随机波动模型的首中时问题

The first hitting time of stochastic volatility models

研究了一类波动率是平方根过程的随机波动CEV模型的首中时问题.利用鞅方法求解首中时和波动率的联合拉普拉斯变换,继而将问题转换为求解一类变系数二阶常微分方程,通过变量代换将此方程转化为经典的Whittaker方程,得到联合拉普拉斯变换表达式.最后,选取不同的参数,使随机波动CEV模型的资产价格过程能够涵盖O-U过程、几何布朗运动、平方根过程等几种常见的扩散过程,画出不同参数下联合拉普拉斯变换函数的三维图像,并分析其变化趋势.

This paper explores the first passage times of stochastic volatility CEV model.We mainly solve the joint Laplace transform of the first hitting time and volatility.Firstly,we use the It8 formula to construct the martingale which can convert the problem into the process of solving a differential equation.Then,we introduce an appropriate second order variable coefficient ordinary differential equation,after a change of variable,it is turned to the Whittaker's equation.It's not difficult to get the general solution of Whittaker's equation.Thus,the explicit expressions for the joint Laplace transformation of the first passage times of stochastic volatility CEV model can be derived.Finally,selecting the parametersγbe 0,1/2and 1,let the asset price process covers the O-U process,geometric Brownian motion and square root process.Under different parameters,we obtain explicit expression of the joint Laplace transformation function,and use Matlab to draw the corresponding diagram and analyze the trend of graph.