混合模型下具有动态违约边界的债券定价
|
摘要
在混合模型下,研究了具有动态违约边界的公司债券定价问题.首先利用风险中性定价原理建立此定价问题的数学模型.然后,应用函数代换技巧和偏微分方程镜像法给出模型的显式解.最后,通过一个算例分析动态违约边界对公司债券价格的影响.结果表明:通过调整违约边界的相关参数值,可以得到不同形状的债券价格曲线,进而控制风险或得到更高的债券收益率.
In this paper, a pricing problem for corporate bond with dynamic default barrier is studied under a hybrid model. Firstly, a mathematical model for the pricing problem is set up by applying risk-free equilibrium principle. Then, a closed-form formula for the pricing model is obtained by using the variable transformation technique and the image method, which extends the relevant literature's results. Finally,a numerical experiment is presented to analyze the effect of the dynamic barrier on the bond price. Our studies show that the different shape curve of a bond's price can be obtained by adjusting the relevant parameter on the default boundary, and then can control the risk or get a higher bond's yield.
In this paper, a pricing problem for corporate bond with dynamic default barrier is studied under a hybrid model. Firstly, a mathematical model for the pricing problem is set up by applying risk-free equilibrium principle. Then, a closed-form formula for the pricing model is obtained by using the variable transformation technique and the image method, which extends the relevant literature's results. Finally,a numerical experiment is presented to analyze the effect of the dynamic barrier on the bond price. Our studies show that the different shape curve of a bond's price can be obtained by adjusting the relevant parameter on the default boundary, and then can control the risk or get a higher bond's yield.
引文
[1] BIELECKI T R, RUTKOWSKI M.信用风险:建模、估值和对冲[M].唐齐鸣,等译.上海:格致出版社/上海人民出版社,2011.
[2] MADAN D, UNAL H. A two-factor hazard rate model for pricing risky debt and the term structure of credit spreads[J]. J Financ Quant Anal, 2000, 35(1):43-65.
[3] CATHCART L, EL-JAHEL L. Pricing defaultable bonds:a middle-way approach between structural and reduced-form models[J]. Quant Financ, 2006, 6(3):243-253.
[4] REALDON M. Credit risk pricing with both expected and unexpected default[J]. Appl Financ Econ Lett, 2007, 3(4):225-230.
[5]毕玉升,边保军.具可料和不可料违约的公司债券定价[J].同济大学学报(自然科学版),2007, 35(7):989-993.
[6] HOCHT S, ZAGST R. Pricing credit derivatives under stochastic recovery in a hybrid model[J]. Appl Stoch Models Bus Ind, 2010, 26(3):254-276.
[7] EMMANUEL F S, HELEN E O. On hybrid model for the valuation of credit risk[J]. Appl Comput Math, 2014, 3(6-1):8-11.
[8] O H C, JO J J, KIM C H. Pricing corporate defaultable bond using declared firm value[J]. Electron J Math Anal Appl, 2014, 2(1):1-11.
[9] BALLESTRA L V, PACELLI G. Valuing risky debt:a new model combining structural information with the reduced-form approach[J]. Insurance Math Econom, 2014, 55:261-271.
[10] PAN J, XIAO Q X. Pricing defaultable bonds with stochastic recovery under a hybrid model[J].Chinese J Engrg Math, 2016, 36(6):631-650.
[11] MERTON R C. On the pricing of corporate debt:the risk structure of interest rates[J]. J Finance,1974, 29(2):449-470.
[12] BLACK F, COX J C. Valuing corporate securities:some effects of bond indenture provisions[J]. J Finance, 1976, 31(2):351-367.
[13] LONGSTAFF F A, SCHWARTZ E S. A simple approach to valuing risky fixed and floating rate debt[J]. J Finance, 1995, 50(3):789-819.
[14] BRIYS E, DE VARENNE F. Valuing risky fixed rate debt:an extension[J]. J Financ Quant Anal,1997, 32(2):239-248.
[15] HUI C H, LO C F, TSANG S W. Pricing corporate bonds with dynamic default barriers[J]. J Risk,2003, 5(3):17-37.
[16] LO C F, HUI C H. Lie-algebraic approach for pricing moving barrier options with time-dependent parameters[J]. J Math Anal Appl, 2006, 323(2):1455-1464.
[17] BERNARD C, LE COURTOIS O, QUITTARD-PINON F. Pricing derivatives with barriers in a stochastic interest rate environment[J]. J Econom Dynam Control, 2008, 32(9):2903-2938.
[18] SHREVE S E. Stochastic Calculus for Finance II:Continuous-Time Models[M]. New York:SpringerVerlag, 2004.
[19] WILMOTT P. Derivatives:The Theory and Practice of Financial Engineering[M]. Chichester:Wiley,1998.
[20] BUCHEN P. Image options and the road to barriers[J]. Risk Manag, 2001, 14(9):127-130.
[21]任学敏,魏嵬,姜礼尚,等.信用风险估值的数学模型与案例分析[M].北京:高等教育出版社,2014.
[2] MADAN D, UNAL H. A two-factor hazard rate model for pricing risky debt and the term structure of credit spreads[J]. J Financ Quant Anal, 2000, 35(1):43-65.
[3] CATHCART L, EL-JAHEL L. Pricing defaultable bonds:a middle-way approach between structural and reduced-form models[J]. Quant Financ, 2006, 6(3):243-253.
[4] REALDON M. Credit risk pricing with both expected and unexpected default[J]. Appl Financ Econ Lett, 2007, 3(4):225-230.
[5]毕玉升,边保军.具可料和不可料违约的公司债券定价[J].同济大学学报(自然科学版),2007, 35(7):989-993.
[6] HOCHT S, ZAGST R. Pricing credit derivatives under stochastic recovery in a hybrid model[J]. Appl Stoch Models Bus Ind, 2010, 26(3):254-276.
[7] EMMANUEL F S, HELEN E O. On hybrid model for the valuation of credit risk[J]. Appl Comput Math, 2014, 3(6-1):8-11.
[8] O H C, JO J J, KIM C H. Pricing corporate defaultable bond using declared firm value[J]. Electron J Math Anal Appl, 2014, 2(1):1-11.
[9] BALLESTRA L V, PACELLI G. Valuing risky debt:a new model combining structural information with the reduced-form approach[J]. Insurance Math Econom, 2014, 55:261-271.
[10] PAN J, XIAO Q X. Pricing defaultable bonds with stochastic recovery under a hybrid model[J].Chinese J Engrg Math, 2016, 36(6):631-650.
[11] MERTON R C. On the pricing of corporate debt:the risk structure of interest rates[J]. J Finance,1974, 29(2):449-470.
[12] BLACK F, COX J C. Valuing corporate securities:some effects of bond indenture provisions[J]. J Finance, 1976, 31(2):351-367.
[13] LONGSTAFF F A, SCHWARTZ E S. A simple approach to valuing risky fixed and floating rate debt[J]. J Finance, 1995, 50(3):789-819.
[14] BRIYS E, DE VARENNE F. Valuing risky fixed rate debt:an extension[J]. J Financ Quant Anal,1997, 32(2):239-248.
[15] HUI C H, LO C F, TSANG S W. Pricing corporate bonds with dynamic default barriers[J]. J Risk,2003, 5(3):17-37.
[16] LO C F, HUI C H. Lie-algebraic approach for pricing moving barrier options with time-dependent parameters[J]. J Math Anal Appl, 2006, 323(2):1455-1464.
[17] BERNARD C, LE COURTOIS O, QUITTARD-PINON F. Pricing derivatives with barriers in a stochastic interest rate environment[J]. J Econom Dynam Control, 2008, 32(9):2903-2938.
[18] SHREVE S E. Stochastic Calculus for Finance II:Continuous-Time Models[M]. New York:SpringerVerlag, 2004.
[19] WILMOTT P. Derivatives:The Theory and Practice of Financial Engineering[M]. Chichester:Wiley,1998.
[20] BUCHEN P. Image options and the road to barriers[J]. Risk Manag, 2001, 14(9):127-130.
[21]任学敏,魏嵬,姜礼尚,等.信用风险估值的数学模型与案例分析[M].北京:高等教育出版社,2014.