随机波动模型的首中时问题

doi:10.3785/j.issn.1008-9497.2017.03.009

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摘要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.

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	张苗
	刘晖
	张飞龙

关键词 ：随机波动CEV模型, 首中时, 鞅方法, 联合拉普拉斯变换, Whittaker方程

Abstract：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 Itô 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 parameters γ be 0, 1/2 and 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.

Key words： stochastic volatility CEV model first passage times martingale method joint Laplace transforms Whittaker's equation

收稿日期: 2016-07-16

ZTFLH:

O211.63

基金资助:国家自然科学基金资助项目（11471254）.

作者简介: 张苗(1993-),ORCID:http://orcid.org/0000-0003-1640-3173,女,硕士研究生,主要从事随机模型研究,E-mail:feilon1001@163.com.

引用本文:

张苗, 刘晖, 张飞龙. 随机波动模型的首中时问题[J]. 浙江大学学报（理学版）, 2017, 44(3): 296-301.
ZHANG Miao, LIU Hui, ZHANG Feilong. The first hitting time of stochastic volatility models. Journal of ZheJIang University(Science Edition), 2017, 44(3): 296-301.

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http://www.zjujournals.com/sci/CN/10.3785/j.issn.1008-9497.2017.03.009 或 http://www.zjujournals.com/sci/CN/Y2017/V44/I3/296

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