基于可违约价格的违约期权和债券的定价研究
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摘要
随着金融创新的发展,市场上金融衍生品是越来越多,这些金融衍生品的出现,金融衍生品为投资者带来了更多的投资机会(工具),同时带来了金融风险。2008年发生波及全球的金融危机就是由抵押债券这类金融衍生品的违约引发的。因此对于金融衍生品的风险定价,比如期权和债券存在违约风险时的定价,就引起了学者们的广泛关注,成为当今金融研究领域的热点问题。
如何刻画投资者面对可违约期权和债券带来的违约风险,就成了必须要解决的急迫问题。本文采用均值方差对冲原理,研究投资者面临的信用违约风险,给出了定价公式。在这个过程中,我们先是针对基础资产具有不同的价格过程的期权定价,比如价格过程为可违约跳扩散过程,价格过程为存在一般跳的可违约过程。然后我们还针对完备市场和不完备市场条件下,可违约债券的定价,给出债券的定价公式。在最后还对均值方差对冲过程中涉及的最优方差鞅测度作出研究,给出该测度的的具体结构,并作了详细的证明。
论文首先研究了可违约跳扩散过程的期权的定价。我们假定期权所依附的股票价格为可违约的跳扩散过程,在此基础上给出了违约过程的鞅表示定理,利用该定理推出了与最优控制过程以及与期权有关的两个倒向随机微分方程,证明了这两个方程的解的存在性和唯一性。利用均值方差对冲原理给出了可违约期权的定价公式,而且还给出了针对跳扩散过程的可违约期权定价的显式解。
其次,我们在上一章研究的基础上考察了具有一般跳过程的可违约期权的定价,即针对可违约半鞅的期权的定价。同样我们还是假定期权所依附的股票价格为可违约的一般跳过程,在此基础上推导出了基于可违约的一般跳过程的鞅表示定理,由此推出了与最优控制过程和与期权有关的两个倒向随机微分方程,并证明了这两个方程的解的存在性和唯一性,然后利用均值方差对冲的原理,给出了具有一般跳过程的可违约期权的定价公式。
接着研究了均值方差对冲原理中所涉及的最优方差鞅测度。假定涉及的股票价格为可违约的指数Levy过程,在此基础上给出了可违约过程的鞅表示定理,给出了最优方差鞅测度存在的鞅条件,和涉及的最优化问题,然后利用Largrange乘子法解出最优化问题,给出了最优方差鞅测度的具体结构。
接着我们还研究了在完备市场下,可违约零息债券的定价。利用等价鞅测度变换,使债券在新的测度下是一个鞅,并由此给出可违约零息债券的价值过程,以及在Vasicek模型下的定价公式。
最后,论文还考察了在不完备市场下,具有可违约风险的零息债券的定价。我们假定相应的债券价格为可违约过程,在此基础上我们给出了与最优控制过程有关的倒向随机微分方程,以及与债券价值有关的倒向随机微分方程,并证明了这两个方程解的存在性和唯一性,然后用均值方差对冲的原理给出可违约零息债券的定价公式,另外还给出了债券在任意时刻的价格的表示方程。
With the development of financial innovation, financial derivatives have been more and more in fiancial markets. These financial derivatives provide important tools for investors to avoid financial risks, and also provide tremendous investment opportunities for investors. And same time, these derivatives also give huge financial risks. The global financial crisis which occurred in 2008 has been caused by financial derivatives, such as the mortgage bonds. The pricing of default derivatives has been caused widespread concern for scholars, and has been a hot problem in modern finance.
How to characterize the risk of options and bonds for investors has become the urgent question which must solve. In this article, we use the mean variance hedging methodology to study the default risk, and give the pricing formula for defaultable options and bonds. Firstly, we aim at the defaultable options based on different price processes, such as defaultable jump diffusion processes and defaultable semimartingales with general jumps. Then we also aim at defaultable bond in complete market and in incomplete market conditions; give the pricing formula for defaultable bonds. In the last, we careful study the optimal-variance martingale measures, and give an explicit construction of the optimal-variance martingale measures.
In chapter two we derive a complete solution for pricing defaultable options based on defaultable jump-difusion process. The approach consists of three steps:First, we set up a defaultable martingale representation theorem for defaultable jump-difusion process. Then, we construct two backward stochastic differential equations for hedging as a stochastic control process and the option. Finally, we find out the optimal investment strategy through minimizing the cost function, which leads to the pricing equation for defaultable options.
In chapter three we propose a methodology for pricing defaultable option for semimartingales with general jumps in incomplete markets. First of all, a default process martingale representation theorem based on semimartingale is proved. Then two backward semimartingale differential equations (BSDEs) about control process and option are constructed, and their solutions are proved existent and unique. Finally a mean-variance hedging methodology is used to derive the pricing formula of defaultable options for semimartingales with general jumps.
In chapter four, we treat defaultable zero coupon bond as a contingent claim in incomplete markets. By setting up an investment portfolio with risky asset, we develop the bond pricing equation by finding the optimal investment strategy with minimum risk through linear-quadratic hedging (LQ) method.
In chapter five, we assume the short-term interest rate of bonds is a diffuse process; in this chapter addresses the pricing process of defaultable zero-coupon bonds and their pricing method. By employing the equivalent martingale measure and the principle of affine term structure of interest rate, an explicit pricing equation of defaultable zero-coupon bonds is derived under the assumptions of Vasicek model.
In chapter six, we proposed a local martingale representation theorem for exponential defaultabale jump-diffusion processes based on the defalutable jump-diffusion price processes. And we construct an optimal problem for optimal-variance martingale measures under the martingale condition, we use Euler-Lagrange equation, give an explicit construction of the optimal-variance martingale measures.
如何刻画投资者面对可违约期权和债券带来的违约风险,就成了必须要解决的急迫问题。本文采用均值方差对冲原理,研究投资者面临的信用违约风险,给出了定价公式。在这个过程中,我们先是针对基础资产具有不同的价格过程的期权定价,比如价格过程为可违约跳扩散过程,价格过程为存在一般跳的可违约过程。然后我们还针对完备市场和不完备市场条件下,可违约债券的定价,给出债券的定价公式。在最后还对均值方差对冲过程中涉及的最优方差鞅测度作出研究,给出该测度的的具体结构,并作了详细的证明。
论文首先研究了可违约跳扩散过程的期权的定价。我们假定期权所依附的股票价格为可违约的跳扩散过程,在此基础上给出了违约过程的鞅表示定理,利用该定理推出了与最优控制过程以及与期权有关的两个倒向随机微分方程,证明了这两个方程的解的存在性和唯一性。利用均值方差对冲原理给出了可违约期权的定价公式,而且还给出了针对跳扩散过程的可违约期权定价的显式解。
其次,我们在上一章研究的基础上考察了具有一般跳过程的可违约期权的定价,即针对可违约半鞅的期权的定价。同样我们还是假定期权所依附的股票价格为可违约的一般跳过程,在此基础上推导出了基于可违约的一般跳过程的鞅表示定理,由此推出了与最优控制过程和与期权有关的两个倒向随机微分方程,并证明了这两个方程的解的存在性和唯一性,然后利用均值方差对冲的原理,给出了具有一般跳过程的可违约期权的定价公式。
接着研究了均值方差对冲原理中所涉及的最优方差鞅测度。假定涉及的股票价格为可违约的指数Levy过程,在此基础上给出了可违约过程的鞅表示定理,给出了最优方差鞅测度存在的鞅条件,和涉及的最优化问题,然后利用Largrange乘子法解出最优化问题,给出了最优方差鞅测度的具体结构。
接着我们还研究了在完备市场下,可违约零息债券的定价。利用等价鞅测度变换,使债券在新的测度下是一个鞅,并由此给出可违约零息债券的价值过程,以及在Vasicek模型下的定价公式。
最后,论文还考察了在不完备市场下,具有可违约风险的零息债券的定价。我们假定相应的债券价格为可违约过程,在此基础上我们给出了与最优控制过程有关的倒向随机微分方程,以及与债券价值有关的倒向随机微分方程,并证明了这两个方程解的存在性和唯一性,然后用均值方差对冲的原理给出可违约零息债券的定价公式,另外还给出了债券在任意时刻的价格的表示方程。
With the development of financial innovation, financial derivatives have been more and more in fiancial markets. These financial derivatives provide important tools for investors to avoid financial risks, and also provide tremendous investment opportunities for investors. And same time, these derivatives also give huge financial risks. The global financial crisis which occurred in 2008 has been caused by financial derivatives, such as the mortgage bonds. The pricing of default derivatives has been caused widespread concern for scholars, and has been a hot problem in modern finance.
How to characterize the risk of options and bonds for investors has become the urgent question which must solve. In this article, we use the mean variance hedging methodology to study the default risk, and give the pricing formula for defaultable options and bonds. Firstly, we aim at the defaultable options based on different price processes, such as defaultable jump diffusion processes and defaultable semimartingales with general jumps. Then we also aim at defaultable bond in complete market and in incomplete market conditions; give the pricing formula for defaultable bonds. In the last, we careful study the optimal-variance martingale measures, and give an explicit construction of the optimal-variance martingale measures.
In chapter two we derive a complete solution for pricing defaultable options based on defaultable jump-difusion process. The approach consists of three steps:First, we set up a defaultable martingale representation theorem for defaultable jump-difusion process. Then, we construct two backward stochastic differential equations for hedging as a stochastic control process and the option. Finally, we find out the optimal investment strategy through minimizing the cost function, which leads to the pricing equation for defaultable options.
In chapter three we propose a methodology for pricing defaultable option for semimartingales with general jumps in incomplete markets. First of all, a default process martingale representation theorem based on semimartingale is proved. Then two backward semimartingale differential equations (BSDEs) about control process and option are constructed, and their solutions are proved existent and unique. Finally a mean-variance hedging methodology is used to derive the pricing formula of defaultable options for semimartingales with general jumps.
In chapter four, we treat defaultable zero coupon bond as a contingent claim in incomplete markets. By setting up an investment portfolio with risky asset, we develop the bond pricing equation by finding the optimal investment strategy with minimum risk through linear-quadratic hedging (LQ) method.
In chapter five, we assume the short-term interest rate of bonds is a diffuse process; in this chapter addresses the pricing process of defaultable zero-coupon bonds and their pricing method. By employing the equivalent martingale measure and the principle of affine term structure of interest rate, an explicit pricing equation of defaultable zero-coupon bonds is derived under the assumptions of Vasicek model.
In chapter six, we proposed a local martingale representation theorem for exponential defaultabale jump-diffusion processes based on the defalutable jump-diffusion price processes. And we construct an optimal problem for optimal-variance martingale measures under the martingale condition, we use Euler-Lagrange equation, give an explicit construction of the optimal-variance martingale measures.
引文
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