VaR风险耦合理论模型、数值模拟技术及应用研究
|
摘要
现有VaR 风险计量方法一般未考虑各种风险相互耦合作用的影响,易导致对金融风险估计不足,欲揭示风险耦合作用的特性则增加了分析问题的难度和复杂性。风险耦合作用使得线性叠加原理不能成立,且线性相关结构不足以描述风险耦合的非线性特征。因此,在现代金融风险管理中迫切需要建立能够综合反映各种金融风险相互耦合作用影响的风险评估体系。
本文对VaR 风险耦合理论以及数值模拟技术进行了深入的研究。为了获取风险耦合理论模型中所需的信用参数,文中充分挖掘我国上市公司的市场信息,引入违约距离的概念,提出了上市公司违约概率模型,建立了信用评级体系; 借助copula相关结构理论描述风险间的耦合作用,建立了VaR 风险耦合理论模型并给出了相关的数值模拟技术,拓展了VaR 风险计量方法的应用范围; 基于强度模型导出了反映风险耦合影响的可转换债券定价方程,通过径向基函数方法求得其数值解,并利用VaR 风险计量方法评价了可转换债券的风险; 提出了非平移收益曲线下的利率风险对冲策略,并利用VaR 风险计量方法检验了策略的有效性; 为了反映收益率的“尖峰厚尾性”和波动率的集聚性,文中采用广义误差分布代替GARCH 模型中的正态分布假设,建立了广义误差分布GARCH 模型,利用copula 相关结构的Monte Carlo数值模拟技术生成了具有copula 相关结构的模拟情景分布,并借助违约概率信息模拟公司违约行为,从而获得了风险耦合作用下资产组合的风险值VaR 及Mean-CVaR有效前沿。
研究表明,违约概率模型能有效的识别上市公司的信用风险; 本文建立的我国上市公司信用评级体系的评估结果与新华远东评级体系相应的评估结果呈显著地正相关关系; VaR 风险耦合评估模型能有效地反映信用风险、市场风险及其耦合作用的影响,更准确地度量金融资产的整体风险,ST 公司具有较大的信用风险,仅度量其市场风险会严重的低估其风险; 考虑信用风险的可转换债券理论价值更接近其市场价值,忽略信用风险影响会低估其金融风险; 非平移收益曲线下的利率风险对冲策略有效地降低了对冲组合的风险值VaR。本文的研究对金融风险管理和投资决策具有重要的参考价值。
At present, value at risk (VaR) approach often ignores risks' coupling influence, which underestimates the financial risk. Since coupling risks haven't linear additivity and linear correlation is not sufficient to describe nonlinear dependence structure for coupling risks, the risk measure problem under risks' coupling effect becomes very complex and difficult. So, it is very important to build coupling risks' measure system for financial risk management.
The work presented in this dissertation extends VaR approach using risks' coupling theory and measures coupling risks using numerical simulation technology. In order to gain the credit risk’s parameters in the risks' coupling model, default probability is obtained from stock's price according to the concept of default distance, default probability model about Chinese listed company is proposed and credit rating system is built. Using the theory of copulas to describe the dependence structure of coupling risks, the paper builds VaR risks' coupling model and proposes numerical simulation technology about the model, extends the application of VaR approach; proposes the pricing model of convertible bond which takes the credit risk into account based on the intensity model, uses radial basis function approach to solve the valuation model and measures the VaR of convertible bond; designs interest rate risk hedge strategy under nonparallel shift of the yield curve, verifies the hedge strategy's effect using VaR approach. In order to describe the characteristics of stock return's distribution which are fat-tails and exaggerate volatility, Normal Distribution is replaced by Generalized Error Distribution in GARCH model, the simulated distribution with copulas dependence is made using Monte Carlo simulation. Through simulating default behaves of Chinese listed companies, capital portfolio's VaR and efficient frontier of Mean-CVaR model under risks' coupling effect are gained.
The result shows that default probability model can identify the credit risk of Chinese listed companies effectively; credit rating system gives Chinese listed companies' credit rating results and the results are positive correlated with the corresponding rating results from Xinhua Finance; VaR risks' coupling model can reflect credit risk, market risk and the coupled effect between risks, and exactly measures the total risk of financial capital. ST listed company's credit risk is bigger, and its total risk can be underestimated if only measuring its market risk; considering credit risk, the theory value of convertible bond be closer to its market price, if ignores credit risk, can underestimates its total risk; interest
rate risk hedge strategy under nonparallel shift of the yield curve reduces the VaR of hedging portfolio effectively. The researches have important reference value for financial risk management and investment decision.
本文对VaR 风险耦合理论以及数值模拟技术进行了深入的研究。为了获取风险耦合理论模型中所需的信用参数,文中充分挖掘我国上市公司的市场信息,引入违约距离的概念,提出了上市公司违约概率模型,建立了信用评级体系; 借助copula相关结构理论描述风险间的耦合作用,建立了VaR 风险耦合理论模型并给出了相关的数值模拟技术,拓展了VaR 风险计量方法的应用范围; 基于强度模型导出了反映风险耦合影响的可转换债券定价方程,通过径向基函数方法求得其数值解,并利用VaR 风险计量方法评价了可转换债券的风险; 提出了非平移收益曲线下的利率风险对冲策略,并利用VaR 风险计量方法检验了策略的有效性; 为了反映收益率的“尖峰厚尾性”和波动率的集聚性,文中采用广义误差分布代替GARCH 模型中的正态分布假设,建立了广义误差分布GARCH 模型,利用copula 相关结构的Monte Carlo数值模拟技术生成了具有copula 相关结构的模拟情景分布,并借助违约概率信息模拟公司违约行为,从而获得了风险耦合作用下资产组合的风险值VaR 及Mean-CVaR有效前沿。
研究表明,违约概率模型能有效的识别上市公司的信用风险; 本文建立的我国上市公司信用评级体系的评估结果与新华远东评级体系相应的评估结果呈显著地正相关关系; VaR 风险耦合评估模型能有效地反映信用风险、市场风险及其耦合作用的影响,更准确地度量金融资产的整体风险,ST 公司具有较大的信用风险,仅度量其市场风险会严重的低估其风险; 考虑信用风险的可转换债券理论价值更接近其市场价值,忽略信用风险影响会低估其金融风险; 非平移收益曲线下的利率风险对冲策略有效地降低了对冲组合的风险值VaR。本文的研究对金融风险管理和投资决策具有重要的参考价值。
At present, value at risk (VaR) approach often ignores risks' coupling influence, which underestimates the financial risk. Since coupling risks haven't linear additivity and linear correlation is not sufficient to describe nonlinear dependence structure for coupling risks, the risk measure problem under risks' coupling effect becomes very complex and difficult. So, it is very important to build coupling risks' measure system for financial risk management.
The work presented in this dissertation extends VaR approach using risks' coupling theory and measures coupling risks using numerical simulation technology. In order to gain the credit risk’s parameters in the risks' coupling model, default probability is obtained from stock's price according to the concept of default distance, default probability model about Chinese listed company is proposed and credit rating system is built. Using the theory of copulas to describe the dependence structure of coupling risks, the paper builds VaR risks' coupling model and proposes numerical simulation technology about the model, extends the application of VaR approach; proposes the pricing model of convertible bond which takes the credit risk into account based on the intensity model, uses radial basis function approach to solve the valuation model and measures the VaR of convertible bond; designs interest rate risk hedge strategy under nonparallel shift of the yield curve, verifies the hedge strategy's effect using VaR approach. In order to describe the characteristics of stock return's distribution which are fat-tails and exaggerate volatility, Normal Distribution is replaced by Generalized Error Distribution in GARCH model, the simulated distribution with copulas dependence is made using Monte Carlo simulation. Through simulating default behaves of Chinese listed companies, capital portfolio's VaR and efficient frontier of Mean-CVaR model under risks' coupling effect are gained.
The result shows that default probability model can identify the credit risk of Chinese listed companies effectively; credit rating system gives Chinese listed companies' credit rating results and the results are positive correlated with the corresponding rating results from Xinhua Finance; VaR risks' coupling model can reflect credit risk, market risk and the coupled effect between risks, and exactly measures the total risk of financial capital. ST listed company's credit risk is bigger, and its total risk can be underestimated if only measuring its market risk; considering credit risk, the theory value of convertible bond be closer to its market price, if ignores credit risk, can underestimates its total risk; interest
rate risk hedge strategy under nonparallel shift of the yield curve reduces the VaR of hedging portfolio effectively. The researches have important reference value for financial risk management and investment decision.
引文
[1] Markowitz, H.M. Portfolio Selection[J]. Journal of Finance, 1952, 7(1): 77-91.
[2] Sharpe, W.F. Capita Asset Prices: A Theory of Market Equilibrium Under Conditions of Risk[J]. Journal of Finance, 1964, 19: 425-442.
[3] Lintner, J. The Valuation of Risk Assets and the Selection of Risky Investments in Stock Portfolios and Capital Budgets[J]. Review of Economics and Statistics, 1965, 47: 13-37.
[4] Mossin, J. Equilibrium in a Capital Asset Market[J]. Econometrica, 1966, 13(3): 768-783.
[5] Ross, S.A. The Arbitrage Theory of Capital Asset Pricing[J]. Journal of Economic Theory, 1976, 13(3): 341-360.
[6] Modigliani, F., Miller, M.H. The cost of Capital,Corporation Finance,and the Theory of Inestment[J]. American Economic Review, 1958, 48(6): 261-297.
[7] Merton, R.C. On the Pricing of Corporate Debt: The Risk Structure of Interest Rates[J]. Journal of Finance, 1974, 29(2): 449-470.
[8] Black, F., Scholes, M. The pricing of options and corporate liabilities[J]. Journal of Political Economy, 1973, 81(3): 637-659.
[9] Merton, R.C. Theory of Rational Option Pricing[J]. Bell Journal of Economics and Management Science, 1973, 4: 141-183.
[10] Engle, R.F. Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation[J]. Econometrica, 1982, 50: 987-1007.
[11] Bollerslev, T. Generalized autoregressive conditional heteroskedasticity[J]. Journal of Econometrics, 1986, 31: 307-327.
[12] Group of Thirty. Derivatives: Practices and Principles. New York:Group of Thirty; 1993.
[13] Giot, P., Laurent, S. Market risk in commodity markets: a VaR approach[J]. Energy Economics, 2003, 25(5): 435-457.
[14] Angelidis, T., Benos, A., Degiannakis, S. The use of GARCH models in VaR estimation[J]. Statistical Methodology, 2004, 1(1-2): 105-128.
[15] Lehnert, T., Wolff, C.C.P. Scale-consistent Value-at-Risk[J]. Finance Research Letters, 2004, 1(2): 127-134.
[16] Gen?ay, R., Sel?uk, F. Extreme value theory and Value-at-Risk: Relative performance in emerging markets[J]. International Journal of Forecasting, 2004, 20(2): 287-303.
[17] Jacobson, T., Roszbach, K. Bank lending policy, credit scoring and value-at-risk[J]. Journal of Banking and Finance, 2003, 27(4): 615-633.
[18] Gordy, M.B. A risk-factor model foundation for ratings-based bank capital rules[J]. Journal of Financial Intermediation, 2003, 12(3): 199-232.
[19] Longstaff, F.A., Schwartz, E.S. A simple approach to valuing risky fixed and floating rate debt[J]. The Journal of Finance, 1995, 50(3): 789.
[20] Eric, B., Francois de, V. On the risk of insurance liabilities: Debunking some common pitfalls[J]. Journal of Risk and Insurance, 1997, 64(4): 673.
[21] Leland, H.E. Corporate debt value, bond covenants, and optimal capital structure[J]. The Journal of Finance, 1994, 49(4): 1213-1252.
[22] Leland, H.E., Toft, K.B. Optimal capital structure, endogenous bankruptcy, and the term structure of credit spreads[J]. The Journal of Finance, 1996, 51(3): 987-1019.
[23] Pierre, M.-B., William, P. Strategic debt service[J]. The Journal of Finance, 1997, 52(2): 531-556.
[24] Jarrow, R.A., Turnbull, S.M. Pricing derivatives on financial securities subject to credit risk[J]. The Journal of Finance, 1995, 50(1): 53-85.
[25] Duffie, D., Singleton, K.J. An econometric model of the term structure of interest-rate swap yields[J]. The Journal of Finance, 1997, 52(4): 1287-1322.
[26] Madan, D.B., Unal, H. Pricing the risks of default[J]. Review of Derivatives Research, 1998, 2(2-3): 121-160.
[27] Duffie, D., Singleton, K.J. Modeling term structures of defaultable bonds[J]. The Review of Financial Studies, 1999, 12(4): 687.
[28] Robert, A.J., David, L., Stuart, M.T. A markov model for the term structure of credit risk spreads[J]. The Review of Financial Studies (1986-1998), 1997, 10(2): 481.
[29] Lando, D. On Cox Processes and Credit Risky Securities[J]. Review of Derivatives Research, 1998, 2(2-3): 99-120.
[30] Bielecki, T.R., Rutkowski, M. Multiple Ratings Model of Defaultable Term Structure[J]. Mathematical Finance, 2000, 10(2): 125-139.
[31] Zhou, C. A jump-diffusion approach to modeling credit risk and valuing defaultable securities. In: Board of Governors of the Federal Reserve System (U.S.), Finance and Economics Discussion Series: 1997-15; 1997.
[32] Giesecke, K. Default and Information. In: http://www.defaultrisk.com, Working Paper; 2003.
[33] Cho-Jieh, C., Harry, P. Unifying discrete structural models and reduced-form models in credit risk using a jump-diffusion process[J]. Insurance, Mathematics & Economics, 2003, 33(2): 357.
[34] Yang, H. An integrated risk management method: VaR approach[J]. Multinational Finance Journal, 2000, 4(3/4): 201.
[35] Theodore, M.B., Jr., William, F.M. Modeling correlated market and credit risk in fixed income portfolios[J]. Journal of Banking & Finance, 2002, 26(2,3): 347.
[36] Zhou, C. The term structure of credit spreads with jump risk[J]. Journal of Banking & Finance, 2001, 25(11): 2015-2040.
[37] Georges, H. The analytic pricing of asymmetric defaultable swaps[J]. Journal of Banking & Finance, 2001, 25(2): 295.
[38] Frank, S.S., Timothy, G.T. An empirical analysis of credit default swaps[J]. International Review of Financial Analysis, 2002, 11(3): 297-309.
[39] Christine, A.B., Sally, W. Credit risk: The case of First Interstate Bankcorp[J]. International Review of Financial Analysis, 2002, 11(2): 229.
[40] Gordy, M.B. A comparative anatomy of credit risk models[J]. Journal of Banking & Finance, 2000, 24(1,2): 119-149.
[41] Michel, C., Dan, G., Robert, M. A comparative analysis of current credit risk models[J]. Journal of Banking & Finance, 2000, 24(1,2): 59.
[42] Pamela, N., William, P., Simone, V. Stability of rating transitions[J]. Journal of Banking & Finance, 2000, 24(1,2): 203.
[43] Treacy, W.F., Carey, M. Credit risk rating systems at large US banks[J]. Journal of Banking & Finance, 2000, 24(1-2): 167-201.
[44] 张维, 李玉霜. 商业银行信用风险分析综述[J]. 管理科学学报, 1998, 1(03): 20-27.
[45] 王春峰, 万海晖, 张维. 基于神经网络技术的商业银行信用风险评估[J]. 系统工程理论与实践, 1999, 09: 24-32.
[46] 陈雄华, 林成德, 叶武. 基于神经网络的企业信用等级评估[J]. 系统工程学报, 2002, 06.
[47] 庞素琳, 王燕鸣, 黎荣舟. 基于BP 算法的信用风险评价模型研究[J]. 数学的实践与认识, 2003, 08.
[48] 王春峰, 康莉. 基于遗传规划方法的商业银行信用风险评估模型[J]. 系统工程理论与实践, 2001, 02.
[49] 何运强, 方兆本. 债券信用评级与信用风险[J]. 管理科学, 2003, 02.
[50] 王焜, 徐枞巍. 企业信用风险管理的系统分析[J]. 管理评论, 2002, 09.
[51] 陈忠阳. 信用风险量化管理模型发展探析[J]. 国际金融研究, 2000, 10.
[52] 张玲, 白效锋. 试论信用风险计量法在金融机构风险管理中的应用[J]. 金融理论与实践, 2000, 05.
[53] 程鹏, 吴冲锋, 李为冰. 信用风险度量和管理方法研究[J]. 管理工程学报, 2002, 16(01): 70-73.
[54] 陶铄, 杨晓光. 商业银行内部信用评级的比较研究[J]. 管理评论, 2002, 09.
[55] 王春峰, 李汶华. 小样本数据信用风险评估研究[J]. 管理科学学报, 2001, 01.
[56] 刘宏峰, 杨晓光. 违约损失率的估计:发达国家的经验及启示[J]. 管理评论, 2003a, 06.
[57] 邓云胜, 沈沛龙, 任若恩. 贷款组合信用风险VaR 的蒙特卡罗仿真[J]. 计算机仿真, 2003, 02.
[58] 刘宏峰, 杨晓光. 信用衍生产品的类型及用途[J]. 管理评论, 2003b, 02.
[59] Robert, F.E., David, M.L., Russell, P.R. Estimating time varying risk premia in the term structure: the ARCH-M model[J]. Econometrica (1986-1998), 1987, 55(2): 391.
[60] David, X.L. On default correlation: A copula function approach[J]. The Journal of Fixed Income, 2000, 9(4): 43.
[61] Giesecke, K. A Simple Exponential Model for Dependent Defaults[J]. The Journal of Fixed Income, 2003, 13(3): 74.
[62] Beatriz Vaz de Melo, M., Rafael Martins de, S. Measuring financial risks with copulas[J]. International Review of Financial Analysis, 2004, 13(1): 27.
[63] Estrella, A., Hendricks, D., Kambhu, J., et al. The price risk of options positions: Measurement and capital requirements[J]. Federal Reserve Bank of New York Quarterly Review, 1994, 19(2): 27.
[64] Rockafeller, T., Uryasev, S. Optimization of conditional value-at-risk[J]. Journal of Risk, 2000, 2(3): 21-24.
[65] Jun-Koo, K., Lee, Y.W. The pricing of convertible debt offerings[J]. Journal of Financial Economics, 1996, 41(2): 231-248.
[66] Franke, C., Schaback, R. Solving partial differential equations by collocation using radial basis functions[J]. Applied Mathematics and Computation, 1998, 93(1): 73-82.
[67] Hon, Y.C. A quasi-radial basis functions method for American options pricing[J]. Computers and Mathematics with Applications, 2002, 43(3-5): 513-524.
[68] Kansa, E.J., Hon, Y.C. Circumventing the Ill-Conditioning Problem with Multiquadric Radial Basis Functions: Applications to Elliptic Partial Differential Equations[J]. Computers and Mathematics with Applications, 2000, 39(7-8): 123-137.
[69] Lianfen, Q. Regularized Radial Basis Function Networks: Theory and Applications[J]. Technometrics, 2002, 44(3): 294.
[70] Zhai, Y., Yao, D. A radial-basis function based surface Laplacian estimate for a realistic head model[J]. Brain Topography [NLM -MEDLINE], 2004, 17(1): 55.
[71] Cooper, I.A. Asset Values, Interest-Rate Changes, and Duration[J]. Journal of Financial and Quantitative Analysis, 1977, 12(5): 701.
[72] Litterman, R., Scheinkman, J. Common Factors Affecting Bond Returns[J]. Journal of Fixed Income, 1991, 1(1): 54-61.
[73] Derman, E. The Future of Modeling[J]. Risk, 1997, (12): 164-167.
[74] Ho, T.S.Y. Key Rate Durations: Measures of Interest Rate Risks[J]. The Journal of Fixed Income, 1992, 2(2): 29.
[75] Reitano, R.R. Non-parallel yield curve shifts and stochastic immunization[J]. Journal of Portfolio Management, 1996, 22(2): 71.
[76] Johnson, B.D., Meyer, K.R. Managing Yield Curve Risk In An Index Environment[J]. Financial Analysts Journal, 1989, 45(6): 51.
[77] Nawalkha, S.K., Chambers, D.R. The M-vector model: Derivation and testing of extensions to M-square[J]. Journal of Portfolio Management, 1997, 23(2): 92.
[78] Gloria, M.S. Immunization derived from a polynomial duration vector in the Spanish bond market[J]. Journal of Banking & Finance, 2001, 25(6): 1037.
[79] Heiko, L. Managing yield-curve risk with combination hedges[J]. Financial Analysts Journal, 2001, 57(3): 63.
[2] Sharpe, W.F. Capita Asset Prices: A Theory of Market Equilibrium Under Conditions of Risk[J]. Journal of Finance, 1964, 19: 425-442.
[3] Lintner, J. The Valuation of Risk Assets and the Selection of Risky Investments in Stock Portfolios and Capital Budgets[J]. Review of Economics and Statistics, 1965, 47: 13-37.
[4] Mossin, J. Equilibrium in a Capital Asset Market[J]. Econometrica, 1966, 13(3): 768-783.
[5] Ross, S.A. The Arbitrage Theory of Capital Asset Pricing[J]. Journal of Economic Theory, 1976, 13(3): 341-360.
[6] Modigliani, F., Miller, M.H. The cost of Capital,Corporation Finance,and the Theory of Inestment[J]. American Economic Review, 1958, 48(6): 261-297.
[7] Merton, R.C. On the Pricing of Corporate Debt: The Risk Structure of Interest Rates[J]. Journal of Finance, 1974, 29(2): 449-470.
[8] Black, F., Scholes, M. The pricing of options and corporate liabilities[J]. Journal of Political Economy, 1973, 81(3): 637-659.
[9] Merton, R.C. Theory of Rational Option Pricing[J]. Bell Journal of Economics and Management Science, 1973, 4: 141-183.
[10] Engle, R.F. Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation[J]. Econometrica, 1982, 50: 987-1007.
[11] Bollerslev, T. Generalized autoregressive conditional heteroskedasticity[J]. Journal of Econometrics, 1986, 31: 307-327.
[12] Group of Thirty. Derivatives: Practices and Principles. New York:Group of Thirty; 1993.
[13] Giot, P., Laurent, S. Market risk in commodity markets: a VaR approach[J]. Energy Economics, 2003, 25(5): 435-457.
[14] Angelidis, T., Benos, A., Degiannakis, S. The use of GARCH models in VaR estimation[J]. Statistical Methodology, 2004, 1(1-2): 105-128.
[15] Lehnert, T., Wolff, C.C.P. Scale-consistent Value-at-Risk[J]. Finance Research Letters, 2004, 1(2): 127-134.
[16] Gen?ay, R., Sel?uk, F. Extreme value theory and Value-at-Risk: Relative performance in emerging markets[J]. International Journal of Forecasting, 2004, 20(2): 287-303.
[17] Jacobson, T., Roszbach, K. Bank lending policy, credit scoring and value-at-risk[J]. Journal of Banking and Finance, 2003, 27(4): 615-633.
[18] Gordy, M.B. A risk-factor model foundation for ratings-based bank capital rules[J]. Journal of Financial Intermediation, 2003, 12(3): 199-232.
[19] Longstaff, F.A., Schwartz, E.S. A simple approach to valuing risky fixed and floating rate debt[J]. The Journal of Finance, 1995, 50(3): 789.
[20] Eric, B., Francois de, V. On the risk of insurance liabilities: Debunking some common pitfalls[J]. Journal of Risk and Insurance, 1997, 64(4): 673.
[21] Leland, H.E. Corporate debt value, bond covenants, and optimal capital structure[J]. The Journal of Finance, 1994, 49(4): 1213-1252.
[22] Leland, H.E., Toft, K.B. Optimal capital structure, endogenous bankruptcy, and the term structure of credit spreads[J]. The Journal of Finance, 1996, 51(3): 987-1019.
[23] Pierre, M.-B., William, P. Strategic debt service[J]. The Journal of Finance, 1997, 52(2): 531-556.
[24] Jarrow, R.A., Turnbull, S.M. Pricing derivatives on financial securities subject to credit risk[J]. The Journal of Finance, 1995, 50(1): 53-85.
[25] Duffie, D., Singleton, K.J. An econometric model of the term structure of interest-rate swap yields[J]. The Journal of Finance, 1997, 52(4): 1287-1322.
[26] Madan, D.B., Unal, H. Pricing the risks of default[J]. Review of Derivatives Research, 1998, 2(2-3): 121-160.
[27] Duffie, D., Singleton, K.J. Modeling term structures of defaultable bonds[J]. The Review of Financial Studies, 1999, 12(4): 687.
[28] Robert, A.J., David, L., Stuart, M.T. A markov model for the term structure of credit risk spreads[J]. The Review of Financial Studies (1986-1998), 1997, 10(2): 481.
[29] Lando, D. On Cox Processes and Credit Risky Securities[J]. Review of Derivatives Research, 1998, 2(2-3): 99-120.
[30] Bielecki, T.R., Rutkowski, M. Multiple Ratings Model of Defaultable Term Structure[J]. Mathematical Finance, 2000, 10(2): 125-139.
[31] Zhou, C. A jump-diffusion approach to modeling credit risk and valuing defaultable securities. In: Board of Governors of the Federal Reserve System (U.S.), Finance and Economics Discussion Series: 1997-15; 1997.
[32] Giesecke, K. Default and Information. In: http://www.defaultrisk.com, Working Paper; 2003.
[33] Cho-Jieh, C., Harry, P. Unifying discrete structural models and reduced-form models in credit risk using a jump-diffusion process[J]. Insurance, Mathematics & Economics, 2003, 33(2): 357.
[34] Yang, H. An integrated risk management method: VaR approach[J]. Multinational Finance Journal, 2000, 4(3/4): 201.
[35] Theodore, M.B., Jr., William, F.M. Modeling correlated market and credit risk in fixed income portfolios[J]. Journal of Banking & Finance, 2002, 26(2,3): 347.
[36] Zhou, C. The term structure of credit spreads with jump risk[J]. Journal of Banking & Finance, 2001, 25(11): 2015-2040.
[37] Georges, H. The analytic pricing of asymmetric defaultable swaps[J]. Journal of Banking & Finance, 2001, 25(2): 295.
[38] Frank, S.S., Timothy, G.T. An empirical analysis of credit default swaps[J]. International Review of Financial Analysis, 2002, 11(3): 297-309.
[39] Christine, A.B., Sally, W. Credit risk: The case of First Interstate Bankcorp[J]. International Review of Financial Analysis, 2002, 11(2): 229.
[40] Gordy, M.B. A comparative anatomy of credit risk models[J]. Journal of Banking & Finance, 2000, 24(1,2): 119-149.
[41] Michel, C., Dan, G., Robert, M. A comparative analysis of current credit risk models[J]. Journal of Banking & Finance, 2000, 24(1,2): 59.
[42] Pamela, N., William, P., Simone, V. Stability of rating transitions[J]. Journal of Banking & Finance, 2000, 24(1,2): 203.
[43] Treacy, W.F., Carey, M. Credit risk rating systems at large US banks[J]. Journal of Banking & Finance, 2000, 24(1-2): 167-201.
[44] 张维, 李玉霜. 商业银行信用风险分析综述[J]. 管理科学学报, 1998, 1(03): 20-27.
[45] 王春峰, 万海晖, 张维. 基于神经网络技术的商业银行信用风险评估[J]. 系统工程理论与实践, 1999, 09: 24-32.
[46] 陈雄华, 林成德, 叶武. 基于神经网络的企业信用等级评估[J]. 系统工程学报, 2002, 06.
[47] 庞素琳, 王燕鸣, 黎荣舟. 基于BP 算法的信用风险评价模型研究[J]. 数学的实践与认识, 2003, 08.
[48] 王春峰, 康莉. 基于遗传规划方法的商业银行信用风险评估模型[J]. 系统工程理论与实践, 2001, 02.
[49] 何运强, 方兆本. 债券信用评级与信用风险[J]. 管理科学, 2003, 02.
[50] 王焜, 徐枞巍. 企业信用风险管理的系统分析[J]. 管理评论, 2002, 09.
[51] 陈忠阳. 信用风险量化管理模型发展探析[J]. 国际金融研究, 2000, 10.
[52] 张玲, 白效锋. 试论信用风险计量法在金融机构风险管理中的应用[J]. 金融理论与实践, 2000, 05.
[53] 程鹏, 吴冲锋, 李为冰. 信用风险度量和管理方法研究[J]. 管理工程学报, 2002, 16(01): 70-73.
[54] 陶铄, 杨晓光. 商业银行内部信用评级的比较研究[J]. 管理评论, 2002, 09.
[55] 王春峰, 李汶华. 小样本数据信用风险评估研究[J]. 管理科学学报, 2001, 01.
[56] 刘宏峰, 杨晓光. 违约损失率的估计:发达国家的经验及启示[J]. 管理评论, 2003a, 06.
[57] 邓云胜, 沈沛龙, 任若恩. 贷款组合信用风险VaR 的蒙特卡罗仿真[J]. 计算机仿真, 2003, 02.
[58] 刘宏峰, 杨晓光. 信用衍生产品的类型及用途[J]. 管理评论, 2003b, 02.
[59] Robert, F.E., David, M.L., Russell, P.R. Estimating time varying risk premia in the term structure: the ARCH-M model[J]. Econometrica (1986-1998), 1987, 55(2): 391.
[60] David, X.L. On default correlation: A copula function approach[J]. The Journal of Fixed Income, 2000, 9(4): 43.
[61] Giesecke, K. A Simple Exponential Model for Dependent Defaults[J]. The Journal of Fixed Income, 2003, 13(3): 74.
[62] Beatriz Vaz de Melo, M., Rafael Martins de, S. Measuring financial risks with copulas[J]. International Review of Financial Analysis, 2004, 13(1): 27.
[63] Estrella, A., Hendricks, D., Kambhu, J., et al. The price risk of options positions: Measurement and capital requirements[J]. Federal Reserve Bank of New York Quarterly Review, 1994, 19(2): 27.
[64] Rockafeller, T., Uryasev, S. Optimization of conditional value-at-risk[J]. Journal of Risk, 2000, 2(3): 21-24.
[65] Jun-Koo, K., Lee, Y.W. The pricing of convertible debt offerings[J]. Journal of Financial Economics, 1996, 41(2): 231-248.
[66] Franke, C., Schaback, R. Solving partial differential equations by collocation using radial basis functions[J]. Applied Mathematics and Computation, 1998, 93(1): 73-82.
[67] Hon, Y.C. A quasi-radial basis functions method for American options pricing[J]. Computers and Mathematics with Applications, 2002, 43(3-5): 513-524.
[68] Kansa, E.J., Hon, Y.C. Circumventing the Ill-Conditioning Problem with Multiquadric Radial Basis Functions: Applications to Elliptic Partial Differential Equations[J]. Computers and Mathematics with Applications, 2000, 39(7-8): 123-137.
[69] Lianfen, Q. Regularized Radial Basis Function Networks: Theory and Applications[J]. Technometrics, 2002, 44(3): 294.
[70] Zhai, Y., Yao, D. A radial-basis function based surface Laplacian estimate for a realistic head model[J]. Brain Topography [NLM -MEDLINE], 2004, 17(1): 55.
[71] Cooper, I.A. Asset Values, Interest-Rate Changes, and Duration[J]. Journal of Financial and Quantitative Analysis, 1977, 12(5): 701.
[72] Litterman, R., Scheinkman, J. Common Factors Affecting Bond Returns[J]. Journal of Fixed Income, 1991, 1(1): 54-61.
[73] Derman, E. The Future of Modeling[J]. Risk, 1997, (12): 164-167.
[74] Ho, T.S.Y. Key Rate Durations: Measures of Interest Rate Risks[J]. The Journal of Fixed Income, 1992, 2(2): 29.
[75] Reitano, R.R. Non-parallel yield curve shifts and stochastic immunization[J]. Journal of Portfolio Management, 1996, 22(2): 71.
[76] Johnson, B.D., Meyer, K.R. Managing Yield Curve Risk In An Index Environment[J]. Financial Analysts Journal, 1989, 45(6): 51.
[77] Nawalkha, S.K., Chambers, D.R. The M-vector model: Derivation and testing of extensions to M-square[J]. Journal of Portfolio Management, 1997, 23(2): 92.
[78] Gloria, M.S. Immunization derived from a polynomial duration vector in the Spanish bond market[J]. Journal of Banking & Finance, 2001, 25(6): 1037.
[79] Heiko, L. Managing yield-curve risk with combination hedges[J]. Financial Analysts Journal, 2001, 57(3): 63.