This paper analyzed and experimented with an importance sampling technique for estimates of value-at-risk and expected shortfall by approximating the portfolio loss using simulated upper-tail probability, and simulated conditional mean of a portfolio loss, ; . First we apply spectral decomposition to the covariance matrix of underlying risk factors to decide the key factor (c1) in relation to the correlated underlying risk factors. Then we propose a two-step procedure to generate
X. We can sample random variates Z first and then find Z
1. Given simulated loss value of the option portfolio driven byZ , portfolio loss L is a strictly monotonic
11
function in relation to Z1. Based on the different relationships, an importance sampling change of measure can then be determined by solving Z1
. The method was
applied to option portfolios with different characteristics that can prove its efficiency.Our numerical results show that the derived VaR and expected shortfall are very close to the naïve Monte Carlo method in all cases considered, and produces large variance reductions; more than two orders of magnitude and often more than three orders of magnitude improvements are obtained, which demonstrate dramatic variance reduction for extreme portfolio loss. Our algorithm shows that the proposed estimator has constant coefficient of variation, which suggests the proposed estimator has bounded relative error.
Under our framework, in practical all parameters necessary for pricing of the instrument and measuring the VaR and expected shortfall are available from the trading and valuation system at the front office in a financial institution. The nature of our algorithm is straightforward and easy for realizing the practical needs of risk management. Therefore, we conclude our algorithm is an efficient vehicle for measuring the option portfolio loss.
7. REFERENCES
1. Acerbi, C. and D. Tasche. "On the Coherence of Expected Shortfall." Journal of Banking and Finance, 26 (2002), pp. 1487-1503.
2. Alexander, C. Market Risk Analysis, value-at-risk Models: Wiley 2009.
3. Artzner, P., F. Delbaen, J. M. Eber, et al. "Think Coherently." Risk, 10 (1997), pp.
68-71.
4. Artzner, P., F. Delbaen, J. M. Eber, et al. "Coherent Measure of Risk."
Mathematical Finance, 9 (1999), pp. 203-228.
5. Basel Committee on Banking Supervision. Amendment to the capital accord to incorporate market risks, 1996. Available from http://www.bis.org (accessed November 2011).
6. Asmussen, S. and P. Glynn. Stochastic Simulation: Algorithms and Analysis:
Springer 2007.
7. Chen, Z. and P. Glasserman. "Fast pricing of basket default swaps." Operations Research Proceedings 2005, 56, 2 (2008), pp. 286-303.
8. Chiang, M.-H., M.-L. Yueh and M.-H. Hsieh. "An efficient algorithm for basket default swap valuation." The Journal of Derivatives, 15, 2 (2007), pp. 8-19.
9. Glasserman, P., P. Heidelberger and P. Shahabuddin. "Asymptotically optimal importance sampling and stratification for pricing path-dependent options."
Mathematical Finance, 9 (1999a), pp. 117-152.
10. Glasserman, P., P. Heidelberger and P. Shahabuddin. Importance Sampling and Stratification for Value-at-Risk. Proceedings of the Sixth International Conference Computational Finance, In Y.S. Abu Mostafa, B. LeBaron, A.W. Lo, A.S.
Weigend, eds., MIT Press, 1999b.
11. Glasserman, P., P. Heidelberger and P. Shahabuddin. "Variance reduction techniques for estimating value-at-risk." Management Science, 46 (2000), pp.
1349-1364.
12. Glasserman, P., P. Heidelberger and P. Shahabuddin. "Portfolio Value-At-Risk with Heavy-Tailed Risk Factors." Mathematical Finance, 12 (2002), pp. 239-269.
13. Glasserman, P. and Y. Wang. "Counterexamples in importance sampling for rare event probabilities." The Annals of Applied Probability, 7 (1997), pp. 731-746.
14. Heidelberger, P. "Fast simulation of rare events in queueing and reliability models." ACM Trans. Model. Comput. Simul., 5, 1 (1995), pp. 43-85.
15. Jorion, P. value-at-risk: The New Benchmark for Controlling Derivatives Risk, New York: McGraw-Hill 1997.
16. Joshi, M. and D. Kainth. "Rapid and accurate development of prices and Greeks for nth to default credit swaps in the Li model." Quantitative Finance, 4, 3 (2004), pp. 266-275.
17. RiskMetrics Technical Document, Morgan, J. P., 1996.
18. Rouvinez, C. "Going Greek with VAR." Risk, 10, 2 (1997), pp. 57-65.
19. Wilson, T. value-at-risk, Wiley, Chichester, England 1999.
20. Yueh, M.-L. and M. C. W. Wong. "Analytical VaR and Expected Shortfall for Quadratic Portfolios." Journal of Derivatives, 2010, 1 (2010), pp. 1-12.
ge 國科會補助專題研究計畫項下出席國際學術會議心得報告
(英文) Longevity 7: Seventh International Longevity Risk andCapital Markets Solutions Conference
發表論 文題目
(中文)使用保單折現進行保險公司之死亡風險避險：資產負債管 理方法
(英文) Using Life Settlements to Hedge the Mortality Risk of the Life Insurers: An Asset-Liability Management Approach"
二、與會心得
在session 1主要是討論風險管理的議題，我聆聽了Elisa Luciano、Luca Regisz、與Elena Vignar共同研究發表的「Delta and Gamma Hedging of Mortality and Interest Rate Risk」，這篇論文研究年金現金流量的避險問題，並假設死亡率 與利率均為隨機過程，他們使用delta-gamma的避險技術來處理死亡風險，並藉 由分析模擬風險因子的變化，來了解年金的價值變化。
在第Session 2，則是由我和政大風管系老師共同發表「Using Life Settlements to Hedge the Mortality Risk of the Life Insurers: An Asset-Liability Management Approach」，我們提出一個基於避險方法基礎，以評估保單貼現避險效率的研究 方法，來分析保險公司保單貼現商品對於死亡風險避險的效果，我們的數值方法 證明保險公司可以有效的利用保單貼現來處理死亡風險，這篇論文使用保單貼現 做為避險工具是較為先進的做法，也因此引起與會的學者不少互動，對於本研究 未來的發展有相當的助益。
在session 3我則選擇聆聽由Rui Zhou，Johnny Siu-Hang Liy，Ken Seng Tan共 同發表的「A Two-Population Mortality Model with Transitory Jump Effects」，這 篇論文討論死亡率的動態隨機過程存在著跳躍＂jump＂的影響，這些jump會顯 著的影響連結死亡率的證券價格，因此也必須在模型中考量。這篇論文強調過去 的模型均僅考量單一族群基差風險，因此在不同族群間的模型適用性未曾有過研 究探討，這篇論文提出兩個版本的模型來分析這些不同族群之間的jumps規則，
對於模型的假設等有不少值得深入了解的地方，我也提出問題與一些模型上的建 議予發表論文的學者共同討論。
三、考察參觀活動(無是項活動者略) 無
四、建議 無
五、攜回資料名稱及內容 (1) 大會議程
(2) 大會論文集光碟 六、其他
無
國科會補助計畫衍生研發成果推廣資料表
日期:2012/01/09
國科會補助計畫
計畫名稱: 投資組合信用風險，投資組合市場風險與路徑相依選擇權評價之快速蒙地卡 羅演算法研究
計畫主持人: 謝明華
計畫編號: 99-2410-H-004-086- 學門領域: 財務
無研發成果推廣資料
99 年度專題研究計畫研究成果彙整表
其他成果
(
無法以量化表達之成果如辦理學術活動、獲 得獎項、重要國際合 作、研究成果國際影響 力及其他協助產業技 術發展之具體效益事 項等，請以文字敘述填 列。)
預計發表於國際期刊之撰寫中的 Working Paper：
1. Hsieh, Ming-hua, Liao, Wei-cheng, Chen, Chung-lung A Fast Algorithm for Estimating
Value-at-Risk and Expected Shortfall, Working Paper, 2011.
2. Hsieh Ming-hua, So-de Shyu, Yi-Hsi Lee, and Yu-Fen Chiu Fast simulation of portfolio credit
risk under general multifactor copula models, Working Paper, 2011.
成果項目 量化 名稱或內容性質簡述
測驗工具(含質性與量性)
0課程/模組
0電腦及網路系統或工具
0教材
0舉辦之活動/競賽
0研討會/工作坊
0電子報、網站
0科 教 處 計 畫 加 填 項
目 計畫成果推廣之參與（閱聽）人數
0國科會補助專題研究計畫成果報告自評表
請就研究內容與原計畫相符程度、達成預期目標情況、研究成果之學術或應用價 值（簡要敘述成果所代表之意義、價值、影響或進一步發展之可能性） 、是否適 合在學術期刊發表或申請專利、主要發現或其他有關價值等，作一綜合評估。
1. 請就研究內容與原計畫相符程度、達成預期目標情況作一綜合評估
■達成目標
□未達成目標（請說明，以 100 字為限）
□實驗失敗
□因故實驗中斷
□其他原因 說明：
2. 研究成果在學術期刊發表或申請專利等情形：
論文：□已發表 □未發表之文稿 ■撰寫中 □無 專利：□已獲得 □申請中 ■無
技轉：□已技轉 □洽談中 ■無 其他：（以 100 字為限）
3. 請依學術成就、技術創新、社會影響等方面，評估研究成果之學術或應用價 值（簡要敘述成果所代表之意義、價值、影響或進一步發展之可能性）（以 500 字為限）
This project analyzed and experimented with an importance sampling technique for estimates of value-at-risk and expected shortfall by approximating the portfolio loss using simulated upper-tail probability, P(L>b) and simulated conditional mean of a portfolio loss, E[L；L>b]. First we apply spectral decomposition to the covariance matrix of underlying risk factors to decide the key factor (c1) in relation to the correlated underlying risk factors. Then we propose a two-step procedure to generate X. We can sample random variates first and then find Z1.
Given simulated loss value of the option portfolio driven by , portfolio loss L is a strictly monotonic function in relation to Z1. Based on the different relationships, an importance sampling change of measure can then be determined by solving Z1. The method was applied to option portfolios with different characteristics that can prove its efficiency.
Our numerical results show that the derived VaR and expected shortfall are very close to the naï；ve Monte Carlo method in all cases considered, and produces large variance reductions； more than two orders of magnitude and often more than three orders of magnitude improvements are obtained, which demonstrate dramatic variance reduction for extreme portfolio loss. Our algorithm shows that the
proposed estimator has bounded relative error.
Under our framework, in practical all parameters necessary for pricing of the instrument and measuring the VaR and expected shortfall are available from the trading and valuation system at the front office in a financial institution. The nature of our algorithm is straightforward and easy for realizing the practical needs of risk management. Therefore, we conclude our algorithm is an efficient vehicle for measuring the option portfolio loss.