多因素可转换债券定价模型及实证研究
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摘要
可转换债券是一种兼具有债券和股票特性的复杂混合金融衍生工具，赋予其持有人在规定时间内，依约定条件将持有债券转换为发行公司股票或者其他公司股票的权利。可转换债券的筹资成本低及可获取股权溢价融资等特点，使得其成为公司在资本市场进行融资的一种重要途径。由于可转换债券的定价直接影响到发行人的融资成本和投资人的获利空间，从而对可转换债券市场的发展产生深远影响，因此其定价理论已成为可转换债券研究的关键问题。
可转换债券定价相对一般的债券和期权更为困难。在现有对可转换债券定价的文献中，大部分是单因素模型，即仅仅考虑到标的股票价格的波动对可转债价格的影响，事实上，利率是金融市场上一个非常重要的因素，所有的证券价格及收益率都与之相关。同时，可转换债券作为一种企业债券，必然存在信用风险。所以，在构建可转换债券模型时，有必要将利率与信用风险的因素加入其中。此外，可转换债券定价问题在本质上仍然属于无套利定价理论的应用范畴，而现有的文献大多是针对完全市场条件下的定价研究。但完全市场的假设有时并不符合实际的投资环境。因此，有必要运用实用性更广的定价理论去探讨可转换债券定价问题。本论文结合解析法和数值法进行理论研究和实证分析，丰富了可转换债券的定价理论，具有重要的学术价值。同时，合理的可转换债券定价，必然会促进金融市场的发展和繁荣，故对于金融市场也具有十分重要的现实指导意义。
本文对可转换债券的定价模型和方法做了深入研究，并利用实际市场数据进行了详细的实证分析，主要的研究成果如下：
1.运用二叉树方法方法对随机利率下考虑信用风险的可回售可赎回可转换债券进行定价研究。考虑到标的股票价格及利率两个因素对可转换债券价格的重要性，分别采用不同的二叉树图对两者的变化过程进行刻画，并进一步考虑信用风险对可转换债券价格的影响，采用违约率和回收率进行描述，得到了含信用风险的两因子二叉树模型。在此基础上，进行了算例模拟，得到了在股票价格服从CRR模型下，利率分别服从常数波动率二叉树模型及时变波动率二叉树模型下的可转换债券定价结果。
2.运用最小二乘随机化拟蒙特卡罗（LSRQM）方法对随机利率下考虑信用风险的可回售可赎回可转换债券进行定价研究。首先建立了标的股票价格与利率的随机微分方程，并将其转换到风险中性概率空间中。考虑到可转换债券具有企业债券的性质，存在一定的信用风险。采用Jarrow和Turnbull (1995)的模型刻画信用风险，得到了包含违约风险的可转换债券的LSM方法的计算步骤。由于LSM方法的局限，本文将其修正为最小二乘随机化拟蒙特卡罗方法。在此基础上，进行了确定参数下的数值模拟，得到了随机利率及信用风险下的可转换债券理论价格，并进一步利用实际市场数据进行实证分析，从而验证了该模型的有效性。
3.运用保险精算方法对可转换债券进行定价研究。在一般的保险精算定价法的基础上，考虑资产价格的实际特点而引入广义O-U过程，由利率的回复特点建立Hull-White随机利率模型，通过利用随机微分方程的相关理论，得到了一般欧式期权及交换期权的精确解，并将一般欧式期权定价结果推广到有红利率的情形，给出了欧式买权卖权在上述模型下的平价公式，进一步得到了附加了赎回条款的可转换债券的保险精算定价的一般表达式。在理论研究的基础上，进行了数值模拟比较分析，显示出保险精算模型与B-S模型下的期权价格确实存在不同程度的差别，并且利用中国金融市场的实际数据进行了实证分析，结果表明保险精算模型的结果优于B-S模型。
4.利用效用无差别定价方法对可转换债券进行定价研究。首先分析了附加可赎回条款的可转换债券的最优投资策略，进而得到了在CIR利率模型下，可转换债券的效用无差别价格的一般表达式，并在该理论基础上，选择中国市场上的三支可赎回可转换债券作样本进行了实证分析，利用最大似然法得到参数估计值，并运用蒙特卡罗方法模拟了可转换债券理论价格过程。实证结果表明，可转债的效用无差别价格比实际市场价格要高0.24%-4.58%，且虚值状态下的可转债的低估程度较大。但与B-S模型下的理论价格相比，效用无差别价格明显要优于B-S价格，能更好地拟合实际价格。
The convertible bond is a complex financial derivative with the characteristic ofthe bond and the stock. It gives the holders the right to transform the bonds to theunderling stocks according to the agreed terms. Because of the lower financing costand the equity premium financing, the convertible bonds have became an importantway for the company to financing in the capital market. The pricing of convertiblebonds directly influences the profit of investors and the financing cost of issuers thathas a profound affect on the development of the convertible bonds market, so thepricing theory has been the critical question of the convertible bonds research.
The convertible bonds pricing is more difficulty than the common bonds andoptions. A lot of literature in convertible bonds pricing is modeled by single factorwhich only considers the influence of the underlying stocks to the convertible bondsprices. In fact, the interest rate is a very important factor in the finance market that allthe securities prices and the yield rates are related to. Meanwhile, as one kind of thecorporate bonds, the convertible bonds should involve credit risks. So we consider theinterest rate and the credit risk in the pricing models of the convertible bonds.Furthermore, the pricing problem of the convertible bonds belongs to the applicationof the no-risk arbitrage pricing theory and most of the existing literature is based onthe complete market condition. However, the hypothesis of complete marketsometimes not fit the actual investment environment. Therefore, it is necessary to usethe general pricing theory to study the pricing problem of convertible bonds. Thispaper makes the theoretical research and the empirical research using the analysismethod and the numerical method, and it enriches the pricing theory and hasimportant academic value. Moreover, the reasonable prices to the convertible bondsmust promote the development and the boom of the finance market, so it has veryimportant realistic guiding meaning.
This paper makes intensive study in the pricing models and the methods of theconvertible bonds, and makes detailed empirical research using the actual market data.The major research results are as follows.
1. This paper uses the binary tree method to pricing the convertible bonds withadditional provisions of call and put considering the stochastic interest rates andcredit risks. Consider that the underling stock prices and the interest rates are very important to the convertible bonds, different binary tree models are built to fit theirvariation processes respectively. Furturemore, consider the influence of credit riskthat is described by the default rate and recovery rate, the two-factor binary treemodel involving credit risk is obtained. On the above model, the simulation exampleis made and gets the convertible bonds prices when the stock prices follow the CRRmodel and the interest rates follow the constant volatility and time-varying volatilitybinary tree models respectively
2. Using the least-squares randomized quasi-Monte Carlo (LSRQM) method toprice the convertible bonds. Firstly, the stochastic differential equations of theunderling stock price and the interest rate are made and are converted to the riskneutral probability space. As the convertible bonds are an kind of corporate bonds thatexist particular credit risks, this paper uses the Jarrow and Turnbull (1995) model todescribe the credit risks, then the calculate procedure of the LSM method onconvertible bonds containing the default risk is obtained. Because of the restriction ofthe LSM method, this paper amends it to the least squares quasi-stochastic MonteCarlo. Moreover, the simulation example is made under the determinate parametersthat reflect the influences of the stochastic interest rates and the credit risks on theconvertible bonds. At last, the empirical analysis is made using the real market data,and the effectiveness of this model is proved.
3. This paper discusses an actuarial approach to the convertible bonds pricing.According to the definition of actuarial pricing approach, build the market modelwhere the interest rates are stochastic and the stock prices are driven by generalizedexp-Ornstein-Uhlenback process, and the exact solutions for the general Europeanoption and the exchange option are obtained with the help of the related theory ofstochastic differential equation. Then the European call-put parity relation isderived naturally, and the new prices of European call option and the put option withcontinuous dividend yield are deduced from the above results. Furthermore, theactuarial prices of convertible bonds are obtained. Based on the theoretical study, acomparative analysis of numerical simulation and an empirical analysis are madebetween the above-mentioned results and the B-S prices, the results indicate theactuarial prices are better than the B-S pricis.
4. This paper proposes a pricing model for convertible bonds based on the utility-indifference method. Firstly, analyse the optimal hedging strategy of the convertiblebonds with the callable provision.Then, the general expression of utility indifferenceprice on convertible bonds is obtained under the CIR interest rate model. Furthermore, using the proposed theoretical model, we present an empirical pricing study of theChina’s market, using3convertible bonds and more than70months of daily marketprices. The parameters value is estimated by the maximum likelihood method, and theprices of convertible bonds are simulated by the Monte Carlo approach. The empiricalresults indicate that the theoretical prices are higher than the actual market prices0.24%-4.58%, and the utility indifference prices are better than the B-S prices.
可转换债券定价相对一般的债券和期权更为困难。在现有对可转换债券定价的文献中，大部分是单因素模型，即仅仅考虑到标的股票价格的波动对可转债价格的影响，事实上，利率是金融市场上一个非常重要的因素，所有的证券价格及收益率都与之相关。同时，可转换债券作为一种企业债券，必然存在信用风险。所以，在构建可转换债券模型时，有必要将利率与信用风险的因素加入其中。此外，可转换债券定价问题在本质上仍然属于无套利定价理论的应用范畴，而现有的文献大多是针对完全市场条件下的定价研究。但完全市场的假设有时并不符合实际的投资环境。因此，有必要运用实用性更广的定价理论去探讨可转换债券定价问题。本论文结合解析法和数值法进行理论研究和实证分析，丰富了可转换债券的定价理论，具有重要的学术价值。同时，合理的可转换债券定价，必然会促进金融市场的发展和繁荣，故对于金融市场也具有十分重要的现实指导意义。
本文对可转换债券的定价模型和方法做了深入研究，并利用实际市场数据进行了详细的实证分析，主要的研究成果如下：
1.运用二叉树方法方法对随机利率下考虑信用风险的可回售可赎回可转换债券进行定价研究。考虑到标的股票价格及利率两个因素对可转换债券价格的重要性，分别采用不同的二叉树图对两者的变化过程进行刻画，并进一步考虑信用风险对可转换债券价格的影响，采用违约率和回收率进行描述，得到了含信用风险的两因子二叉树模型。在此基础上，进行了算例模拟，得到了在股票价格服从CRR模型下，利率分别服从常数波动率二叉树模型及时变波动率二叉树模型下的可转换债券定价结果。
2.运用最小二乘随机化拟蒙特卡罗（LSRQM）方法对随机利率下考虑信用风险的可回售可赎回可转换债券进行定价研究。首先建立了标的股票价格与利率的随机微分方程，并将其转换到风险中性概率空间中。考虑到可转换债券具有企业债券的性质，存在一定的信用风险。采用Jarrow和Turnbull (1995)的模型刻画信用风险，得到了包含违约风险的可转换债券的LSM方法的计算步骤。由于LSM方法的局限，本文将其修正为最小二乘随机化拟蒙特卡罗方法。在此基础上，进行了确定参数下的数值模拟，得到了随机利率及信用风险下的可转换债券理论价格，并进一步利用实际市场数据进行实证分析，从而验证了该模型的有效性。
3.运用保险精算方法对可转换债券进行定价研究。在一般的保险精算定价法的基础上，考虑资产价格的实际特点而引入广义O-U过程，由利率的回复特点建立Hull-White随机利率模型，通过利用随机微分方程的相关理论，得到了一般欧式期权及交换期权的精确解，并将一般欧式期权定价结果推广到有红利率的情形，给出了欧式买权卖权在上述模型下的平价公式，进一步得到了附加了赎回条款的可转换债券的保险精算定价的一般表达式。在理论研究的基础上，进行了数值模拟比较分析，显示出保险精算模型与B-S模型下的期权价格确实存在不同程度的差别，并且利用中国金融市场的实际数据进行了实证分析，结果表明保险精算模型的结果优于B-S模型。
4.利用效用无差别定价方法对可转换债券进行定价研究。首先分析了附加可赎回条款的可转换债券的最优投资策略，进而得到了在CIR利率模型下，可转换债券的效用无差别价格的一般表达式，并在该理论基础上，选择中国市场上的三支可赎回可转换债券作样本进行了实证分析，利用最大似然法得到参数估计值，并运用蒙特卡罗方法模拟了可转换债券理论价格过程。实证结果表明，可转债的效用无差别价格比实际市场价格要高0.24%-4.58%，且虚值状态下的可转债的低估程度较大。但与B-S模型下的理论价格相比，效用无差别价格明显要优于B-S价格，能更好地拟合实际价格。
The convertible bond is a complex financial derivative with the characteristic ofthe bond and the stock. It gives the holders the right to transform the bonds to theunderling stocks according to the agreed terms. Because of the lower financing costand the equity premium financing, the convertible bonds have became an importantway for the company to financing in the capital market. The pricing of convertiblebonds directly influences the profit of investors and the financing cost of issuers thathas a profound affect on the development of the convertible bonds market, so thepricing theory has been the critical question of the convertible bonds research.
The convertible bonds pricing is more difficulty than the common bonds andoptions. A lot of literature in convertible bonds pricing is modeled by single factorwhich only considers the influence of the underlying stocks to the convertible bondsprices. In fact, the interest rate is a very important factor in the finance market that allthe securities prices and the yield rates are related to. Meanwhile, as one kind of thecorporate bonds, the convertible bonds should involve credit risks. So we consider theinterest rate and the credit risk in the pricing models of the convertible bonds.Furthermore, the pricing problem of the convertible bonds belongs to the applicationof the no-risk arbitrage pricing theory and most of the existing literature is based onthe complete market condition. However, the hypothesis of complete marketsometimes not fit the actual investment environment. Therefore, it is necessary to usethe general pricing theory to study the pricing problem of convertible bonds. Thispaper makes the theoretical research and the empirical research using the analysismethod and the numerical method, and it enriches the pricing theory and hasimportant academic value. Moreover, the reasonable prices to the convertible bondsmust promote the development and the boom of the finance market, so it has veryimportant realistic guiding meaning.
This paper makes intensive study in the pricing models and the methods of theconvertible bonds, and makes detailed empirical research using the actual market data.The major research results are as follows.
1. This paper uses the binary tree method to pricing the convertible bonds withadditional provisions of call and put considering the stochastic interest rates andcredit risks. Consider that the underling stock prices and the interest rates are very important to the convertible bonds, different binary tree models are built to fit theirvariation processes respectively. Furturemore, consider the influence of credit riskthat is described by the default rate and recovery rate, the two-factor binary treemodel involving credit risk is obtained. On the above model, the simulation exampleis made and gets the convertible bonds prices when the stock prices follow the CRRmodel and the interest rates follow the constant volatility and time-varying volatilitybinary tree models respectively
2. Using the least-squares randomized quasi-Monte Carlo (LSRQM) method toprice the convertible bonds. Firstly, the stochastic differential equations of theunderling stock price and the interest rate are made and are converted to the riskneutral probability space. As the convertible bonds are an kind of corporate bonds thatexist particular credit risks, this paper uses the Jarrow and Turnbull (1995) model todescribe the credit risks, then the calculate procedure of the LSM method onconvertible bonds containing the default risk is obtained. Because of the restriction ofthe LSM method, this paper amends it to the least squares quasi-stochastic MonteCarlo. Moreover, the simulation example is made under the determinate parametersthat reflect the influences of the stochastic interest rates and the credit risks on theconvertible bonds. At last, the empirical analysis is made using the real market data,and the effectiveness of this model is proved.
3. This paper discusses an actuarial approach to the convertible bonds pricing.According to the definition of actuarial pricing approach, build the market modelwhere the interest rates are stochastic and the stock prices are driven by generalizedexp-Ornstein-Uhlenback process, and the exact solutions for the general Europeanoption and the exchange option are obtained with the help of the related theory ofstochastic differential equation. Then the European call-put parity relation isderived naturally, and the new prices of European call option and the put option withcontinuous dividend yield are deduced from the above results. Furthermore, theactuarial prices of convertible bonds are obtained. Based on the theoretical study, acomparative analysis of numerical simulation and an empirical analysis are madebetween the above-mentioned results and the B-S prices, the results indicate theactuarial prices are better than the B-S pricis.
4. This paper proposes a pricing model for convertible bonds based on the utility-indifference method. Firstly, analyse the optimal hedging strategy of the convertiblebonds with the callable provision.Then, the general expression of utility indifferenceprice on convertible bonds is obtained under the CIR interest rate model. Furthermore, using the proposed theoretical model, we present an empirical pricing study of theChina’s market, using3convertible bonds and more than70months of daily marketprices. The parameters value is estimated by the maximum likelihood method, and theprices of convertible bonds are simulated by the Monte Carlo approach. The empiricalresults indicate that the theoretical prices are higher than the actual market prices0.24%-4.58%, and the utility indifference prices are better than the B-S prices.
引文
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