带跳扩散的信用风险模型相关问题的研究
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摘要
随着金融的全球化趋势及金融市场的波动性加剧,各国银行和投资者受到了前所未有的信用风险的挑战。世界银行对全球银行业危机的研究表明,导致银行破产的主要原因就是信用风险。因此信用风险及其衍生品成为了经济社会的一个引人关注的课题,我们有必要对当前国际上流行的信用风险模型和技术方法进行研究。
本文在经典信用风险模型的基础上,对于带跳扩散的信用风险模型,当价格和信用价差与违约时间及违约时刻公司的期望折现值有关时,研究该模型的违约概率、盈余分布及零票息债券的价格和信用价差。通过“首次到达时刻”的方法,利用It?’s公式给出违约概率的Laplace变换的积分微分方程。进一步,分析违约时公司的盈余分布,得到其积分微分方程;最后对确定的跳分布,计算违约时刻的数值解。
With the financial globalization trend and the intensifying undulation of the financial markets, commercial banks all over the world and investors have all been faced the unprecedented challenge by credit risk. The World Bank studied the crises of the global banking, which showed that the main cause of the bank bankruptcy is credit risk. Therefore, the credit risk and its derivatives have become interest economic social issues. It is necessary for us to consider the model and the method of the credit risk.
This paper studies some characterizes of the credit risk model with jump-diffusion. We discuss the default probability and the surplus distribution at default. Applying the first passage time approach, we present the integro-differential equation of the default probability and the surplus distribution at default by the It?’s formula. Finally, an example is given when the distribution of the jump size is known.
本文在经典信用风险模型的基础上,对于带跳扩散的信用风险模型,当价格和信用价差与违约时间及违约时刻公司的期望折现值有关时,研究该模型的违约概率、盈余分布及零票息债券的价格和信用价差。通过“首次到达时刻”的方法,利用It?’s公式给出违约概率的Laplace变换的积分微分方程。进一步,分析违约时公司的盈余分布,得到其积分微分方程;最后对确定的跳分布,计算违约时刻的数值解。
With the financial globalization trend and the intensifying undulation of the financial markets, commercial banks all over the world and investors have all been faced the unprecedented challenge by credit risk. The World Bank studied the crises of the global banking, which showed that the main cause of the bank bankruptcy is credit risk. Therefore, the credit risk and its derivatives have become interest economic social issues. It is necessary for us to consider the model and the method of the credit risk.
This paper studies some characterizes of the credit risk model with jump-diffusion. We discuss the default probability and the surplus distribution at default. Applying the first passage time approach, we present the integro-differential equation of the default probability and the surplus distribution at default by the It?’s formula. Finally, an example is given when the distribution of the jump size is known.
引文
[1] Lundberg F I. Approxmerad Framst?lling av Scannolik-hetsfunktionen.Ⅱ.àterf?rs?kring av Kollektivrisker. Uppsala: Almqvist & Wiksell, 1903.
[2] Bohlmann G. Die Theorie des Mittleren Risikos in des Lebensve-sicherung. Transactions of the International Congress of Actuaries, 1909, 1: 593~683.
[3] Lundberg F. F?rs?kringsteknisk Riskutjamming. F. Englunds, A. B. Boktryckeri, Stockholm, 1926.
[4] Cramér H. On the mathematical theory of risk. Stockholm: Skandia Jubilee Volume, 1930.
[5] Cramér H. On some questions connected with mathematical risk. California: University of California Press, 1954.
[6] Cramér H. Collective risk theory. Stockholm: Skandia Jubilee Volume, 1955.
[7] Cramér H. Half a century with probability theory: some personal recollection. The Annals of Probability, 1976, 4: 509~547.
[8] Saunders, Anthony.信用风险度量:风险估值的新方法与其他范式(刘宇飞译).北京:机械工业出版社,2001.
[9] Deventer, Donald R.van.信用风险模型与巴塞尔协议(燕清联合,周天芸译).北京:中国人民大学出版社,2005.
[10]Belmont, David P.金融机构的增值风险管理(洪凯,李华罡,余黎峰译).北京:中国人民大学出版社,2009.
[11] Black, Scholes. The Pricing of Options and Corporate Liabilities, Journal of Political Economy, 1973, 81, no. 3.
[12] Merton, R.. On the pricing of corporate debt: The risk structure if interest rates [J], J. Finance, 1974, 29:449-470.
[13] Robert A. Jarrow, Stuart Turnbull. Pricing Derivatives on Financial Securities Subject to Credit Risk. Journal of Finance, 1995, vol. 50, March.
[14] Jarrow, R., Lando, D., Turnbull, S.. A Markov model for the term structure of credit spreads. Review of Financial Studies, 1997, 10, 481-523.
[15] Duffie, D., Singleton, K.. Modeling term structure of defaultable bond. Review of Financial Studies, 1999, 12, 687-720.
[16] Shimko, D. C., N. Tejima, and D. R. van Deventer. The Pricing of Risky Debtwhen Interest Rates are Stochastic. The Journal of Fixed Income, 1993, pages 58–65.
[17] Jarrow, R. A. and F. Yu. Counterparty Risk and the Pricing of Defaultable Securities. Journal ofFinance,2001, 56(5): 1765-99.
[18] Black, F., Cox, J.. Valuing corporate securities liabilities: Some effects of bond indenture provisions. Journal of Finance, 1976, 31, 351-367.
[19] Longstaff, F., Schwartz, E.. A simple approach to valuing risky and floating rate debt. Journal of Finance, September 1995, 50(3): 789-819.
[20] Collin-Dufresne, P., Goldstein, R. S.. Do credit spreads reflect stationary leverage ratios? Journal of Finance, 2001, 56, 1929-1957.
[21] Zhou, C. S.. The term structure of credit spreads with jump risk. Journal of Banking and Finance, 2001, 25, 2015-2040.
[22] Hilberink, B., Rogers, L. C. G.. Optimal capital structure and endogenous default. Finance and Stochastics, 2002, 6, 237-263.
[23] Le Courtois, O., Quittard-Pinon, F.. Risk-neutral and default probabilities with an Endogenous bankruptcy jump-diffusion model. Asia Pacific Finance Markets, 2006, 13, 11-39.
[24] Kiesel, R., Scherer, M.. Dynamic credit portfolio modeling in structure models with jumps. At www.defaultrisk.com=pp?model170.htm, 2007.
[25] Duffie, D., Singleton, K.. An econometric model of the term structure of interest-rate swap yield [J]. Journal of Finance, 1997, September, 52(1):1287-1321
[26] D. Lando. On cox processes and credit risk securities[J]. Review of Derivatives research, 1998, 2:99-120.
[27] Madan, D. B., Unal, H.. Pricing the risks of default. The review of Derivatives Research, 1998, 2, 121-160.
[28] Guojing Wang, Rong Wu. Distributions for the risk process with a stochastic return on investments Stochastic Processes and their Applications, 2001, 95, 329–341.
[29] Hailiang Yang. Ruin theory in a financial corporation model with credit risk. Insurance: Mathematics and Economics, 2003, 33, 135–145.
[30] Guojing Wang, Rong Wu. Pricing Zero-coupon bond and its fair premium under a structural credit risk model with jumps. Preprint, 2009.
[31] Danilo M. and Spokoinyi V..“Estimation of time dependent volatility via local change point analysis”WIAS Preprint, 2004, No. 904.
[32] Goldentyer L., Klebaner F.C. and Liptser R..“Tracking volatility”Problems of Information Transmition forthcoming, 2005.
[33]Jones, E. Philip, Scott P. Mason, and Eric Rosenfeld. Contingent Claims Analysis of Corporate Capital Structures: an Empirical Investigation. Journal of Finance, 1984, 39, No. 3, 611-625.
[34] Leland, Hayne E.. Corporate Debt Value, Bond Covenants, and Optimal Capital Structure. The Journal of Finance, 1994, 49, No. 4, 1213-1252.
[35] Anderson, R. W., Sundaresan, S.. Design and valuation of debt contracts. Review of Financial Studies, 1996, 9, 37–68.
[36] Gerber, H.U., Shiu, E.S.W..The time value of ruin in a Sparre Andersen model. North American Actuarial Journal, 2005, 9 (2), 49-84.
[37] Yuen, K.C., Lu, L., Wu, R.. The compound Poisson process perturbed by a diffusion with a threshold dividend strategy. Applied Stochastic Models in Business and Industry, 2009, 25(1), 73-93.
[2] Bohlmann G. Die Theorie des Mittleren Risikos in des Lebensve-sicherung. Transactions of the International Congress of Actuaries, 1909, 1: 593~683.
[3] Lundberg F. F?rs?kringsteknisk Riskutjamming. F. Englunds, A. B. Boktryckeri, Stockholm, 1926.
[4] Cramér H. On the mathematical theory of risk. Stockholm: Skandia Jubilee Volume, 1930.
[5] Cramér H. On some questions connected with mathematical risk. California: University of California Press, 1954.
[6] Cramér H. Collective risk theory. Stockholm: Skandia Jubilee Volume, 1955.
[7] Cramér H. Half a century with probability theory: some personal recollection. The Annals of Probability, 1976, 4: 509~547.
[8] Saunders, Anthony.信用风险度量:风险估值的新方法与其他范式(刘宇飞译).北京:机械工业出版社,2001.
[9] Deventer, Donald R.van.信用风险模型与巴塞尔协议(燕清联合,周天芸译).北京:中国人民大学出版社,2005.
[10]Belmont, David P.金融机构的增值风险管理(洪凯,李华罡,余黎峰译).北京:中国人民大学出版社,2009.
[11] Black, Scholes. The Pricing of Options and Corporate Liabilities, Journal of Political Economy, 1973, 81, no. 3.
[12] Merton, R.. On the pricing of corporate debt: The risk structure if interest rates [J], J. Finance, 1974, 29:449-470.
[13] Robert A. Jarrow, Stuart Turnbull. Pricing Derivatives on Financial Securities Subject to Credit Risk. Journal of Finance, 1995, vol. 50, March.
[14] Jarrow, R., Lando, D., Turnbull, S.. A Markov model for the term structure of credit spreads. Review of Financial Studies, 1997, 10, 481-523.
[15] Duffie, D., Singleton, K.. Modeling term structure of defaultable bond. Review of Financial Studies, 1999, 12, 687-720.
[16] Shimko, D. C., N. Tejima, and D. R. van Deventer. The Pricing of Risky Debtwhen Interest Rates are Stochastic. The Journal of Fixed Income, 1993, pages 58–65.
[17] Jarrow, R. A. and F. Yu. Counterparty Risk and the Pricing of Defaultable Securities. Journal ofFinance,2001, 56(5): 1765-99.
[18] Black, F., Cox, J.. Valuing corporate securities liabilities: Some effects of bond indenture provisions. Journal of Finance, 1976, 31, 351-367.
[19] Longstaff, F., Schwartz, E.. A simple approach to valuing risky and floating rate debt. Journal of Finance, September 1995, 50(3): 789-819.
[20] Collin-Dufresne, P., Goldstein, R. S.. Do credit spreads reflect stationary leverage ratios? Journal of Finance, 2001, 56, 1929-1957.
[21] Zhou, C. S.. The term structure of credit spreads with jump risk. Journal of Banking and Finance, 2001, 25, 2015-2040.
[22] Hilberink, B., Rogers, L. C. G.. Optimal capital structure and endogenous default. Finance and Stochastics, 2002, 6, 237-263.
[23] Le Courtois, O., Quittard-Pinon, F.. Risk-neutral and default probabilities with an Endogenous bankruptcy jump-diffusion model. Asia Pacific Finance Markets, 2006, 13, 11-39.
[24] Kiesel, R., Scherer, M.. Dynamic credit portfolio modeling in structure models with jumps. At www.defaultrisk.com=pp?model170.htm, 2007.
[25] Duffie, D., Singleton, K.. An econometric model of the term structure of interest-rate swap yield [J]. Journal of Finance, 1997, September, 52(1):1287-1321
[26] D. Lando. On cox processes and credit risk securities[J]. Review of Derivatives research, 1998, 2:99-120.
[27] Madan, D. B., Unal, H.. Pricing the risks of default. The review of Derivatives Research, 1998, 2, 121-160.
[28] Guojing Wang, Rong Wu. Distributions for the risk process with a stochastic return on investments Stochastic Processes and their Applications, 2001, 95, 329–341.
[29] Hailiang Yang. Ruin theory in a financial corporation model with credit risk. Insurance: Mathematics and Economics, 2003, 33, 135–145.
[30] Guojing Wang, Rong Wu. Pricing Zero-coupon bond and its fair premium under a structural credit risk model with jumps. Preprint, 2009.
[31] Danilo M. and Spokoinyi V..“Estimation of time dependent volatility via local change point analysis”WIAS Preprint, 2004, No. 904.
[32] Goldentyer L., Klebaner F.C. and Liptser R..“Tracking volatility”Problems of Information Transmition forthcoming, 2005.
[33]Jones, E. Philip, Scott P. Mason, and Eric Rosenfeld. Contingent Claims Analysis of Corporate Capital Structures: an Empirical Investigation. Journal of Finance, 1984, 39, No. 3, 611-625.
[34] Leland, Hayne E.. Corporate Debt Value, Bond Covenants, and Optimal Capital Structure. The Journal of Finance, 1994, 49, No. 4, 1213-1252.
[35] Anderson, R. W., Sundaresan, S.. Design and valuation of debt contracts. Review of Financial Studies, 1996, 9, 37–68.
[36] Gerber, H.U., Shiu, E.S.W..The time value of ruin in a Sparre Andersen model. North American Actuarial Journal, 2005, 9 (2), 49-84.
[37] Yuen, K.C., Lu, L., Wu, R.. The compound Poisson process perturbed by a diffusion with a threshold dividend strategy. Applied Stochastic Models in Business and Industry, 2009, 25(1), 73-93.