政大機構典藏-National Chengchi University Institutional Repository(NCCUR):Item 140.119/124689

政大機構典藏-National Chengchi University Institutional Repository(NCCUR):Item 140.119/124689

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政大典藏 > College of Commerce > Department of Statistics > Theses > Item 140.119/124689

Please use this identifier to cite or link to this item: https://nccur.lib.nccu.edu.tw/handle/140.119/124689

Title:	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
Authors:	鄭昕東 Cheng, Hsin-Tung
Contributors:	劉惠美 Liu, Hui-Mei 鄭昕東 Cheng, Hsin-Tung
Keywords:	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
Date:	2019
Issue Date:	2019-08-07 16:02:43 (UTC+8)
Abstract:	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.
Reference:	[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 of Skew-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 Multivariate Families.”, 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 for Basket 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). “A multivariate skew normal distribution.”, Journal of Multivariate Analysis, 89(1), 181-190. [16] Han, C.H, and Wu, C.T. (2010). “Efficient importance sampling for estimating lower 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.
Description:	碩士國立政治大學統計學系 1063540231
Source URI:	http://thesis.lib.nccu.edu.tw/record/#G1063540231
Data Type:	thesis
DOI:	10.6814/NCCU201900189
Appears in Collections:	[Department of Statistics] Theses

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