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題名 雙峰常態因子模型違約風險估計之研究
Risk default estimation with bimodal normal factor model作者 鄭昕東
Cheng, Hsin-Tung貢獻者 劉惠美
Liu, Hui-Mei
鄭昕東
Cheng, Hsin-Tung關鍵詞 投資組合信用風險
重要性取樣法
偏斜常態分配
雙峰常態分配
變異數縮減
Portfolio credit risk
Importance sampling
Skew normal distribution
Bimodal normal distribution
Variance reduction日期 2019 上傳時間 7-Aug-2019 16:02:43 (UTC+8) 摘要 對投資組合信用風險進行量化分析時，其因子模型不一定有封閉性型式，且因為投資組合間具相依關係，計算困難，大多數情況無法直接進行計算，因此使用模擬方法去估計其值，其中較常使用蒙地卡羅法，但在投資組合違約風險估計中，我們主要討論重大損失之風險違約事件，其屬於稀有事件，在蒙地卡羅法中對於估計稀有事件之模擬計算較缺乏效率，因此我們在此使用Glasserman and Li(2005) 所提之二階段重要性取樣法、 Chiang et al.(2007) 所提之改良式重要性取樣法，相較於傳統蒙地卡羅法，可以更有效率估計及達到變異數縮減之效果。分配假設上，常使用常態分配當作系統風險因子之分配去進行模擬計算，但實際資料並非都符合此分配假設，像是匯率報酬率資料有些是雙峰或是偏態形式，因此我們使用偏斜常態分配(Skew Normal)、雙峰常態分配(Bimodal Normal)，去進行估計，同時也探討在此兩種方法是否可以推廣到非常態關聯結構之因子模型或其在不同模型下有何種限制。在數值估計結果方面，改良式重要性取樣法在不同分配之因子模型都能使用，二階段重要性取樣法則是受到指數扭轉法(Exponential Twisting)之限制不能在雙峰常態分配之因子模型使用，兩方法應用在估計偏斜常態分配上，相較於蒙地卡羅法，皆能有效估計、達到變異數縮減，節省模擬時間，其中以改良式重要性取樣法為最佳。
The factor model uncertainly has a closed form when the portfolio credit risk is quantitatively analyzed. Because the investment portfolio has a dependent relationship, the calculation is difficult and most of the cases cannot be directly calculated. Therefore, the simulation method is applied to estimate the value, the most often applied method is Monte Carlo method. However, in the portfolio default risk estimation, we mainly discuss the risk default event of major losses, which belong to rare events. In the Monte Carlo method, the simulation calculation for estimating rare events is inefficient, so we apply the two-steps importance sampling method proposed by Glasserman and Li (2005) and the modified importance sampling method proposed by Chiang et al. (2007). These two methods can more effectively estimate and achieve better result of variance reduction than the traditional Monte Carlo method.In the past, normal distribution was often applied as the distribution of system risk factors to perform simulation calculations. However, the actual data did not all fit this allocation hypothesis. Therefore, we also tried to use skew normal distribution and bimodal normal distribution to estimate, and also discussed Whether these two methods can be generalized to the factor model of the non-normal copula or what restrictions are there under different models.In terms of numerical estimation results, the modified importance sampling method can be used in different distribution factor models. The two-steps importance sampling method is restricted by the exponential twisting method and cannot be applied in the bimodal normal distribution factor model. Comparing with the Monte Carlo method, the above two methods ,which are applied in the skew normal distribution, can effectively estimate and achieve better result of variance reduction, and save the simulation time. The modified importance sampling method is the best one among these methods.參考文獻 [1] 陳家丞 (2016).“極值相依模型下投資組合之重要性取樣法”,國立政治大學統計研究所碩士論文,台北.[2] Ara, A., and Louzada, F. (2019). “The Multivariate Alpha Skew Gaussian Distribution. ” Bulletin of the Brazilian Mathematical Society, New Series, 1-21.[3] Arellano-Valle, R.B., Gómez, H.W. and Quintana, F. A. (2004). “A New Class ofSkew-Normal Distributions.”, Communications in Statistics-Theory and Methods,33(7), 1465-1480.[4] Azzalini, A. (1985). “A class of distributions which includes the normal ones.”, Scandinavian Journal of Statistics, 171-178.[5] Azzalini, A. (1986). “Further results on a class of distributions which includes the normal ones.”, Statistica. 46(2), 199-208.[6] Azzalini, A. (2005). “The Skew-normal Distribution and Related MultivariateFamilies.”, Scandinavian Journal of Statistics, 32(2), 159-188.[7] Azzalini, A. and Capitanio, A. (1999). “Statistical Applications of the Multivariate Skew Normal Distribution.”, Journal of the Royal Statistical Society: Series B (Statistical Methodology), 61(3), 579-602.[8] Azzalini A. and Dalla-Valle A. (1996). “The multivariate skew-normal distribution.”, Biometrika, 83(4), 715–726.[9] Chiang, M.H., Yueh, M.L., and Hsieh, M.H. (2007). “An Efficient Algorithm forBasket Default Swap Valuation.”, Journal of Derivatives, 15(2), 8-19.[10] Elal-Olivero, D. (2010). “ Alpha-skew-normal distribution. ” Proyecciones(Antofagasta), 29(3), 224-240.[11] Glasserman‚ P. (2004). “Tail Approximations for Portfolio Credit Risk.”‚ Journal of Derivatives, 12(2), 24-42.[12] Glasserman, P. and Li, J. (2005). “Importance Sampling for Portfolio Credit Risk.”, Management Science, 51(11),1643-1656.[13] Glasserman, P., Heidelberger, P. and Shahabuddin, P. (2000). “Variance Reduction Techniques for Estimating Value-at-Risk. ”, Management Science, 46(10), 1349-1364.[14] González-Farías, G., Domínguez-Molina, J.A. and Gupta, A.K. (2004). “Additive properties of skew normal random vectors.”, Journal of Statistical Planning and Inference, 126(2), 521-534.[15] González-Farías, G., Domínguez-Molina, J.A. and Gupta, A.K. (2004). “Amultivariate skew normal distribution.”, Journal of Multivariate Analysis, 89(1), 181-190.[16] Han, C.H, and Wu, C.T. (2010). “Efficient importance sampling for estimatinglower tail probabilities under Gaussian and Student’s t distributions.”, Preprint.National Tsing-Hua University.[17] Jorion, P. (1997). “Value at risk: the new benchmark for controlling market risk: ” Irwin Professional Pub.[18] Li, D.X. (1999). “On default correlation: a copula function approach.” , Journal of Fixed Income , 9(4), 43-54.[19] Louzada, F., Ara, A., and Fernandes, G. (2017). “The bivariate alpha-skew-normal distribution. ” Communications in statistics-Theory and Methods, 46(14), 7147-7156.[20] Sharafi, M., Sajjadnia, Z., and Behboodian, J. (2017). “A new generalization of alpha-skew-normal distribution. ” Communications in statistics-Theory and Methods, 46(12), 6098-6111.[21] Wilson, T. (1999). “Value at risk. ” Risk Management and Analysis, 1, 61-124. 描述 碩士
國立政治大學
統計學系
1063540231資料來源 http://thesis.lib.nccu.edu.tw/record/#G1063540231 資料類型 thesis dc.contributor.advisor 劉惠美 zh_TW dc.contributor.advisor Liu, Hui-Mei en_US dc.contributor.author (Authors) 鄭昕東 zh_TW dc.contributor.author (Authors) Cheng, Hsin-Tung en_US dc.creator (作者) 鄭昕東 zh_TW dc.creator (作者) Cheng, Hsin-Tung en_US dc.date (日期) 2019 en_US dc.date.accessioned 7-Aug-2019 16:02:43 (UTC+8) - dc.date.available 7-Aug-2019 16:02:43 (UTC+8) - dc.date.issued (上傳時間) 7-Aug-2019 16:02:43 (UTC+8) - dc.identifier (Other Identifiers) G1063540231 en_US dc.identifier.uri (URI) http://nccur.lib.nccu.edu.tw/handle/140.119/124689 - dc.description (描述) 碩士 zh_TW dc.description (描述) 國立政治大學 zh_TW dc.description (描述) 統計學系 zh_TW dc.description (描述) 1063540231 zh_TW dc.description.abstract (摘要) 對投資組合信用風險進行量化分析時，其因子模型不一定有封閉性型式，且因為投資組合間具相依關係，計算困難，大多數情況無法直接進行計算，因此使用模擬方法去估計其值，其中較常使用蒙地卡羅法，但在投資組合違約風險估計中，我們主要討論重大損失之風險違約事件，其屬於稀有事件，在蒙地卡羅法中對於估計稀有事件之模擬計算較缺乏效率，因此我們在此使用Glasserman and Li(2005) 所提之二階段重要性取樣法、 Chiang et al.(2007) 所提之改良式重要性取樣法，相較於傳統蒙地卡羅法，可以更有效率估計及達到變異數縮減之效果。分配假設上，常使用常態分配當作系統風險因子之分配去進行模擬計算，但實際資料並非都符合此分配假設，像是匯率報酬率資料有些是雙峰或是偏態形式，因此我們使用偏斜常態分配(Skew Normal)、雙峰常態分配(Bimodal Normal)，去進行估計，同時也探討在此兩種方法是否可以推廣到非常態關聯結構之因子模型或其在不同模型下有何種限制。在數值估計結果方面，改良式重要性取樣法在不同分配之因子模型都能使用，二階段重要性取樣法則是受到指數扭轉法(Exponential Twisting)之限制不能在雙峰常態分配之因子模型使用，兩方法應用在估計偏斜常態分配上，相較於蒙地卡羅法，皆能有效估計、達到變異數縮減，節省模擬時間，其中以改良式重要性取樣法為最佳。 zh_TW dc.description.abstract (摘要) The factor model uncertainly has a closed form when the portfolio credit risk is quantitatively analyzed. Because the investment portfolio has a dependent relationship, the calculation is difficult and most of the cases cannot be directly calculated. Therefore, the simulation method is applied to estimate the value, the most often applied method is Monte Carlo method. However, in the portfolio default risk estimation, we mainly discuss the risk default event of major losses, which belong to rare events. In the Monte Carlo method, the simulation calculation for estimating rare events is inefficient, so we apply the two-steps importance sampling method proposed by Glasserman and Li (2005) and the modified importance sampling method proposed by Chiang et al. (2007). These two methods can more effectively estimate and achieve better result of variance reduction than the traditional Monte Carlo method.In the past, normal distribution was often applied as the distribution of system risk factors to perform simulation calculations. However, the actual data did not all fit this allocation hypothesis. Therefore, we also tried to use skew normal distribution and bimodal normal distribution to estimate, and also discussed Whether these two methods can be generalized to the factor model of the non-normal copula or what restrictions are there under different models.In terms of numerical estimation results, the modified importance sampling method can be used in different distribution factor models. The two-steps importance sampling method is restricted by the exponential twisting method and cannot be applied in the bimodal normal distribution factor model. Comparing with the Monte Carlo method, the above two methods ,which are applied in the skew normal distribution, can effectively estimate and achieve better result of variance reduction, and save the simulation time. The modified importance sampling method is the best one among these methods. en_US dc.description.tableofcontents 誌謝 i摘要 iiAbstract iii目錄 iv圖目錄 vi表目錄 viii第一章 緒論 1第二節 研究背景與動機 1第二節 風險價值(Value at Risk,VAR) 1第三節 研究目的 2第二章 文獻回顧 4第三章 分配介紹 7第一節 偏斜常態(SN)分配介紹 7第二節 雙峰常態(BN)分配介紹 10第三節 Alpha偏斜常態(ASN)分配介紹 12第四節 對稱分量Alpha偏斜常態(SCASN)分配介紹 16第四章 研究方法 20第一節 基本假設 20第二節 因子模型介紹 21第三節 分配應用於因子模型 22I. 系統風險因子服從常態分配 22II. 系統風險因子服從偏斜常態分配(SN) 22III. 系統風險因子服從雙峰常態分配(BN) 24第四節 兩階段重要性取樣法 25I. 債務人之間彼此相互獨立 26II. 債務人之間彼此相依 27第五節 改良式重要性取樣法 31第五章 不同因子模型之模擬方法 34第一節 偏斜常態模型 34I. 單因子模型 34II. 多因子模型 37第二節 雙峰常態模型 41I. 單因子模型 41第六章 估計結果比較與分析 44第一節 偏斜常態因子模型估計結果 44I. 單因子模型之估計結果 44II. 多因子模型之估計結果 56第二節 雙峰常態因子模型估計結果 59I. 單因子模型之估計結果 59第三節 不同分配因子模型之比較 68I. 系統風險因子分別服從N(0,1)、SN(0,1,1)、BN之比較 68第四節 總結分析與建議 72參考文獻 74 zh_TW dc.format.extent 2769220 bytes - dc.format.mimetype application/pdf - dc.source.uri (資料來源) http://thesis.lib.nccu.edu.tw/record/#G1063540231 en_US dc.subject (關鍵詞) 投資組合信用風險 zh_TW dc.subject (關鍵詞) 重要性取樣法 zh_TW dc.subject (關鍵詞) 偏斜常態分配 zh_TW dc.subject (關鍵詞) 雙峰常態分配 zh_TW dc.subject (關鍵詞) 變異數縮減 zh_TW dc.subject (關鍵詞) Portfolio credit risk en_US dc.subject (關鍵詞) Importance sampling en_US dc.subject (關鍵詞) Skew normal distribution en_US dc.subject (關鍵詞) Bimodal normal distribution en_US dc.subject (關鍵詞) Variance reduction en_US dc.title (題名) 雙峰常態因子模型違約風險估計之研究 zh_TW dc.title (題名) Risk default estimation with bimodal normal factor model en_US dc.type (資料類型) thesis en_US dc.relation.reference (參考文獻) [1] 陳家丞 (2016).“極值相依模型下投資組合之重要性取樣法”,國立政治大學統計研究所碩士論文,台北.[2] Ara, A., and Louzada, F. (2019). “The Multivariate Alpha Skew Gaussian Distribution. ” Bulletin of the Brazilian Mathematical Society, New Series, 1-21.[3] Arellano-Valle, R.B., Gómez, H.W. and Quintana, F. A. (2004). “A New Class ofSkew-Normal Distributions.”, Communications in Statistics-Theory and Methods,33(7), 1465-1480.[4] Azzalini, A. (1985). “A class of distributions which includes the normal ones.”, Scandinavian Journal of Statistics, 171-178.[5] Azzalini, A. (1986). “Further results on a class of distributions which includes the normal ones.”, Statistica. 46(2), 199-208.[6] Azzalini, A. (2005). “The Skew-normal Distribution and Related MultivariateFamilies.”, Scandinavian Journal of Statistics, 32(2), 159-188.[7] Azzalini, A. and Capitanio, A. (1999). “Statistical Applications of the Multivariate Skew Normal Distribution.”, Journal of the Royal Statistical Society: Series B (Statistical Methodology), 61(3), 579-602.[8] Azzalini A. and Dalla-Valle A. (1996). “The multivariate skew-normal distribution.”, Biometrika, 83(4), 715–726.[9] Chiang, M.H., Yueh, M.L., and Hsieh, M.H. (2007). “An Efficient Algorithm forBasket Default Swap Valuation.”, Journal of Derivatives, 15(2), 8-19.[10] Elal-Olivero, D. (2010). “ Alpha-skew-normal distribution. ” Proyecciones(Antofagasta), 29(3), 224-240.[11] Glasserman‚ P. (2004). “Tail Approximations for Portfolio Credit Risk.”‚ Journal of Derivatives, 12(2), 24-42.[12] Glasserman, P. and Li, J. (2005). “Importance Sampling for Portfolio Credit Risk.”, Management Science, 51(11),1643-1656.[13] Glasserman, P., Heidelberger, P. and Shahabuddin, P. (2000). “Variance Reduction Techniques for Estimating Value-at-Risk. ”, Management Science, 46(10), 1349-1364.[14] González-Farías, G., Domínguez-Molina, J.A. and Gupta, A.K. (2004). “Additive properties of skew normal random vectors.”, Journal of Statistical Planning and Inference, 126(2), 521-534.[15] González-Farías, G., Domínguez-Molina, J.A. and Gupta, A.K. (2004). “Amultivariate skew normal distribution.”, Journal of Multivariate Analysis, 89(1), 181-190.[16] Han, C.H, and Wu, C.T. (2010). “Efficient importance sampling for estimatinglower tail probabilities under Gaussian and Student’s t distributions.”, Preprint.National Tsing-Hua University.[17] Jorion, P. (1997). “Value at risk: the new benchmark for controlling market risk: ” Irwin Professional Pub.[18] Li, D.X. (1999). “On default correlation: a copula function approach.” , Journal of Fixed Income , 9(4), 43-54.[19] Louzada, F., Ara, A., and Fernandes, G. (2017). “The bivariate alpha-skew-normal distribution. ” Communications in statistics-Theory and Methods, 46(14), 7147-7156.[20] Sharafi, M., Sajjadnia, Z., and Behboodian, J. (2017). “A new generalization of alpha-skew-normal distribution. ” Communications in statistics-Theory and Methods, 46(12), 6098-6111.[21] Wilson, T. (1999). “Value at risk. ” Risk Management and Analysis, 1, 61-124. zh_TW dc.identifier.doi (DOI) 10.6814/NCCU201900189 en_US
