股价服从跳扩散的可转换债券定价模型
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摘要
可转换债券是一种复合性融资工具,可由债券在一定的时间按照一定的比例转换成股票,因此兼具债券、股票和期权的特征。对于可转换债券投资者而言,既可以选择持有债券,要求公司还本付息;也可选择在约定的时间内将债券转换成标的股票,享受股利分配或资本增值。可转换债券作为一种金融衍生品种,具有风险低、收益高的特点,近年来在我国发展十分迅速,市场日益成熟和繁荣。
但是,由于可转换债券引入我国时间不长,市场投资者对其价值还不是很了解,相关的理论研究还集中在定性分析和条款设计上。在这个背景下本文研究可转换债券价值,对我国可转换债券市场以及金融产品的创新都有非常重要的理论意义和现实意义。
本文在传统的Black-Scholes可转换债券定价模型的基础上加以一定的改进,使可转换债券的定价更准确。本文用跳扩散模型代替传统的几何布朗运动来描述股票价格运动的过程,把跳扩散模型下的期权定价方法运用到可转债定价中去。
本文的核心内容由两部分组成。第一部分建立了以股票价格为基础变量的单因素可转债定价模型。首先利用无套利原理导出可转债满足的偏微分方程,然后通过迭代法得出定价公式,最后以南山转债为例进行应用分析。第二部分对第一部分进行拓展,引入随机利率模型,建立以股票价格和利率为基础变量的双因素定价模型,利用鞅方法得出定价公式。但是本文仍没有考虑可转债的转股价格调整条款和转股价格修正条款,由于这些因素具有极大的不确定性,所以本文也就没有考虑,这将是未来有待进一步研究方向。
Convertible bonds(CBs) are a kind of complex financial instruments, that endow investors the rights to hold the bonds till maturity or to transform the bonds into stocks in accordance with a certain proportion, since they combine the features of bonds、stocks and options. As CBs holders, they can choose to hold bonds and ask the issuing company to pay interest and capital back or to convert the bonds to the underlying stocks to gain the dividend and premium. In China CBs as low-risk and high-return financial derivatives developed rapidly, convertible market becomes more mature and prosperous in recent years.
However because the time from convertible bond introduced in our national financial market was still short, investors can't understand their value enough, and there are only a few system studies on convertible bond pricing, the primarily research on the convertible bond still focused on the qualitative analysis and clause design, pricing study was not enough. So the research on the pricing of convertible bond has very important theoretical and practical meaning to China's financial market and product innovation.
This paper improves the traditional Black-Scholes pricing model to make the pricing of CBs more accurate. The jump-diffusion model instead of the traditional geometric Brown motion to describe the movement of stock price, and the pricing method of options based on the jump-diffusion model is applied to price CBs.
The core of this paper consists of two parts. Firstly based upon a underlying variables: stock, the single-factor pricing model of CBs is derived through no-arbitrage principle, then the iterative method is applied for a pricing formula, finally an empirical research on Nanshan' CBs is carried out. The second part introduces the random interest rate model, and establishes the two-factor pricing model of CBs based on underlying variables: interest rate and stack, using the Martingale method for solution. But this paper still did not consider the clauses of adjusting convertible price and some other factors that influence the value of CBs. This is a task for future study.
但是,由于可转换债券引入我国时间不长,市场投资者对其价值还不是很了解,相关的理论研究还集中在定性分析和条款设计上。在这个背景下本文研究可转换债券价值,对我国可转换债券市场以及金融产品的创新都有非常重要的理论意义和现实意义。
本文在传统的Black-Scholes可转换债券定价模型的基础上加以一定的改进,使可转换债券的定价更准确。本文用跳扩散模型代替传统的几何布朗运动来描述股票价格运动的过程,把跳扩散模型下的期权定价方法运用到可转债定价中去。
本文的核心内容由两部分组成。第一部分建立了以股票价格为基础变量的单因素可转债定价模型。首先利用无套利原理导出可转债满足的偏微分方程,然后通过迭代法得出定价公式,最后以南山转债为例进行应用分析。第二部分对第一部分进行拓展,引入随机利率模型,建立以股票价格和利率为基础变量的双因素定价模型,利用鞅方法得出定价公式。但是本文仍没有考虑可转债的转股价格调整条款和转股价格修正条款,由于这些因素具有极大的不确定性,所以本文也就没有考虑,这将是未来有待进一步研究方向。
Convertible bonds(CBs) are a kind of complex financial instruments, that endow investors the rights to hold the bonds till maturity or to transform the bonds into stocks in accordance with a certain proportion, since they combine the features of bonds、stocks and options. As CBs holders, they can choose to hold bonds and ask the issuing company to pay interest and capital back or to convert the bonds to the underlying stocks to gain the dividend and premium. In China CBs as low-risk and high-return financial derivatives developed rapidly, convertible market becomes more mature and prosperous in recent years.
However because the time from convertible bond introduced in our national financial market was still short, investors can't understand their value enough, and there are only a few system studies on convertible bond pricing, the primarily research on the convertible bond still focused on the qualitative analysis and clause design, pricing study was not enough. So the research on the pricing of convertible bond has very important theoretical and practical meaning to China's financial market and product innovation.
This paper improves the traditional Black-Scholes pricing model to make the pricing of CBs more accurate. The jump-diffusion model instead of the traditional geometric Brown motion to describe the movement of stock price, and the pricing method of options based on the jump-diffusion model is applied to price CBs.
The core of this paper consists of two parts. Firstly based upon a underlying variables: stock, the single-factor pricing model of CBs is derived through no-arbitrage principle, then the iterative method is applied for a pricing formula, finally an empirical research on Nanshan' CBs is carried out. The second part introduces the random interest rate model, and establishes the two-factor pricing model of CBs based on underlying variables: interest rate and stack, using the Martingale method for solution. But this paper still did not consider the clauses of adjusting convertible price and some other factors that influence the value of CBs. This is a task for future study.
引文
[1]陈超等.股票价格服从跳扩散过程的期权定价模型[J].管理工程学报.2001(2):74-75
[2]陈守红.可转换债券投融资-理论与实务[M].上海财经大学出版社.2005,10
[3]龚朴,赵海滨.双因素可转换债券定价模型FEM解[J].系统工程理论方法应用.2004(10):451-454
[4]龚朴,赵海滨,司继文.可转换债券定价的有限元方法[J].数量经济技术经济研究.2004(2):104-110
[5]姜礼尚、徐承龙等.金融衍生产品的偏微分方程模型和案例分析[M].高等教育出版社.2007,11
[6]姜礼尚.期权定价的数学模型和方法[M].高等教育出版社.2003,1
[7]赖其男.关于我国可转换债券定价的实证研究[J].金融研究.2005(9):105-121
[8]林义湘、何媛媛等.可转换债券定价理论分析[N].中国证券报.1998年2月21日
[9]刘立喜.可转换公司债券[M],上海财经大学出版社.1999
[10]宁丽娟,刘新平.股票价格服从跳-扩散过程的期权定价模型[J].陕西师范大学学报(自然科学版).2003,31(4):16-19
[11]钱晓松.跳扩散模型的测度变换与期权定价[J].应用概率统计.2004,(2):91-99
[12]邵宇.微观金融学及其数学基础[M].清华大学出版社.2005,8
[13]史树中.金融经济学,北京大学光华管理学院讲义.2001
[14]王青流,邹辉文.可转债定价理论的文献综述及基于B-S模型的应用研究[J].哈尔滨商业大学学报(社会科学版).2007,3:58-61
[15]危慧惠.中国可转换债券定价机制研究[M].中国财政经济出版社.2007,4
[16]杨如彦,魏刚等.可转债及其绩效评价[M].北京:中国人民大学出版社.2002
[17]郑小迎,陈军,陈金贤.可转换债券定价模型探讨[J].系统工程理论与发展.2000,8:24-28
[18]郑振龙,林海.中国可转债定价研究[J].厦门大学学报.2004(2):93-99
[19]约翰.赫尔(美).期权、期货和其他衍生品种[M].华夏出版社.2004,7
[20]http://www.Chinabond.com.cn,债券投资、资料检索
[21]http://www.p5w.net,全景网络
[22]Black F,Scholes M.The pricing of options and corporate liabilities[J],Journal of Political Economy.1973,81(3):637-659
[23]Brennan Michael J.,E.S.Schwarz.Convertible Bond:Valuation and Optional Strategies for Call and Conversion[J].Journal of Finance,December 1977,17(5):1699-1715
[24]Brennan M.J.and Schwartz E.S..Analyzing Convertible Bonds[J].The Journal of Financial and Quantitative Analysis.Volume 15,907-929,1980
[25]Cox J,Ross S.,Rubinstein M.Option Pricing:A Simplified Approach[J],Journal of Financial Economics,1979,10(7):22 9-264
[26]Hull J.,White A.Using Hull-White Interest Rate Trees[J].The Journal of Derivatives.1996,3(3):26-36
[27]John C.Cox,Jonathan E.Ingersoll,Jr.,Stephen A.Ross,A Theory of the Term Structure of Interest Rates[J],Econometrica,Volume53,Issue2,385-408,Mar.,1985
[28]K.G.Nyborg.The Use and Pricing of Convertible Bonds[J].Applied Mathematical Finance.1996,3:167-190
[29]F.Andre-le Pogamp,F.Moraux.Valuing Callable Convertible Bonds:A Reduced Approach[J].Applied Financial Economics.2004,14:743-749
[30]Merton R.Theory of rational option pricing[J].Journal of Financial and Management Science.1973,4(1):141-184
[2]陈守红.可转换债券投融资-理论与实务[M].上海财经大学出版社.2005,10
[3]龚朴,赵海滨.双因素可转换债券定价模型FEM解[J].系统工程理论方法应用.2004(10):451-454
[4]龚朴,赵海滨,司继文.可转换债券定价的有限元方法[J].数量经济技术经济研究.2004(2):104-110
[5]姜礼尚、徐承龙等.金融衍生产品的偏微分方程模型和案例分析[M].高等教育出版社.2007,11
[6]姜礼尚.期权定价的数学模型和方法[M].高等教育出版社.2003,1
[7]赖其男.关于我国可转换债券定价的实证研究[J].金融研究.2005(9):105-121
[8]林义湘、何媛媛等.可转换债券定价理论分析[N].中国证券报.1998年2月21日
[9]刘立喜.可转换公司债券[M],上海财经大学出版社.1999
[10]宁丽娟,刘新平.股票价格服从跳-扩散过程的期权定价模型[J].陕西师范大学学报(自然科学版).2003,31(4):16-19
[11]钱晓松.跳扩散模型的测度变换与期权定价[J].应用概率统计.2004,(2):91-99
[12]邵宇.微观金融学及其数学基础[M].清华大学出版社.2005,8
[13]史树中.金融经济学,北京大学光华管理学院讲义.2001
[14]王青流,邹辉文.可转债定价理论的文献综述及基于B-S模型的应用研究[J].哈尔滨商业大学学报(社会科学版).2007,3:58-61
[15]危慧惠.中国可转换债券定价机制研究[M].中国财政经济出版社.2007,4
[16]杨如彦,魏刚等.可转债及其绩效评价[M].北京:中国人民大学出版社.2002
[17]郑小迎,陈军,陈金贤.可转换债券定价模型探讨[J].系统工程理论与发展.2000,8:24-28
[18]郑振龙,林海.中国可转债定价研究[J].厦门大学学报.2004(2):93-99
[19]约翰.赫尔(美).期权、期货和其他衍生品种[M].华夏出版社.2004,7
[20]http://www.Chinabond.com.cn,债券投资、资料检索
[21]http://www.p5w.net,全景网络
[22]Black F,Scholes M.The pricing of options and corporate liabilities[J],Journal of Political Economy.1973,81(3):637-659
[23]Brennan Michael J.,E.S.Schwarz.Convertible Bond:Valuation and Optional Strategies for Call and Conversion[J].Journal of Finance,December 1977,17(5):1699-1715
[24]Brennan M.J.and Schwartz E.S..Analyzing Convertible Bonds[J].The Journal of Financial and Quantitative Analysis.Volume 15,907-929,1980
[25]Cox J,Ross S.,Rubinstein M.Option Pricing:A Simplified Approach[J],Journal of Financial Economics,1979,10(7):22 9-264
[26]Hull J.,White A.Using Hull-White Interest Rate Trees[J].The Journal of Derivatives.1996,3(3):26-36
[27]John C.Cox,Jonathan E.Ingersoll,Jr.,Stephen A.Ross,A Theory of the Term Structure of Interest Rates[J],Econometrica,Volume53,Issue2,385-408,Mar.,1985
[28]K.G.Nyborg.The Use and Pricing of Convertible Bonds[J].Applied Mathematical Finance.1996,3:167-190
[29]F.Andre-le Pogamp,F.Moraux.Valuing Callable Convertible Bonds:A Reduced Approach[J].Applied Financial Economics.2004,14:743-749
[30]Merton R.Theory of rational option pricing[J].Journal of Financial and Management Science.1973,4(1):141-184