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題名 偏常態因子信用組合下之效率估計值模擬
Efficient Simulation in Credit Portfolio with Skew Normal Factor作者 林永忠
Lin, Yung Chung貢獻者 劉惠美
Liu, Hui Mei
林永忠
Lin, Yung Chung關鍵詞 蒙地卡羅模擬
重點採樣法
信用風險組合
變異縮減
Monte Carlo Simulation
Importance Sampling
Portfolio credit risk
Variance reduction日期 2011 上傳時間 30-十月-2012 11:20:44 (UTC+8) 摘要 在因子模型下，損失分配函數的估算取決於混合型聯合違約分配。蒙地卡羅是一個經常使用的計算工具。然而，一般蒙地卡羅模擬是一個不具有效率的方法，特別是在稀有事件與複雜的債務違約模型的情形下，因此，找尋可以增進效率的方法變成了一件迫切的事。對於這樣的問題，重點採樣法似乎是一個可以採用且吸引人的方法。透過改變抽樣的機率測度，重點採樣法使估計量變得更有效率，尤其是針對相對複雜的模型。因此，我們將應用重點採樣法來估計偏常態關聯結構模型的尾部機率。這篇論文包含兩個部分。Ⅰ：應用指數扭轉法---一個經常使用且為較佳的終點採樣技巧---於條件機率。然而，這樣的程序無法確保所得的估計量有足夠的變異縮減。此結果指出，對於因子在選擇重點採樣上，我們需要更進一步的考慮。Ⅱ：進一步應用重點採樣法於因子；在這樣的問題上，已經有相當多的方法在文獻中被提出。在這些文獻中，重點採樣的方法可大略區分成兩種策略。第一種策略主要在選擇一個最好的位移。最佳的位移值可透過操作不同的估計法來求得，這樣的策略出現在Glasserman等(1999)或Glasserman與Li (2005)。第二種策略則如同在Capriotti (2008)中的一樣，則是考慮擁有許多參數的因子密度函數作為重點採樣的候選分配。透過解出非線性優化問題，就可確立一個未受限於位移的重點採樣分配。不過，這樣的方法在尋找最佳的參數當中，很容易引起另一個效率上的問題。為了要讓此法有效率，就必須在使用此法前，對參數的穩健估計上，投入更多的工作，這將造成問題更行複雜。本文中，我們說明了另一種簡單且具有彈性的策略。這裡，我們所提的演算法不受限在如同Gaussian模型下決定最佳位移的作法，也不受限於因子分配函數參數的估計。透過Chiang, Yueh與Hsie (2007)文章中的主要概念，我們提供了重點採樣密度函數一個合理的推估並且找出了一個不同於使用隨機近似的演算法來加速模擬的進行。最後，我們提供了一些單因子的理論的證明。對於多因子模型，我們也因此有了一個較有效率的估計演算法。我們利用一些數值結果來凸顯此法在效率上，是遠優於蒙地卡羅模擬。
Under a factor model, computation of the loss density function relies on the estimates of some mixture of the joint default probability and joint survival probability. Monte Carlo simulation is among the most widely used computational tools in such estimation. Nevertheless, general Monte Carlo simulation is an ineffective simulation approach, in particular for rare event aspect and complex dependence between defaults of multiple obligors. So a method to increase efficiency of estimation is necessary. Importance sampling (IS) seems to be an attractive method to address this problem. Changing the measure of probabilities, IS makes an estimator to be efficient especially for complicated model. Therefore, we consider IS for estimation of tail probability of skew normal copula model. This paper consists of two parts. First, we apply exponential twist, a usual and better IS technique, to conditional probabilities and the factors. However, this procedure does not always guarantee enough variance reduction. Such result indicates the further consideration of choosing IS factor density.Faced with this problem, a variety of approaches has recently been proposed in the literature ( Capriotti 2008, Glasserman et al 1999, Glasserman and Li 2005). The better choices of IS density can be roughly classified into two kinds of strategies. The first strategy depends on choosing optimal shift. The optimal drift is decided by using different approximation methods. Such strategy is shown in Glasserman et al 1999, or Glasserman and Li 2005.The second strategy, as shown in Capriotti (2008), considers a family of factor probability densities which depend on a set of real parameters. By formulating in terms of a nonlinear optimization problem, IS density which is not limited the determination of drift is then determinate. The method that searches for the optimal parameters, however, incurs another efficiency problem. To keep the method efficient, particular care for robust parameters estimation needs to be taken in preliminary Monte Carlo simulation. This leads method to be more complicated.In this paper, we describe an alternative strategy that is straightforward and flexible enough to be applied in Monte Carlo setting. Indeed, our algorithm is not limited to the determination of optimal drift in Gaussian copula model, nor estimation of parameters of factor density. To exploit the similar concept developed for basket default swap valuation in Chiang, Yueh, and Hsie (2007), we provide a reasonable guess of the optimal sampling density and then establish a way different from stochastic approximation to speed up simulation.Finally, we provide theoretical support for single factor model and take this approach a step further to multifactor case. So we have a rough but fast approximation that execute entirely with Monte Carlo in general situation. We support our approach by some portfolio examples. Numerical results show that such algorithm is more efficient than general Monte Carlo simulation.參考文獻 [1] Anderson, L, and J. Sidenius., “ Extensions to the Gaussian Copula: Random Recovery and Random Factor Loadings.” Journal of Credit Risk, 1, No.1, pp. 29-70, 2005.[2] Arnold, B. C, and G. D. Lin, “Characterizations of the Skew-Normal and Generalized Chi Distribution.” Sankhyã : The Indian Journal of Statistics, 66, No.4, pp. 593-606, 2004.[3] Azzalini, A, “A Class of Distributions Which Include the Normal Ones.” Scandinavian Journal of Statistics, 12, pp. 171-178, 1985.[4] Bassamboo, A, S. Juneja, and A. Zeevi, “Portfolio Credit Risk with Extremal Dependence: Asymptotic Analysis and Efficient Simulation.” Operations Reseach, 56, pp. 593-606, 2008.[5] Bluhm, C, L. Overbeck, and C. Wagner, “An Introduction to Credit Risk Modeling.” CRC Press/Chapman & Hall, 2002.[6] Bucklew, J. A, Large Deviation Techniques in Decision, Simulation, and Estimation, John Wiley, New York, 1990.[7] Capriotti, L, “Least-Square Importance Sampling for Monte Carlo Security Pricing.” Quantitative Finance, 8, No.5, pp 485-497, 2008.[8] Chiang, M. H, M. L. Yueh, and M.H. Hsieh, “An Efficient Algorithm for Basket Default Swap Valuation.” Journal of Derivatives, pp. 8-19, 2007.[9] Crouhy, M, D. Galai, and R. Mark, “A Comparative Analysis of Current Credit Risk Models.” Journal of Banking and Finance, 24, pp. 59-117, 2000.[10] Glasserman, P, "Tail Approximations for Portfolio Credit Risk." Journal of Derivatives, 12, No. 2, pp. 24-42, 2004.[11] Glasserman, P, and J. Li, “Importance Sampling for Portfolio Credit Risk.” Management Science, 55, No.11, pp. 1643-1656, 2005.[12] Glasserman, P, P. Heidelberger, and P. Shahabuddin, "Asymptotically Optimal Importance Sampling and Stratification for Pricing Path-Dependent Options." Mathematical Finance. 9, No. 2, pp. 117-152, 1999.[13] Glasserman, P, W. Kang, and P. Shahabuddin, "Fast Simulation of Multifactor Portfolio Credit Risk." Operations Research. 56, No.5, pp. 1200-1217, 2008.[14] Graciela González-Farı́as., D. M. Armando, A. K. Gupta, “Additive Properties of Skew Normal Random Vectors.” Journal of Statistical Planning and Inference. 126, pp. 521-534, 2004.[15] Gupta, A. K, T. T. Nguyen, and J. A. Sanqui, “Characterization of the Skew- Normal Distribution.” Annals of the Institute of Statistical Mathematics, 56, pp. 351-360, 2004.[16] Gupta, G, C. Finger, M. Bhatia. CreditMetrics Technical Document. J.P.Morgan & CO, New York, 1997.[17] Hull, J, and A. White, “Valuation of CDO and nth to Default CDS without Monte Carlo Simulation.” Journal of Derivarives, 12, No. 2, pp. 8-23, 2004.[18] Johnson, N. L, S. Kotz, A. W. Kemp, Univariate Discrete Distributions, 2nd ed. John Wiley and Sons, New York, 1992.[19] Kostadinov, K, “Tail Approximation for Credit Risk Portfolios with Heavy-Tailed Risk Factors.” Technique Report, The Munich University of Technology, 2005.[20] Li, D,”On Default Correlation: A Copula Function Approach.” Journal of Fixed Income, 9, pp. 43-54, 2000.[21] Lucas, A, P. Klaassen, P. Spreij, and S. Straetmans, “Tail Behavior of Credit Loss Distributions for General Latent Factor Models.“ Applied Mathematical Finance. 10, No. 4, pp. 337-357, 2003.[22] McNeil, A.J, R. Frey, and P. Embrechts, Quantitative Risk Management: Concept, Techniques, and Tools. Princeton University Press, 2005.[23] Ney, P, “Dominating Points and the Asymptotics of the Large Deviations for Random Walk on Rd.” Annals of Probability. 11, No. 1, pp. 158-167, 1983.[24] O’Hagan and T. Leonard, “Bayes Estimation Subject to Uncertainty about Parameter Constraints.” Biometrika, 63, pp. 201-202, 1976.[25] Sadowsky, J. S., J. A. Bucklew, “On Large Deviations Theory and Asymptotically Efficient Monte Carlo Estimation.” IEEE Transactions on Information Theory, 36, No.3, May, pp. 579-588, 1990.[26] Schloegl, L, and D. O’Kane, “A Note on the Large Homogenous Portfolio Approximation with the Student-t Copula.” Finance and Stochastics, 9, pp. 577-584, 2005.[27] Sen, P. K, J. M. Singer, Large Sample Methods in Statistics. Chapman & Hall, London, 1993.[28] Shao, J, Mathematical Statistics. Spring-Verlag, New York, 1998.[29] Williams,D, Probability with Martingales. Cambridge: Cambridge University Press, 1991. 描述 博士
國立政治大學
統計研究所
92354502
100資料來源 http://thesis.lib.nccu.edu.tw/record/#G0923545022 資料類型 thesis dc.contributor.advisor 劉惠美 zh_TW dc.contributor.advisor Liu, Hui Mei en_US dc.contributor.author (作者) 林永忠 zh_TW dc.contributor.author (作者) Lin, Yung Chung en_US dc.creator (作者) 林永忠 zh_TW dc.creator (作者) Lin, Yung Chung en_US dc.date (日期) 2011 en_US dc.date.accessioned 30-十月-2012 11:20:44 (UTC+8) - dc.date.available 30-十月-2012 11:20:44 (UTC+8) - dc.date.issued (上傳時間) 30-十月-2012 11:20:44 (UTC+8) - dc.identifier (其他 識別碼) G0923545022 en_US dc.identifier.uri (URI) http://nccur.lib.nccu.edu.tw/handle/140.119/54550 - dc.description (描述) 博士 zh_TW dc.description (描述) 國立政治大學 zh_TW dc.description (描述) 統計研究所 zh_TW dc.description (描述) 92354502 zh_TW dc.description (描述) 100 zh_TW dc.description.abstract (摘要) 在因子模型下，損失分配函數的估算取決於混合型聯合違約分配。蒙地卡羅是一個經常使用的計算工具。然而，一般蒙地卡羅模擬是一個不具有效率的方法，特別是在稀有事件與複雜的債務違約模型的情形下，因此，找尋可以增進效率的方法變成了一件迫切的事。對於這樣的問題，重點採樣法似乎是一個可以採用且吸引人的方法。透過改變抽樣的機率測度，重點採樣法使估計量變得更有效率，尤其是針對相對複雜的模型。因此，我們將應用重點採樣法來估計偏常態關聯結構模型的尾部機率。這篇論文包含兩個部分。Ⅰ：應用指數扭轉法---一個經常使用且為較佳的終點採樣技巧---於條件機率。然而，這樣的程序無法確保所得的估計量有足夠的變異縮減。此結果指出，對於因子在選擇重點採樣上，我們需要更進一步的考慮。Ⅱ：進一步應用重點採樣法於因子；在這樣的問題上，已經有相當多的方法在文獻中被提出。在這些文獻中，重點採樣的方法可大略區分成兩種策略。第一種策略主要在選擇一個最好的位移。最佳的位移值可透過操作不同的估計法來求得，這樣的策略出現在Glasserman等(1999)或Glasserman與Li (2005)。第二種策略則如同在Capriotti (2008)中的一樣，則是考慮擁有許多參數的因子密度函數作為重點採樣的候選分配。透過解出非線性優化問題，就可確立一個未受限於位移的重點採樣分配。不過，這樣的方法在尋找最佳的參數當中，很容易引起另一個效率上的問題。為了要讓此法有效率，就必須在使用此法前，對參數的穩健估計上，投入更多的工作，這將造成問題更行複雜。本文中，我們說明了另一種簡單且具有彈性的策略。這裡，我們所提的演算法不受限在如同Gaussian模型下決定最佳位移的作法，也不受限於因子分配函數參數的估計。透過Chiang, Yueh與Hsie (2007)文章中的主要概念，我們提供了重點採樣密度函數一個合理的推估並且找出了一個不同於使用隨機近似的演算法來加速模擬的進行。最後，我們提供了一些單因子的理論的證明。對於多因子模型，我們也因此有了一個較有效率的估計演算法。我們利用一些數值結果來凸顯此法在效率上，是遠優於蒙地卡羅模擬。 zh_TW dc.description.abstract (摘要) Under a factor model, computation of the loss density function relies on the estimates of some mixture of the joint default probability and joint survival probability. Monte Carlo simulation is among the most widely used computational tools in such estimation. Nevertheless, general Monte Carlo simulation is an ineffective simulation approach, in particular for rare event aspect and complex dependence between defaults of multiple obligors. So a method to increase efficiency of estimation is necessary. Importance sampling (IS) seems to be an attractive method to address this problem. Changing the measure of probabilities, IS makes an estimator to be efficient especially for complicated model. Therefore, we consider IS for estimation of tail probability of skew normal copula model. This paper consists of two parts. First, we apply exponential twist, a usual and better IS technique, to conditional probabilities and the factors. However, this procedure does not always guarantee enough variance reduction. Such result indicates the further consideration of choosing IS factor density.Faced with this problem, a variety of approaches has recently been proposed in the literature ( Capriotti 2008, Glasserman et al 1999, Glasserman and Li 2005). The better choices of IS density can be roughly classified into two kinds of strategies. The first strategy depends on choosing optimal shift. The optimal drift is decided by using different approximation methods. Such strategy is shown in Glasserman et al 1999, or Glasserman and Li 2005.The second strategy, as shown in Capriotti (2008), considers a family of factor probability densities which depend on a set of real parameters. By formulating in terms of a nonlinear optimization problem, IS density which is not limited the determination of drift is then determinate. The method that searches for the optimal parameters, however, incurs another efficiency problem. To keep the method efficient, particular care for robust parameters estimation needs to be taken in preliminary Monte Carlo simulation. This leads method to be more complicated.In this paper, we describe an alternative strategy that is straightforward and flexible enough to be applied in Monte Carlo setting. Indeed, our algorithm is not limited to the determination of optimal drift in Gaussian copula model, nor estimation of parameters of factor density. To exploit the similar concept developed for basket default swap valuation in Chiang, Yueh, and Hsie (2007), we provide a reasonable guess of the optimal sampling density and then establish a way different from stochastic approximation to speed up simulation.Finally, we provide theoretical support for single factor model and take this approach a step further to multifactor case. So we have a rough but fast approximation that execute entirely with Monte Carlo in general situation. We support our approach by some portfolio examples. Numerical results show that such algorithm is more efficient than general Monte Carlo simulation. en_US dc.description.tableofcontents 中文摘要 iiABSTRACT ivCONTENTS viLIST OF FIGURES viiLIST OF TABLES viiiChapter 1 Introduction 1Chapter 2 Portfolio Credit Risk Models 42.1 The Portfolio Loss Distribution 42.2 Skew Normal Distribution and Its Properties 6Chapter 3 Variance Reduction Methodology 93.1 IS Method 93.2 IS Conditional on SN Factor 123.3 IS For SN Factor 17Chapter 4 The New Method for SN Factor 394.1 Extension of CYH Importance Sampling Algorithm 394.2 The Proposed algorithm for Skew Factor Model 464.3 Asymptotic Optimality 51Chapter 5 Implementation Issues 55Chapter 6 Concluding Remarks 59REFERENCE 61 zh_TW dc.language.iso en_US - dc.source.uri (資料來源) http://thesis.lib.nccu.edu.tw/record/#G0923545022 en_US dc.subject (關鍵詞) 蒙地卡羅模擬 zh_TW dc.subject (關鍵詞) 重點採樣法 zh_TW dc.subject (關鍵詞) 信用風險組合 zh_TW dc.subject (關鍵詞) 變異縮減 zh_TW dc.subject (關鍵詞) Monte Carlo Simulation en_US dc.subject (關鍵詞) Importance Sampling en_US dc.subject (關鍵詞) Portfolio credit risk en_US dc.subject (關鍵詞) Variance reduction en_US dc.title (題名) 偏常態因子信用組合下之效率估計值模擬 zh_TW dc.title (題名) Efficient Simulation in Credit Portfolio with Skew Normal Factor en_US dc.type (資料類型) thesis en dc.relation.reference (參考文獻) [1] Anderson, L, and J. Sidenius., “ Extensions to the Gaussian Copula: Random Recovery and Random Factor Loadings.” Journal of Credit Risk, 1, No.1, pp. 29-70, 2005.[2] Arnold, B. C, and G. D. Lin, “Characterizations of the Skew-Normal and Generalized Chi Distribution.” Sankhyã : The Indian Journal of Statistics, 66, No.4, pp. 593-606, 2004.[3] Azzalini, A, “A Class of Distributions Which Include the Normal Ones.” Scandinavian Journal of Statistics, 12, pp. 171-178, 1985.[4] Bassamboo, A, S. Juneja, and A. Zeevi, “Portfolio Credit Risk with Extremal Dependence: Asymptotic Analysis and Efficient Simulation.” Operations Reseach, 56, pp. 593-606, 2008.[5] Bluhm, C, L. Overbeck, and C. Wagner, “An Introduction to Credit Risk Modeling.” CRC Press/Chapman & Hall, 2002.[6] Bucklew, J. A, Large Deviation Techniques in Decision, Simulation, and Estimation, John Wiley, New York, 1990.[7] Capriotti, L, “Least-Square Importance Sampling for Monte Carlo Security Pricing.” Quantitative Finance, 8, No.5, pp 485-497, 2008.[8] Chiang, M. H, M. L. Yueh, and M.H. Hsieh, “An Efficient Algorithm for Basket Default Swap Valuation.” Journal of Derivatives, pp. 8-19, 2007.[9] Crouhy, M, D. Galai, and R. Mark, “A Comparative Analysis of Current Credit Risk Models.” Journal of Banking and Finance, 24, pp. 59-117, 2000.[10] Glasserman, P, "Tail Approximations for Portfolio Credit Risk." Journal of Derivatives, 12, No. 2, pp. 24-42, 2004.[11] Glasserman, P, and J. Li, “Importance Sampling for Portfolio Credit Risk.” Management Science, 55, No.11, pp. 1643-1656, 2005.[12] Glasserman, P, P. Heidelberger, and P. Shahabuddin, "Asymptotically Optimal Importance Sampling and Stratification for Pricing Path-Dependent Options." Mathematical Finance. 9, No. 2, pp. 117-152, 1999.[13] Glasserman, P, W. Kang, and P. Shahabuddin, "Fast Simulation of Multifactor Portfolio Credit Risk." Operations Research. 56, No.5, pp. 1200-1217, 2008.[14] Graciela González-Farı́as., D. M. Armando, A. K. Gupta, “Additive Properties of Skew Normal Random Vectors.” Journal of Statistical Planning and Inference. 126, pp. 521-534, 2004.[15] Gupta, A. K, T. T. Nguyen, and J. A. Sanqui, “Characterization of the Skew- Normal Distribution.” Annals of the Institute of Statistical Mathematics, 56, pp. 351-360, 2004.[16] Gupta, G, C. Finger, M. Bhatia. CreditMetrics Technical Document. J.P.Morgan & CO, New York, 1997.[17] Hull, J, and A. White, “Valuation of CDO and nth to Default CDS without Monte Carlo Simulation.” Journal of Derivarives, 12, No. 2, pp. 8-23, 2004.[18] Johnson, N. L, S. Kotz, A. W. Kemp, Univariate Discrete Distributions, 2nd ed. John Wiley and Sons, New York, 1992.[19] Kostadinov, K, “Tail Approximation for Credit Risk Portfolios with Heavy-Tailed Risk Factors.” Technique Report, The Munich University of Technology, 2005.[20] Li, D,”On Default Correlation: A Copula Function Approach.” Journal of Fixed Income, 9, pp. 43-54, 2000.[21] Lucas, A, P. Klaassen, P. Spreij, and S. Straetmans, “Tail Behavior of Credit Loss Distributions for General Latent Factor Models.“ Applied Mathematical Finance. 10, No. 4, pp. 337-357, 2003.[22] McNeil, A.J, R. Frey, and P. Embrechts, Quantitative Risk Management: Concept, Techniques, and Tools. Princeton University Press, 2005.[23] Ney, P, “Dominating Points and the Asymptotics of the Large Deviations for Random Walk on Rd.” Annals of Probability. 11, No. 1, pp. 158-167, 1983.[24] O’Hagan and T. Leonard, “Bayes Estimation Subject to Uncertainty about Parameter Constraints.” Biometrika, 63, pp. 201-202, 1976.[25] Sadowsky, J. S., J. A. Bucklew, “On Large Deviations Theory and Asymptotically Efficient Monte Carlo Estimation.” IEEE Transactions on Information Theory, 36, No.3, May, pp. 579-588, 1990.[26] Schloegl, L, and D. O’Kane, “A Note on the Large Homogenous Portfolio Approximation with the Student-t Copula.” Finance and Stochastics, 9, pp. 577-584, 2005.[27] Sen, P. K, J. M. Singer, Large Sample Methods in Statistics. Chapman & Hall, London, 1993.[28] Shao, J, Mathematical Statistics. Spring-Verlag, New York, 1998.[29] Williams,D, Probability with Martingales. Cambridge: Cambridge University Press, 1991. zh_TW
