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本文在快速随机波动率变换的情况下研究了单个信用衍生品定价的风险规避问题。本文首先在Merton最优化的问题中添加随机波动率,将其应用到信用风险的研究中。假设波动率过程是一种不同时间尺度的扰动过程,对其进行近似估计。由于价值过程可看成是一个不完全市场围绕着完全市场常数波动率的一种扰动,故这种近似方法是可行的。其次,当波动率变换是快速均值回归过程时,这种扰动过程是一种关于Hamilton-Jacobi-Bellman偏微分方程的奇异扰动过程。当波动率变换非常缓慢时,这种扰动过程是一般的扰动过程。两种扰动过程叠加在一起便是多元时间维度随机波动率过程。再者,本文提出的算法与线性期权定价问题具有较高的一致性,因而也发现了风险容忍度方程的一些新性质。本文最后还在快速时间维度的框架下研究了波动率对于信用风险的投资优化问题。由于新建模型是一个非线性PDE,故不能直接给出其解析解,于是本文引进一种扰动逼近的方法,该方法的关键中间步骤是把问题分成可解和扰动两个部分。通过扰动逼近的方法,可得一个近似的解析解。
In this paper, we study the risk aversion of single credit derivatives pricing under the condition of fast stochastic volatility transformation. This paper first adds stochastic volatility to the Merton optimization problem and applies it to the study of credit risk. It is assumed that the volatility process is a disturbance process of different time scales and its approximate estimation is made. Since the value process can be viewed as a perturbation of the volatility of a complete market around an incomplete market, this approximation is feasible. Second, when the volatility transformation is a fast-mean-value regression process, this perturbation process is a singular perturbation process for the Hamilton-Jacobi-Bellman partial differential equations. When the volatility transformation is very slow, this disturbance process is a general disturbance process. Stacking two disturbing processes together is a process of multiple volatility of time dimension. Furthermore, the proposed algorithm has a higher consistency with the linear option pricing problem, and thus some new properties of the risk tolerance equation have also been found out. Finally, this paper also studies the optimization of investment volatility for credit risk under the framework of fast time dimension. Since the new model is a nonlinear PDE, its analytical solution can not be directly given. Therefore, a perturbation approximation method is introduced in this paper. The key intermediate step of this method is to divide the problem into two parts: solvable and perturbation. By perturbation approximation method, an approximate analytic solution can be obtained.