跳扩散模型在寿险合同与信用衍生品定价中的应用

发布时间：2018-05-31 20:23

本文选题：跳扩散模型 + 违约概率　；参考：《苏州大学》2012年博士论文

【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.
[Abstract]:Since the subprime mortgage crisis , the quantitative analysis of the risk of default has been paid more and more attention . The structured credit risk model is a comparative frontier breach risk measure model . In fact , in the real market , the value of the asset is assumed to be subject to geometric Brownian motion . In fact , in the actual market , the asset ' s investment return has " asymmetric peak thickness tail " and " fluctuation rate smile " . In fact , in the framework of the jump diffusion model , we can describe the value change process of assets . In order to describe this phenomenon , we can quantify the company ' s default risk .

It is well known that the probability of default is the core of the measure of default risk , and it is one of the most critical factors in product pricing analysis . Fortunately , when the jump - diffusion model is special , such as double exponential distribution , we can give a numerical solution to the probability of default .

In the first of the structured model , the default time of the company is defined as the time when the company value is first lower than a certain obstacle level , and the obstacle level can be set as a function of certainty related to time , and it can be set as a stochastic process . First , when the value of the company is assumed to obey the geometric jump diffusion process , we give the Laplace transform of the default time and the closed formula of the expected discounted value of the company .

Secondly , we consider using a random obstacle that is independent of the company ' s value process to replace the constant obstacle . Therefore , we have established the correlation structure between the asset value and the default obstacle , especially when the company value and random obstacle are described by two related geometric double exponential jump diffusion models , we obtain the closed formula of the expected discount rate of the default time and the company ' s value and obstacle level at the time of default .

Finally , we consider the fair pricing problem of the red life insurance contract in the life insurance market . The insurance company often provides some life insurance products with the guaranteed benefits , and the pricing theory of the commonly used interest is used to consider the fair pricing problem of these life insurance contracts . In this paper , we present a value process of the company that operates multiple related business contracts . When the density function of the jump scale distribution has rational Laplace transform , we use Laplace transform to obtain the semi - analytic formula of the price of the red life insurance contract .
【学位授予单位】：苏州大学
【学位级别】：博士
【学位授予年份】：2012
【分类号】：F224;F830

【共引文献】