Dynamics in Stock Market Returns, Journal of Business Economics Statistics, 20, 377-389.