带有违约风险的可转债定价及实证分析
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摘要
本文研究了带有违约风险的一类信用衍生证券(可转换债券)的定价理论,并对中国证券市场的若干可转换债券进行了实证研究。
本文首先设定股票价格和市场利率两个因素所满足的市场模型,并对可转债发行公司的信用风险进行了合理假设,在此基础上,用无套利定价的方法建立可转换债券在持有期间所满足的偏微分方程,然后结合可转换债券赎回、转换特征及信用风险的假设,将可转换债券的定价问题转化为一个线性互补问题。
由于可转债的条款结构复杂,以及可转换债券的价格对股票价格过程的强路径依赖特性,对基于此定价模型的线性互补问题一般无法求出解析解,只能寻求数值方法进行求解。本文用基于有限差分的算子分裂方法和不动点迭代算法对可转换债券的定价问题进行数值求解,并对本文提出的若干差分格式算法的稳定性、收敛性以及解的存在唯一性进行证明。得到了这类强路径依赖的可转换债券定价理论的可行数值解法。
最后,本文对我国证券市场的若干可转债品种进行实证研究。对可转债的实际价格与股票价格和市场利率的对比分析显示,可转债价值与股票价格高度正相关,与市场利率高度负相关；用本文所建模型计算出的可转债的理论价格和实际价格的对比分析显示,本文提出的模型比较适合我国可转债市场,能较好的模拟可转债价格的市场变化。通过实证研究发现,我国可转债在证券市场处于合理状况时,可转债价格处于被低估的状态,而且在可转债处于实值状态时,被低估的程度较小,在可转债处于虚值状态时,低估程度较大。通过分析,本文认为,我国可转债市场尚不完善,应该通过降低可转债发行限制、扩大发行规模、完善可转债信息、改革可转债市场制度、建立卖空机制、简化可转债条款等办法促进可转债市场的发展。
This thesis studies the pricing theory of a class of credit derivative securities with default risk and does empirical researches on some convertible bonds of our securities market.
First, we set some models that can meet the requirements of stock prices and the market interest rates, and we make some reasonable assumptions of the company's default risk. On the basis of this, we set up a partial differential equation using no-arbitrage theory for pricing convertible bonds. Then, combining with the call provisions, conversion provisions and the default risk assumptions of the convertible bonds, we convert the equation to a linear complementary problem.
In consequence of the fact that the value of convertible bonds is strong path dependent on the stock price process and that the terms of convertible bonds are very complex, so we can not find out its analytical solution and we can only solve the problem using numerical algorithm. Therefore, we solve the problem using operator splitting and fixed point iterative method based on finite difference method and analyze the stability, convergence, existence and uniqueness of the solution. In this way, we get the feasible numerical method of the convertible bonds pricing theory.
At last, we do empirical researches on some convertible bonds of our securities market. The comparative analysis of the convertible bonds value with the stock price and market interest rates shows that the convertible bonds value is highly positively related with the stock price but is negatively related with the market interest rates. Furthermore the comparative analysis of the actual price with the theoretical price indicates that our model is suitable for our market and can better simulate the changing trend of the convertible bonds price. Based on the results of our empirical research, we find that the convertible bonds are undervalued in normal circumstances. When the convertible bonds in the in-the-value conditions, the underestimate degree is low, otherwise, the underestimate degree is high. By analysis, we think our convertible bonds market is not perfect enough and our suggestion is that we should promote its development by reducing the issuance restrictions, perfecting information, reforming the institutions, simplifying the terms and expanding the issuance size of convertible bonds.
本文首先设定股票价格和市场利率两个因素所满足的市场模型,并对可转债发行公司的信用风险进行了合理假设,在此基础上,用无套利定价的方法建立可转换债券在持有期间所满足的偏微分方程,然后结合可转换债券赎回、转换特征及信用风险的假设,将可转换债券的定价问题转化为一个线性互补问题。
由于可转债的条款结构复杂,以及可转换债券的价格对股票价格过程的强路径依赖特性,对基于此定价模型的线性互补问题一般无法求出解析解,只能寻求数值方法进行求解。本文用基于有限差分的算子分裂方法和不动点迭代算法对可转换债券的定价问题进行数值求解,并对本文提出的若干差分格式算法的稳定性、收敛性以及解的存在唯一性进行证明。得到了这类强路径依赖的可转换债券定价理论的可行数值解法。
最后,本文对我国证券市场的若干可转债品种进行实证研究。对可转债的实际价格与股票价格和市场利率的对比分析显示,可转债价值与股票价格高度正相关,与市场利率高度负相关；用本文所建模型计算出的可转债的理论价格和实际价格的对比分析显示,本文提出的模型比较适合我国可转债市场,能较好的模拟可转债价格的市场变化。通过实证研究发现,我国可转债在证券市场处于合理状况时,可转债价格处于被低估的状态,而且在可转债处于实值状态时,被低估的程度较小,在可转债处于虚值状态时,低估程度较大。通过分析,本文认为,我国可转债市场尚不完善,应该通过降低可转债发行限制、扩大发行规模、完善可转债信息、改革可转债市场制度、建立卖空机制、简化可转债条款等办法促进可转债市场的发展。
This thesis studies the pricing theory of a class of credit derivative securities with default risk and does empirical researches on some convertible bonds of our securities market.
First, we set some models that can meet the requirements of stock prices and the market interest rates, and we make some reasonable assumptions of the company's default risk. On the basis of this, we set up a partial differential equation using no-arbitrage theory for pricing convertible bonds. Then, combining with the call provisions, conversion provisions and the default risk assumptions of the convertible bonds, we convert the equation to a linear complementary problem.
In consequence of the fact that the value of convertible bonds is strong path dependent on the stock price process and that the terms of convertible bonds are very complex, so we can not find out its analytical solution and we can only solve the problem using numerical algorithm. Therefore, we solve the problem using operator splitting and fixed point iterative method based on finite difference method and analyze the stability, convergence, existence and uniqueness of the solution. In this way, we get the feasible numerical method of the convertible bonds pricing theory.
At last, we do empirical researches on some convertible bonds of our securities market. The comparative analysis of the convertible bonds value with the stock price and market interest rates shows that the convertible bonds value is highly positively related with the stock price but is negatively related with the market interest rates. Furthermore the comparative analysis of the actual price with the theoretical price indicates that our model is suitable for our market and can better simulate the changing trend of the convertible bonds price. Based on the results of our empirical research, we find that the convertible bonds are undervalued in normal circumstances. When the convertible bonds in the in-the-value conditions, the underestimate degree is low, otherwise, the underestimate degree is high. By analysis, we think our convertible bonds market is not perfect enough and our suggestion is that we should promote its development by reducing the issuance restrictions, perfecting information, reforming the institutions, simplifying the terms and expanding the issuance size of convertible bonds.
引文
[1]A.R.Mitchell and D:F.Griffiths. The finite difference method in partial differential equations [J]. John & Sons Ltd., Chichester,1980.
[2]Beneish M.D. and E. Press. Interrelation among events of default [J]. Contemporary Accounting Research,1995,12:57-84
[3]Black F. and Cox J. C. Valuing corporate securities:some effects of bond indenture provisions [J]. Journal of Finance,1976,31.
[4]Carauannopoulos P. Valuing Convertible Bonds under the Assumptions of Stochastic Interest Rates:An Empirical Investigation [J]. Quarterly Journal of Businessand Economics,1996,35:17-31.
[5]Cheung W. and I. Nelken. Costing the converts risk [J],1994,7:47-49.
[6]Clark, T. A. and M. I. Weinstein. The behavior of the common stock of bankrupt firms [J]. Journal of Finance,1983,38:489-504.
[7]Davis M. and F. R. Lischka. Convertible bonds with market risk and credit risk [J]. Working paper, Tokyo Mitsubishi International plc,1999.
[8]E.Ayache, P.A.Foryth and K.R.Vetzal. Valuation of Convertible Bonds with Credit Risk [J]. The Journal of Derivatives,2003,5:9-28.
[9]Eaves B C. On the basic theorem of complementarity [J]. Mathematlcal Pmgramming,1978,1:68-75.
[10]F.Black and M.Scholes. The pricing of options and Corporate Liabilities [J]. The Journal of Political Economy,1973,5:637-654.
[11]Ho T. S. Y. and D. M. Pfeffer. Convertible bonds:Model, value attribution, and analytics [J]. Financial Analysts Journal,1996,52:35-44.
[12]Ingersoll Jr J. A contingent claims valuation of convertible securities [J]. Journal of Financial Economics,1977,4.
[13]Kwok Y K. Mathematical Models of Financial Derivatives [M]. Berlin: Springer-Finance,1998.
[14]Longstaff F. A. and E. S. Schwartz. A simple approach to valuing risky fixed and floating rate debt [J]. Journal of Finance,1995.50:789-819.
[15]M.J.Brennan and E.S.Schwartz. Analyzing convertible bonds [J]. Journal of Finance and Quantitative analysis,1980,15:907-929.
[16]M.J.Brennan and E.S.Schwartz. The valuation of American put options [J]. Finance,1977,32:449-462.
[17]McConnell J. J. and E. S. Schwartz. LYON taming. Journal of Finance [J]. 1986,41:561-576.
[18]Merton R C. On the pricing of corporate debt:the risk structure of interest rates [J]. Journal of Finance,1974,4.
[19]Nyborg K.G.. The use and pricing of convertible bonds [J]. Applied Mathematical Finance,1996,3:167-190.
[20]S. Ikonen and J. Toivanen. Efficient numerical methods for pricing American options under stochastic volatility [J]. Reports of the department of mathematical information technology series B.Scientific computing No.B,2005,12.
[21]S. Ikonen and J. Toivanen. Operator Splitting Methods for pricing American options with stochastic volatility [J]. Reports of the department of mathematical information technology series B.Scientific computing No.B,2004,11.
[22]Tsiveriotis K. and Fernandes C. Valuing convertible bonds with credit risk [J]. Journal of Fixed Income,1998,8:95-102.
[23]Yigitbasioglu A. B. Pricing convertible bonds with interest rate, equity, credit and fix risk [J]. Discussion paper 2001-14, ISM A Center, University of Reading.
[24]埃里克.布里斯,蒙齐尔.贝莱拉赫等.期权、期货和特种衍生证券[M].机械工业出版社,2002.
[25]陈守红.可转换债券投融资—理论与实务,上海财经大学出版社,2005(10)
[26]范辛亭,方兆本.随机利率条件下可转换债券定价模型的经验检验[J].中国管理科学,2001.
[27]何运强,方兆本.债券信用评级与信用风险[M].管理科学,2003.
[28]黄健柏,钟美瑞.考虑了信用风险的可转换债券定价模型[J].系统工程,2003,7.
[29]黄靖贵,杨善朝,冯霞.具有动态信用风险的可转债的定价研究[J].数理统计与管理,2008,11.
[30]李荣华.偏微分方程数值解法[M].高等教育出版社,2005.
[31]陆金甫,关治.偏微分方程数值解法(第二版)[M].清华大学出版社,2004.
[32]麦强,胡运权.基于信用风险模型的可转换债券定价研究[J].哈尔滨工业大学学报,2006,3.
[33]王声望,郑维行.实变函数与泛函分析概要(第二册)[M].高等教育出版社,2005.
[34]王维生,李竹香,曾宪庸.实正定阵的若干判定准则[J].哈尔滨工业大学学报,1996,8.
[35]王竹芳,潘德惠,宋俊清.随机利率条件下可转换债券的数值解法及实证检验[J].数量经济技术经济研究,2005,1.
[36]吴彦强,曹德欣,韩超.关于线性互补问题的一个迭代算法[J].淮北煤炭师范学院学报,2005,3.
[37]徐翠薇,孙绳武.计算方法引论[M].高等教育出版社,2002.
[38]雍龙泉.正定矩阵的推广及其应用[J].宝鸡文理学院学报(自然科学版),2007,6：113-115.
[39]于瑾.利率期限结构研究[J].经济日报出版社,2004.
[40]于孝建.可转换债券的定价改进模型与实证分析[J].华南理工大学学报,2006,10.
[41]张培爱、何素艳等.工程力学中的互补问题算法[J].计算力学学报,2006,12.
[42]郑小迎,陈金贤.关于可转换债券定价模型的研究[J].系统工程,1999,3.
[2]Beneish M.D. and E. Press. Interrelation among events of default [J]. Contemporary Accounting Research,1995,12:57-84
[3]Black F. and Cox J. C. Valuing corporate securities:some effects of bond indenture provisions [J]. Journal of Finance,1976,31.
[4]Carauannopoulos P. Valuing Convertible Bonds under the Assumptions of Stochastic Interest Rates:An Empirical Investigation [J]. Quarterly Journal of Businessand Economics,1996,35:17-31.
[5]Cheung W. and I. Nelken. Costing the converts risk [J],1994,7:47-49.
[6]Clark, T. A. and M. I. Weinstein. The behavior of the common stock of bankrupt firms [J]. Journal of Finance,1983,38:489-504.
[7]Davis M. and F. R. Lischka. Convertible bonds with market risk and credit risk [J]. Working paper, Tokyo Mitsubishi International plc,1999.
[8]E.Ayache, P.A.Foryth and K.R.Vetzal. Valuation of Convertible Bonds with Credit Risk [J]. The Journal of Derivatives,2003,5:9-28.
[9]Eaves B C. On the basic theorem of complementarity [J]. Mathematlcal Pmgramming,1978,1:68-75.
[10]F.Black and M.Scholes. The pricing of options and Corporate Liabilities [J]. The Journal of Political Economy,1973,5:637-654.
[11]Ho T. S. Y. and D. M. Pfeffer. Convertible bonds:Model, value attribution, and analytics [J]. Financial Analysts Journal,1996,52:35-44.
[12]Ingersoll Jr J. A contingent claims valuation of convertible securities [J]. Journal of Financial Economics,1977,4.
[13]Kwok Y K. Mathematical Models of Financial Derivatives [M]. Berlin: Springer-Finance,1998.
[14]Longstaff F. A. and E. S. Schwartz. A simple approach to valuing risky fixed and floating rate debt [J]. Journal of Finance,1995.50:789-819.
[15]M.J.Brennan and E.S.Schwartz. Analyzing convertible bonds [J]. Journal of Finance and Quantitative analysis,1980,15:907-929.
[16]M.J.Brennan and E.S.Schwartz. The valuation of American put options [J]. Finance,1977,32:449-462.
[17]McConnell J. J. and E. S. Schwartz. LYON taming. Journal of Finance [J]. 1986,41:561-576.
[18]Merton R C. On the pricing of corporate debt:the risk structure of interest rates [J]. Journal of Finance,1974,4.
[19]Nyborg K.G.. The use and pricing of convertible bonds [J]. Applied Mathematical Finance,1996,3:167-190.
[20]S. Ikonen and J. Toivanen. Efficient numerical methods for pricing American options under stochastic volatility [J]. Reports of the department of mathematical information technology series B.Scientific computing No.B,2005,12.
[21]S. Ikonen and J. Toivanen. Operator Splitting Methods for pricing American options with stochastic volatility [J]. Reports of the department of mathematical information technology series B.Scientific computing No.B,2004,11.
[22]Tsiveriotis K. and Fernandes C. Valuing convertible bonds with credit risk [J]. Journal of Fixed Income,1998,8:95-102.
[23]Yigitbasioglu A. B. Pricing convertible bonds with interest rate, equity, credit and fix risk [J]. Discussion paper 2001-14, ISM A Center, University of Reading.
[24]埃里克.布里斯,蒙齐尔.贝莱拉赫等.期权、期货和特种衍生证券[M].机械工业出版社,2002.
[25]陈守红.可转换债券投融资—理论与实务,上海财经大学出版社,2005(10)
[26]范辛亭,方兆本.随机利率条件下可转换债券定价模型的经验检验[J].中国管理科学,2001.
[27]何运强,方兆本.债券信用评级与信用风险[M].管理科学,2003.
[28]黄健柏,钟美瑞.考虑了信用风险的可转换债券定价模型[J].系统工程,2003,7.
[29]黄靖贵,杨善朝,冯霞.具有动态信用风险的可转债的定价研究[J].数理统计与管理,2008,11.
[30]李荣华.偏微分方程数值解法[M].高等教育出版社,2005.
[31]陆金甫,关治.偏微分方程数值解法(第二版)[M].清华大学出版社,2004.
[32]麦强,胡运权.基于信用风险模型的可转换债券定价研究[J].哈尔滨工业大学学报,2006,3.
[33]王声望,郑维行.实变函数与泛函分析概要(第二册)[M].高等教育出版社,2005.
[34]王维生,李竹香,曾宪庸.实正定阵的若干判定准则[J].哈尔滨工业大学学报,1996,8.
[35]王竹芳,潘德惠,宋俊清.随机利率条件下可转换债券的数值解法及实证检验[J].数量经济技术经济研究,2005,1.
[36]吴彦强,曹德欣,韩超.关于线性互补问题的一个迭代算法[J].淮北煤炭师范学院学报,2005,3.
[37]徐翠薇,孙绳武.计算方法引论[M].高等教育出版社,2002.
[38]雍龙泉.正定矩阵的推广及其应用[J].宝鸡文理学院学报(自然科学版),2007,6：113-115.
[39]于瑾.利率期限结构研究[J].经济日报出版社,2004.
[40]于孝建.可转换债券的定价改进模型与实证分析[J].华南理工大学学报,2006,10.
[41]张培爱、何素艳等.工程力学中的互补问题算法[J].计算力学学报,2006,12.
[42]郑小迎,陈金贤.关于可转换债券定价模型的研究[J].系统工程,1999,3.