在金融商品的衡量中，風險值(VaR)成為了近年來大家關注的一項指標。用以了解投資風險以便做好風險的規避。在風險值模擬中，無母數方法裡的蒙地卡羅模擬法(Monte Carlo Method , MC)為電腦隨機抽取的亂數，加入到價格模擬的隨機過程裡，且無任何模型上的假設，故須承擔模型之風險，較能因應市場的變化。但由於電腦隨機抽取的亂數，容易發生亂數聚集性，而影響了估計的穩定性。為改善此問題，在亂數模擬的部份改以低差異性數列去產生亂數值，稱之為準蒙地卡羅模擬法(Quasi-Monte Carlo Method , Q-MC)，並舉出常見的兩個低差異性數列Halton數列及Sobol數列。在給定不同的衡量準則下，比較其差異。本文模擬的結果顯示，低差異性數列中之Sobol數列，其亂數本身的差異性小，在低差異數列中為較適合的估計風險值的模擬法，且與風險值真值的差距也是最接近的。有效的改進了傳統蒙地卡羅模擬法的缺點，使應用電腦模擬風險值更為穩定和精確。

VaR (Value-at-Risk) has been used as an indicator to respond to the market risk and certainly caused a revolution in risk management.  It has drawn a lot of attention especially after the Orange County and many others events. Therefore how to estimate the true VaR has become an important issue.
Monte Carlo method is one of the methods to estimate VaR. It is done by computer simulation. Though it is the most powerful method, Monte Carlo method is always accompanied with lengthy computation time and subject to model risk of stochastic processes assumed. Quasi-Monte Carlo Method can be another alternative method to overcome this efficiency disadvantage by incorporating the Low Discrepancy Sequences in generating random number. Two commonly used sequences, Halton and Sobol, along with naive Monte Carlo Method are used to study the VaR estimation problem in this research.
It is found that Sobol Sequences of the Low Discrepancy Sequences has smaller MSE and better effientcy among three estimation methods.

目 錄	I
圖 目 錄	III
表 目 錄	III
第一章 緒論	1
1.1 風險值研究背景與動機	1
1.2 研究目的	2
1.3 研究架構	3
第二章 文獻探討	4
2.1風險值(Value-at-Risk ,VaR)	4
2.2 VaR的估計方式	7
2.2.1 變異數-共變異數法(Variance-Covariance Method)	9
2.2.2 歷史模擬法(Historical Simulation Method)	11
2.2.3 蒙地卡羅模擬法(Monte Carlo Method , MC)	13
2.2.4 準蒙地卡羅模擬法(Quasi-Monte Carlo Method , Q-MC)	14
第三章 研究方法	15
3.1 資產組合風險值(Portfolio VaR)	15
3.2 幾何布朗運動(Geometric Brownian Motion,GBM)	15
3.3 以蒙地卡羅模擬法估計資產組合風險值	20
3.3.1 蒙地卡羅模擬法模擬步驟	20
3.3.2 蒙地卡羅模擬法估計資產組合風險值演算法	21
3.4 以準蒙地卡羅模擬法估計資產組合風險值	22
3.4.1 Halton數列及其VaR估計演算法	23
3.4.2 Sobol數列及其VaR演算法	25
3.4.3 低差異性數列的改進	30
3.5 方法評估準則(Methods Assessment)	31
3.5.1 相對有效性(Relative Efficiency , RE)	31
3.5.2 誤差均方根(Root-Mean-Square Error , RMSE)	32
3.5.3 回溯測試(Back Testing)	33
第四章 研究結果	36
4.1 問題設定	36
4.2 模擬結果與比較	36
4.3 回溯測試	43
第五章 結論	45
參考文獻	47
附錄	49
附錄一 VaR 模擬程式	49
附錄二 回溯測試30天風險值之模擬	52
圖 目 錄
圖2.1  VaR定義示意圖	5
圖2.2  相對風險值(relative VaR)示意圖	6
圖2.3  VaR估計方法	8
圖3.1  GBM路徑模擬	19
圖4.1  RMSE模擬圖(m=10,20)	41
圖4.2  RMSE模擬圖(m=30,50)	42
圖4.3  RMSE模擬圖(m=80,100)	42

表 目 錄
表3.1  GBM價格路徑模擬	19
表3.2  b=2，前15個Halton模擬值	24
表3.3  前五個direction number產生表	27
表3.4  一維Sobol數列模擬值	29
表4.1  MC數列模擬值	38
表4.2  Halton數列模擬值	39
表4.3  Sobol數列模擬值	40
表4.4  MC、Sobol數列與Halton數列之RMSE	41
表4.5  回溯測試表	44

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