跳扩散模型在寿险合同与信用衍生品定价中的应用
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摘要
自次贷危机以来,对违约风险的量化分析越来越受到人们的重视.结构化信用风险模型则是一种比较前沿的违约风险度量模型.在基于Merton的经典的结构化模型中,假定资产的价值过程服从几何布朗运动,然而这个假设与从市场数据中观察到,资产的投资回报具有“非对称尖峰厚尾性”和“波动率微笑”这两个特征相矛盾.事实上,在实际市场中,资产的价值并不是连续变化的,突发事件会使得资产价值出现跳跃,为了描述这种现象,我们可以用几何跳扩散过程来刻画资产的价值变动过程.本文则是在跳扩散模型框架下,对公司的违约风险进行了量化分析,并对信用衍生品市场和寿险市场中交易非常活跃的产品,可违约零息债券与分红型寿险合同,进行了定价分析.
众所周知,违约概率是违约风险度量的核心内容,也是产品定价分析中最为关键的因素之一.然而,在跳扩散模型下,一般很难给出违约时间分布的显式表达公式.幸运的是,当跳尺度分布比较特殊时,如双指数分布,我们可以给出违约时间的拉普拉斯变换的闭型公式,利用数值方法反演拉普拉斯变换,即可给出违约概率的数值解.本文则推广了双指数跳扩散模型,考虑一类更广的跳扩散模型,并提供了一个给可违约零息债券和分红型寿险合同定价的方法.本文主要作了三方面的研究工作,其中前两部分工作主要是在结构化框架下,考虑了可违约零息票债券的定价问题,另一部分工作则是研究了跳扩散模型下分红型寿险保单的公平定价问题.
在结构化模型的首中时方法中,公司的违约时刻被定义为公司价值首次低于某个障碍水平的时刻,该障碍水平可以设置为一个与时间有关的确定性的函数,也可设置为一个随机过程.首先,在假定违约障碍是一个确定的常数的条件下,我们考虑了可违约零息债券的价格和公平保费问题.当假定公司的价值服从几何跳扩散过程时,我们给出了违约时间的拉普变换以及违约时公司的期望折现价值所满足的积分-微分方程.特别地,当跳尺度服从超指数分布时,我们得到了违约时间的拉普变换以及违约时公司期望折现值的闭型公式.因此,利用数值方法反演所得到的闭型公式,我们即可给出违约概率,可违约零息债券的价格与公平保费的数值解.
其次,我们考虑用一个与公司价值过程不独立的随机障碍来替代常数障碍.因此,我们建立了资产价值与违约障碍之间的相关结构,特别地,当公司价值与随机障碍用两个相关的几何双指数跳扩散模型来描述时,我们得到了违约时间的拉普变换以及违约时公司价值与障碍水平的期望折现比率的闭型公式.因此,在资产与违约障碍相关的跳扩散模型下,我们也可以给出违约概率,可违约零息债券的价差的数值解.
最后,我们考虑了寿险市场中分红型寿险合同的公平定价问题.保险公司常提供一些带有保证收益的寿险产品,人们常用未定权益的定价理论来考虑这些寿险合同的公平定价问题.本文提出用一个带相关跳的跳扩散模型来描述一个经营多个相关类业务的公司的价值过程,当跳尺度分布的密度函数具有有理的拉普拉斯变换时,我们利用拉普拉斯变换的方法,得到了分红型寿险合同价格的半解析公式.
Since subprime crisis, a quantitative analysis of default risk has been attractinga lot of attention. The structural model is one of the most popular credit risk mea-surement models. In Merton’s classical structural model, the frm’s value process isassumed to follow a geometric Brownian motion. But empirical studies invalidate suchassumptions by suggesting two observations for asset returns:“the asymmetric lep-tokurtic feature” and “the volatility smile.” In fact, the frm’s value does not evolvescontinously in the market. Special events can make the frm’s value jump. In orderto incorporating this phenomenon, we can use geometric jump-difusion processes todescribe the frm’s value dynamics. This paper makes a quantitative analysis of defaultrisk, and investigates the valuation of a defaultable zero-coupon bond and a partici-pating life insurance contract, which are two commonly traded products in the creditderivatives and the life insurance markets, under the jump-difusion models.
It is well known, default probability is the core of the credit risk measurement,and is also one of the most important factors in the analysis of the valuation. However,it is very difcult to give the closed form expression for the distribuion of the defaulttime under a general jump-difusion model. Fortunately, when the jumps have somespecial distributions, such as a double exponential distribution, we can give the formulafor the Laplace transform of the default time. By inverting the Laplace transform,we can obtain the numerical solution for the default probability. Hence, extendingthe double exponential jump-difusion model, this paper considers some more generaljump-difusion models, and provides a method for the pricing of the defaultable zero-coupon bond and the participating life insurance contract. This paper includes threeparts: in the frst two parts, we consider the pricing of the defaultable zero-couponbond within the structural framework, in the last part of this paper, we investigate thevaluation of the participating life insurance contract under the jump-difusion model.
The frst passage time approach in structural form models specifes the default asthe frst time that the frm value falls below a threshold. The default level can be set tobe an exponential function of time or a stochastic process. Firstly, under the assump-tion of the constant default level, we consider the price and the fair premium of the defaultable zero-coupon bond. Under the two-sided jump-difusion models, we give theintegro-diferential equations for the Laplace transform of default time and the frm’sexpected present market value at default. Closed form expressions for them are ob-tained when the jumps have a hyper-exponential distribution. Hence, we could obtainthe numerical solutions for the default probability, the price and the premium of thedefaultable zero-coupon bond by numerically inverting those closed form expressions.
Secondly, we propose a stochastic default barrier which is dependent of the frm’svalue to replace the constant default barrier. We construct a dependence structurebetween the frm’s value and the stochastic default barrier. Especially, when the frm’svalue and the default barrier are modeled by two dependent double exponential jump-difusion processes, we can give the closed form expressions for the Laplace transformof the default time and the expected discounted ratio of the frm value to the defaultbarrier at default. Therefore, under the stochastic default barrier model which isdependent of the frm’s value, we can also obtain the numerical solutions for the defaultprobability and the spread of the defaultable zero-coupon bond.
Finally, we consider the valuation of the participating life insurance contract in thelife insurance market. Life insurance companies usually ofer some life insurance policieswith the guaranteed interest rate, hence, the contingent claim valuation approachsare often used to study the fair value of the life insurance contracts. We propose ajump-difusion process with dependent jumps to model the value of the frm dealingin several dependent classes of businesses. Especially, when the density of the jumpshas a rational Laplace transform, we use the Laplace transform approach to obtain thequasi-closed form formula for the fair value of the participating life insurance contract.
众所周知,违约概率是违约风险度量的核心内容,也是产品定价分析中最为关键的因素之一.然而,在跳扩散模型下,一般很难给出违约时间分布的显式表达公式.幸运的是,当跳尺度分布比较特殊时,如双指数分布,我们可以给出违约时间的拉普拉斯变换的闭型公式,利用数值方法反演拉普拉斯变换,即可给出违约概率的数值解.本文则推广了双指数跳扩散模型,考虑一类更广的跳扩散模型,并提供了一个给可违约零息债券和分红型寿险合同定价的方法.本文主要作了三方面的研究工作,其中前两部分工作主要是在结构化框架下,考虑了可违约零息票债券的定价问题,另一部分工作则是研究了跳扩散模型下分红型寿险保单的公平定价问题.
在结构化模型的首中时方法中,公司的违约时刻被定义为公司价值首次低于某个障碍水平的时刻,该障碍水平可以设置为一个与时间有关的确定性的函数,也可设置为一个随机过程.首先,在假定违约障碍是一个确定的常数的条件下,我们考虑了可违约零息债券的价格和公平保费问题.当假定公司的价值服从几何跳扩散过程时,我们给出了违约时间的拉普变换以及违约时公司的期望折现价值所满足的积分-微分方程.特别地,当跳尺度服从超指数分布时,我们得到了违约时间的拉普变换以及违约时公司期望折现值的闭型公式.因此,利用数值方法反演所得到的闭型公式,我们即可给出违约概率,可违约零息债券的价格与公平保费的数值解.
其次,我们考虑用一个与公司价值过程不独立的随机障碍来替代常数障碍.因此,我们建立了资产价值与违约障碍之间的相关结构,特别地,当公司价值与随机障碍用两个相关的几何双指数跳扩散模型来描述时,我们得到了违约时间的拉普变换以及违约时公司价值与障碍水平的期望折现比率的闭型公式.因此,在资产与违约障碍相关的跳扩散模型下,我们也可以给出违约概率,可违约零息债券的价差的数值解.
最后,我们考虑了寿险市场中分红型寿险合同的公平定价问题.保险公司常提供一些带有保证收益的寿险产品,人们常用未定权益的定价理论来考虑这些寿险合同的公平定价问题.本文提出用一个带相关跳的跳扩散模型来描述一个经营多个相关类业务的公司的价值过程,当跳尺度分布的密度函数具有有理的拉普拉斯变换时,我们利用拉普拉斯变换的方法,得到了分红型寿险合同价格的半解析公式.
Since subprime crisis, a quantitative analysis of default risk has been attractinga lot of attention. The structural model is one of the most popular credit risk mea-surement models. In Merton’s classical structural model, the frm’s value process isassumed to follow a geometric Brownian motion. But empirical studies invalidate suchassumptions by suggesting two observations for asset returns:“the asymmetric lep-tokurtic feature” and “the volatility smile.” In fact, the frm’s value does not evolvescontinously in the market. Special events can make the frm’s value jump. In orderto incorporating this phenomenon, we can use geometric jump-difusion processes todescribe the frm’s value dynamics. This paper makes a quantitative analysis of defaultrisk, and investigates the valuation of a defaultable zero-coupon bond and a partici-pating life insurance contract, which are two commonly traded products in the creditderivatives and the life insurance markets, under the jump-difusion models.
It is well known, default probability is the core of the credit risk measurement,and is also one of the most important factors in the analysis of the valuation. However,it is very difcult to give the closed form expression for the distribuion of the defaulttime under a general jump-difusion model. Fortunately, when the jumps have somespecial distributions, such as a double exponential distribution, we can give the formulafor the Laplace transform of the default time. By inverting the Laplace transform,we can obtain the numerical solution for the default probability. Hence, extendingthe double exponential jump-difusion model, this paper considers some more generaljump-difusion models, and provides a method for the pricing of the defaultable zero-coupon bond and the participating life insurance contract. This paper includes threeparts: in the frst two parts, we consider the pricing of the defaultable zero-couponbond within the structural framework, in the last part of this paper, we investigate thevaluation of the participating life insurance contract under the jump-difusion model.
The frst passage time approach in structural form models specifes the default asthe frst time that the frm value falls below a threshold. The default level can be set tobe an exponential function of time or a stochastic process. Firstly, under the assump-tion of the constant default level, we consider the price and the fair premium of the defaultable zero-coupon bond. Under the two-sided jump-difusion models, we give theintegro-diferential equations for the Laplace transform of default time and the frm’sexpected present market value at default. Closed form expressions for them are ob-tained when the jumps have a hyper-exponential distribution. Hence, we could obtainthe numerical solutions for the default probability, the price and the premium of thedefaultable zero-coupon bond by numerically inverting those closed form expressions.
Secondly, we propose a stochastic default barrier which is dependent of the frm’svalue to replace the constant default barrier. We construct a dependence structurebetween the frm’s value and the stochastic default barrier. Especially, when the frm’svalue and the default barrier are modeled by two dependent double exponential jump-difusion processes, we can give the closed form expressions for the Laplace transformof the default time and the expected discounted ratio of the frm value to the defaultbarrier at default. Therefore, under the stochastic default barrier model which isdependent of the frm’s value, we can also obtain the numerical solutions for the defaultprobability and the spread of the defaultable zero-coupon bond.
Finally, we consider the valuation of the participating life insurance contract in thelife insurance market. Life insurance companies usually ofer some life insurance policieswith the guaranteed interest rate, hence, the contingent claim valuation approachsare often used to study the fair value of the life insurance contracts. We propose ajump-difusion process with dependent jumps to model the value of the frm dealingin several dependent classes of businesses. Especially, when the density of the jumpshas a rational Laplace transform, we use the Laplace transform approach to obtain thequasi-closed form formula for the fair value of the participating life insurance contract.
引文
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