基于Copula的信用违约互换定价模型应用研究
摘要
2008年美国次债危机爆发,并在全球迅速蔓延。信用衍生产品的运用在其中起了关键性作用,尤其是信用违约互换(CDS)。因此,CDS定价模型的研究显得极为重要。其中一篮子CDS定价模型的重点在于合理描述资产之间的相关性。Copula理论成为了当前的热点问题。用copula函数来描述资产之间的相关性的主要优点在于,可以将多资产的联合分布分解为边际分布函数与copula的组合,极大的方便了模型的建立、参数的估计和数据的模拟。
本文研究了一篮子信用违约互换合约定价的基本理论。首先引入了违约强度的概念来建立资产的违约时间分布,然后采用copula函数来描述资产间的非线性相关结构,通过Monte?Carlo模拟生成标的资产违约时间的联合分布函数,从而实现对一篮子信用违约互换合约的定价。本文主要贡献:采用动态copula函数来描述标的资产间违约相关性结构的时变特征,进一步提高了信用违约互换合约定价的精度。
实证部分讨论了最优copula函数的选择问题,通过AIC统计指标选出静态条件最优拟合的copula函数,结果显示不同特征的金融样本数据的最优拟合copula函数是不同的。引入相关系数的动态演化方程估计时变相关系数路径,结果显示动态copula能够较好的反映出外部经济环境的变化,并能够突破静态copula描述资产相关关系的局限性:相关系数变动剧烈的时定价不精确。
最后,通过动态copula模拟标的资产回报的市场数据,估计静态copula相关性矩阵。比较两种模型后发现:在合约到期日临近、标的资产数目的增多(N比较大时)、违约资产个数n占总标的资产个数N比例下降的情况下,第n次一篮子违约互换合约价值的波动性相对较大,此时动态copula模型相对静态copula模型的优势更为明显,用动态copula模型描述标的资产间相关性也更有价值。
As U.S. subprime mortgage crisis in the 2008 slowly evolved into a global financial crisis. Credit derivatives play a primary role in the current financial crisis such as CDS. Therefore, the pricing CDS become a hot issue. For pricing CDS, it is very important how to define the correlation between the assets. Recently it is very popular using copula function to describe the dependent structure between the assets since copula function has some advantages. For example, the joint distribution of assets can be decomposed into the marginal distribution function and the copula function, which greatly facilitates the modeling, parameter estimation and numerical simulation.
In this paper, we study the basic theory of pricing basket default swaps. First, we introduce the concept of default intensity to describe a default time distribution of assets, and then use copula functions to describe the nonlinear dependent structure among the assets. Monte Carlo simulation is widely used to measure the default risk in basket credit swap. The main contribution of this paper in theory is that a dynamic copula function is introduced to capture the change of dependent structure of the underlying assets. The time-varying characteristics of the dynamic copula model improve the accuracy of pricing credit default swaps.
Subsequently, in the empirical application, we choose the optimal copula function by AIC statistical indicators for the static model. The results show that the best fitting copula is different for different the sample data. Then we extend the static copula to the dynamic case, thus allowing us to use copula theory in the analysis of time-varying dependence structure. We examine daily log-return of the index in china over a long period, and find the dependent structure described by dynamic copula can capture the change of of external economic environment, which cannot be caught by the static copula model. This is meaning the time-varying copula model can remedy the static copula model’s defect and the pricing of credit default swap is more accurate.
Finally, this paper deals with the comparison of procedures for credit default times’simulation on the pricing of nth to default swap. We simulate the data of the return of the underlying assets under dynamic copula, and then estimate the static correlation matrix using these data. Based on the comparison, when the maturity is approaching, the number of underlying assets is larger or the number of default assets in basket is smaller, the volatility of CDS’s contract value is relatively large and dynamic copula copula function’s advantage is more obvious for this case. Thus, the dependent structure described by dynamic copula model is more valuable on pricing of nth to default swaps.
本文研究了一篮子信用违约互换合约定价的基本理论。首先引入了违约强度的概念来建立资产的违约时间分布,然后采用copula函数来描述资产间的非线性相关结构,通过Monte?Carlo模拟生成标的资产违约时间的联合分布函数,从而实现对一篮子信用违约互换合约的定价。本文主要贡献:采用动态copula函数来描述标的资产间违约相关性结构的时变特征,进一步提高了信用违约互换合约定价的精度。
实证部分讨论了最优copula函数的选择问题,通过AIC统计指标选出静态条件最优拟合的copula函数,结果显示不同特征的金融样本数据的最优拟合copula函数是不同的。引入相关系数的动态演化方程估计时变相关系数路径,结果显示动态copula能够较好的反映出外部经济环境的变化,并能够突破静态copula描述资产相关关系的局限性:相关系数变动剧烈的时定价不精确。
最后,通过动态copula模拟标的资产回报的市场数据,估计静态copula相关性矩阵。比较两种模型后发现:在合约到期日临近、标的资产数目的增多(N比较大时)、违约资产个数n占总标的资产个数N比例下降的情况下,第n次一篮子违约互换合约价值的波动性相对较大,此时动态copula模型相对静态copula模型的优势更为明显,用动态copula模型描述标的资产间相关性也更有价值。
As U.S. subprime mortgage crisis in the 2008 slowly evolved into a global financial crisis. Credit derivatives play a primary role in the current financial crisis such as CDS. Therefore, the pricing CDS become a hot issue. For pricing CDS, it is very important how to define the correlation between the assets. Recently it is very popular using copula function to describe the dependent structure between the assets since copula function has some advantages. For example, the joint distribution of assets can be decomposed into the marginal distribution function and the copula function, which greatly facilitates the modeling, parameter estimation and numerical simulation.
In this paper, we study the basic theory of pricing basket default swaps. First, we introduce the concept of default intensity to describe a default time distribution of assets, and then use copula functions to describe the nonlinear dependent structure among the assets. Monte Carlo simulation is widely used to measure the default risk in basket credit swap. The main contribution of this paper in theory is that a dynamic copula function is introduced to capture the change of dependent structure of the underlying assets. The time-varying characteristics of the dynamic copula model improve the accuracy of pricing credit default swaps.
Subsequently, in the empirical application, we choose the optimal copula function by AIC statistical indicators for the static model. The results show that the best fitting copula is different for different the sample data. Then we extend the static copula to the dynamic case, thus allowing us to use copula theory in the analysis of time-varying dependence structure. We examine daily log-return of the index in china over a long period, and find the dependent structure described by dynamic copula can capture the change of of external economic environment, which cannot be caught by the static copula model. This is meaning the time-varying copula model can remedy the static copula model’s defect and the pricing of credit default swap is more accurate.
Finally, this paper deals with the comparison of procedures for credit default times’simulation on the pricing of nth to default swap. We simulate the data of the return of the underlying assets under dynamic copula, and then estimate the static correlation matrix using these data. Based on the comparison, when the maturity is approaching, the number of underlying assets is larger or the number of default assets in basket is smaller, the volatility of CDS’s contract value is relatively large and dynamic copula copula function’s advantage is more obvious for this case. Thus, the dependent structure described by dynamic copula model is more valuable on pricing of nth to default swaps.
引文
[1] J?nsson, H. and W. Schoutens. Pricing Constant Maturity Credit Default Swaps Under Jump Dynamics. Journal of Credit Risk. 2009, 5(1), 1-21.
[2] Hull, J., M. Predescu, and A. White, Valuation of CDO and nth to default CDS without Monte Carlo simulation. Journal of Derivatives, 2004, 12(2), p.8-23.
[3] Hull, J., M. Predescu, and A. White. The relationship between credit default swap spreads, bond yields, and credit rating announcements. Journal of Banking & Finance, 2005.28(11), 2789-2811.
[4] Norden, L. and M. Weber. Informational efficiency of credit default swap and stock markets: The impact of credit rating announcements. Journal of Banking & Finance, 2004, 8(12), 2813-2843.
[5] Ericsson, J., K. Jacobs and R.A. Oviedo. The determinants of Credit Default Swap Premia. CIRANO Montréal, 2004, 18(2), 40-55.
[6] Ericsson, J., K. Jacobs, and R. Oviedo. The Determinants of Credit Default Swap Premia: a copula study. Journal of Financial and Quantitative Analysis, 2009, 44(1), 109-132.
[7] Abid, F and N. Naifar. The determinants of credit default swap rates: An explanatory study. International Journal of Theoretical and Applied Finance, 2006(a), 9(1), 23-42.
[8] Li D. On default correlation: a copula function approach. Journal of Fixed Income. 2000, 2(8), 43-54.
[9] Mashal, R., Naldi, M. and Zeevi, A. On the dependence of equity and asset returns. RISK. 2003, 16(1), 83–87.
[10] Duffie D., Garleanu, N. Risk and Valuation of Collateralized Debt Obligations. CFA Institute, 2001, 57(1), 41-59.
[11] Wang, Z.R., et al.. Estimating risk of foreign exchange portfolio: Using VaR and CVaR based on GARCH-EVT-Copula model. Physica a-Statistical Mechanics and Its Applications. 2000, 38(21), 4918-4928.
[12] Lindskog, F., McNeil, A. and Schmock, U.. Kendall's tau for elliptical distributions.Credit Risk. 2003, 9(11), 149–156.
[13] Jobst, A.A.. Correlation, price discovery and co-movement of asset-backed securities and equity. Derivatives Use, Trading Regulation. 2006, 12(42), 60-101.
[14] Tavares, P.A.C., Nguyen, T.U., Chapovsky, A., Vaysburd, I.,“Composite basket model”, working Paper. 2004, Merrill Lynch Credit Derivatives.
[15] Burtschell, X., Gregory, J., and Laurent, J.-P.. Beyond the Gaussian Cop-ula: Stochastic and Local Correlation. Journal of Credit Risk. 2007, 3(1), 31-62.
[16] D. B. Madan, M. Konikov, and M. Marinescu,“Credit and basket default swaps,”The Journal of Credit Risk. 2006, 2(2), 103-115.
[17] Brigo D. and A. Alfonsi. Credit default swap calibration and derivatives pricing with the SSRD stochastic intensity model. Finance and Stochastics. 2005, 9(1), 29-42.
[18] Sircar, R. and T. Zariphopoulou. Utility Valuation of Credit Derivatives: Single and Two-Name Cases. Advances in Mathematical Finance. 2007, 14(3), 279-301.
[19]王琼,基于跳-扩散过程的信用违约互换定价模型.系统工程. 2003, 21(5), 79-84.
[20]周鹏,信用违约互换定价分析.优秀硕士学位论文,同济大学, 2007, 1-60.
[21]马俊美,一篮子信用违约互换定价的偏微分方程方法.高校应用数学学报A辑, 2008, 23(4), 427-437.
[22]田明,一个多因子信用违约互换定价模型.数学的实践与认识, 2008, 38(23), 47-57.
[23]王乐乐,基于信用等级迁移的信用违约互换定价.同济大学学报(自然科学版), 2010, 38(4), 613-619.
[24]高巍,信用违约互换及其定价模型.科技与管理, 2008, 10(1), 72-75.
[25]何海鹰,基于Copula理论的信用风险研究.博士学位论文. 2009,厦门大学, 1-75.
[26]詹原瑞,基于copula函数族的信用违约互换组合定价模型.中国管理科学, 2008, 16(1), 1-6.
[27]蒋晓琴,对于信用违约互换的利益及风险分析.科技信息, 2009. 13 (11), 356.
[28] Sebastian Galiani and Pablo Sanguinetti. The impact of trade liberalization on wage inequality: evidence from Argentina. Journal of Development Economics. 2003, 72(2), 497-513.
[29] Fathi Abid, Nader Naifar. Credit-default swap rates and equity volatility: a nonlinear relationship. Journal of Risk Finance. 2006, 7(4), 348 - 371.
[30] Tenney. Introduction to Copulas. Lectures Notes in Statistics. 9. Springer Verlag. New York. 2003.
[31] Sklar, A.. Fonctions de r′epartition `a n dimensions et leurs marges. Publications de l’Institut de Statistique de L’Universit′e de Paris. 1974, 12(8), 229-231.
[32] Genest S, Van Zand K, Legius E, Dom R, Malfroid M, Baro F, Godderis J, Cassiman JJ. Correlations between triplet repeat expansion and clinical features in Huntington's disease. Arch Neurol. 1995, 52(8), 749–753.?
[33] Akaike, H. Trabajos de estadística y de investigación operativa. Extraordinario. 1974, 14(2), 497-513
[34] Guegan, D. and J. Zang. Pricing bivariate option under GARCH-GH model with dynamic copula: application for Chinese market. European Journal of Finance. 2009, 15(7-8), 777-795.
[35] Guegan, D. and C. Caillault. Change analysis of a dynamic copula for measuring dependence in multivariate financial data. Quantitative Finance. 2010, 10(4), 421-430.
[36] C. Caillault, D. Gu′egan. Empirical estimation of tail dependence using copulas. Application to Asian markets. Quantitative Finance. 2005, 5(24), 489-501.
[37] Patton, A. J. Modelling Asymmetric Exchange Rate Dependence. International Economic Review. 2006, 47(2), 527-556.
[38] Mark Joshi, Dherminder Kainth. Rapid computation of prices and deltas of nth to default swaps in the Li Model. QUARC RBS Group Risk Management. 2004, 9(12), 1-3.
[39] Glasserman and Li. Importance sampling for portfolio credit risk. Management science. 2005, working paper, Citeseer.
[40] Dias, A., Embrechts, P. Modeling exchange rate dependence dynamics at different time horizons. Journal of International Money and Finance. 2010, 29(8),1687-1705.
[41] Jondeau, E., Rockinger, M.. The copula-GARCH model of conditional dependencies: An international stock market application. Journal of International Money and Finance. 2006, 25(6), 827-853.
[42] Granger, C.W.J., TerΥasvirta, T., Patton, A. J.. Common factors in conditional d istributions f or b ivariate t ime s eries. J ournal of E conometrics. 2006, 13(2), 43-57.
[2] Hull, J., M. Predescu, and A. White, Valuation of CDO and nth to default CDS without Monte Carlo simulation. Journal of Derivatives, 2004, 12(2), p.8-23.
[3] Hull, J., M. Predescu, and A. White. The relationship between credit default swap spreads, bond yields, and credit rating announcements. Journal of Banking & Finance, 2005.28(11), 2789-2811.
[4] Norden, L. and M. Weber. Informational efficiency of credit default swap and stock markets: The impact of credit rating announcements. Journal of Banking & Finance, 2004, 8(12), 2813-2843.
[5] Ericsson, J., K. Jacobs and R.A. Oviedo. The determinants of Credit Default Swap Premia. CIRANO Montréal, 2004, 18(2), 40-55.
[6] Ericsson, J., K. Jacobs, and R. Oviedo. The Determinants of Credit Default Swap Premia: a copula study. Journal of Financial and Quantitative Analysis, 2009, 44(1), 109-132.
[7] Abid, F and N. Naifar. The determinants of credit default swap rates: An explanatory study. International Journal of Theoretical and Applied Finance, 2006(a), 9(1), 23-42.
[8] Li D. On default correlation: a copula function approach. Journal of Fixed Income. 2000, 2(8), 43-54.
[9] Mashal, R., Naldi, M. and Zeevi, A. On the dependence of equity and asset returns. RISK. 2003, 16(1), 83–87.
[10] Duffie D., Garleanu, N. Risk and Valuation of Collateralized Debt Obligations. CFA Institute, 2001, 57(1), 41-59.
[11] Wang, Z.R., et al.. Estimating risk of foreign exchange portfolio: Using VaR and CVaR based on GARCH-EVT-Copula model. Physica a-Statistical Mechanics and Its Applications. 2000, 38(21), 4918-4928.
[12] Lindskog, F., McNeil, A. and Schmock, U.. Kendall's tau for elliptical distributions.Credit Risk. 2003, 9(11), 149–156.
[13] Jobst, A.A.. Correlation, price discovery and co-movement of asset-backed securities and equity. Derivatives Use, Trading Regulation. 2006, 12(42), 60-101.
[14] Tavares, P.A.C., Nguyen, T.U., Chapovsky, A., Vaysburd, I.,“Composite basket model”, working Paper. 2004, Merrill Lynch Credit Derivatives.
[15] Burtschell, X., Gregory, J., and Laurent, J.-P.. Beyond the Gaussian Cop-ula: Stochastic and Local Correlation. Journal of Credit Risk. 2007, 3(1), 31-62.
[16] D. B. Madan, M. Konikov, and M. Marinescu,“Credit and basket default swaps,”The Journal of Credit Risk. 2006, 2(2), 103-115.
[17] Brigo D. and A. Alfonsi. Credit default swap calibration and derivatives pricing with the SSRD stochastic intensity model. Finance and Stochastics. 2005, 9(1), 29-42.
[18] Sircar, R. and T. Zariphopoulou. Utility Valuation of Credit Derivatives: Single and Two-Name Cases. Advances in Mathematical Finance. 2007, 14(3), 279-301.
[19]王琼,基于跳-扩散过程的信用违约互换定价模型.系统工程. 2003, 21(5), 79-84.
[20]周鹏,信用违约互换定价分析.优秀硕士学位论文,同济大学, 2007, 1-60.
[21]马俊美,一篮子信用违约互换定价的偏微分方程方法.高校应用数学学报A辑, 2008, 23(4), 427-437.
[22]田明,一个多因子信用违约互换定价模型.数学的实践与认识, 2008, 38(23), 47-57.
[23]王乐乐,基于信用等级迁移的信用违约互换定价.同济大学学报(自然科学版), 2010, 38(4), 613-619.
[24]高巍,信用违约互换及其定价模型.科技与管理, 2008, 10(1), 72-75.
[25]何海鹰,基于Copula理论的信用风险研究.博士学位论文. 2009,厦门大学, 1-75.
[26]詹原瑞,基于copula函数族的信用违约互换组合定价模型.中国管理科学, 2008, 16(1), 1-6.
[27]蒋晓琴,对于信用违约互换的利益及风险分析.科技信息, 2009. 13 (11), 356.
[28] Sebastian Galiani and Pablo Sanguinetti. The impact of trade liberalization on wage inequality: evidence from Argentina. Journal of Development Economics. 2003, 72(2), 497-513.
[29] Fathi Abid, Nader Naifar. Credit-default swap rates and equity volatility: a nonlinear relationship. Journal of Risk Finance. 2006, 7(4), 348 - 371.
[30] Tenney. Introduction to Copulas. Lectures Notes in Statistics. 9. Springer Verlag. New York. 2003.
[31] Sklar, A.. Fonctions de r′epartition `a n dimensions et leurs marges. Publications de l’Institut de Statistique de L’Universit′e de Paris. 1974, 12(8), 229-231.
[32] Genest S, Van Zand K, Legius E, Dom R, Malfroid M, Baro F, Godderis J, Cassiman JJ. Correlations between triplet repeat expansion and clinical features in Huntington's disease. Arch Neurol. 1995, 52(8), 749–753.?
[33] Akaike, H. Trabajos de estadística y de investigación operativa. Extraordinario. 1974, 14(2), 497-513
[34] Guegan, D. and J. Zang. Pricing bivariate option under GARCH-GH model with dynamic copula: application for Chinese market. European Journal of Finance. 2009, 15(7-8), 777-795.
[35] Guegan, D. and C. Caillault. Change analysis of a dynamic copula for measuring dependence in multivariate financial data. Quantitative Finance. 2010, 10(4), 421-430.
[36] C. Caillault, D. Gu′egan. Empirical estimation of tail dependence using copulas. Application to Asian markets. Quantitative Finance. 2005, 5(24), 489-501.
[37] Patton, A. J. Modelling Asymmetric Exchange Rate Dependence. International Economic Review. 2006, 47(2), 527-556.
[38] Mark Joshi, Dherminder Kainth. Rapid computation of prices and deltas of nth to default swaps in the Li Model. QUARC RBS Group Risk Management. 2004, 9(12), 1-3.
[39] Glasserman and Li. Importance sampling for portfolio credit risk. Management science. 2005, working paper, Citeseer.
[40] Dias, A., Embrechts, P. Modeling exchange rate dependence dynamics at different time horizons. Journal of International Money and Finance. 2010, 29(8),1687-1705.
[41] Jondeau, E., Rockinger, M.. The copula-GARCH model of conditional dependencies: An international stock market application. Journal of International Money and Finance. 2006, 25(6), 827-853.
[42] Granger, C.W.J., TerΥasvirta, T., Patton, A. J.. Common factors in conditional d istributions f or b ivariate t ime s eries. J ournal of E conometrics. 2006, 13(2), 43-57.