Please wait a minute...
浙江大学学报（理学版）
浙江大学学报（理学版）  2017, Vol. 44 Issue (3): 296-301    DOI: 10.3785/j.issn.1008-9497.2017.03.009
数学与计算机科学     
随机波动模型的首中时问题
张苗1, 刘晖2, 张飞龙3
1. 西安电子科技大学 数学与统计学院, 陕西 西安 710126;
2. 北京大学 地球与空间科学学院, 北京 100871;
3. 西安电子科技大学 物理与光电工程学院, 陕西 西安 710126
The first hitting time of stochastic volatility models
ZHANG Miao1, LIU Hui2, ZHANG Feilong3
1. School of Mathematics and Statistics, Xidian University, Xi'an 710126, China;
2. School of Earth and Space Scienecs, Peking University, Beijing 100871, China;
3. School of Physics and Optoelectronic Engineering, Xidian University, Xi'an 710126, China
 全文: PDF(1368 KB)   HTML  
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.
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 出版日期: 2017-03-01
CLC:  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.
服务　　
把本文推荐给朋友
加入引用管理器
E-mail Alert
RSS
作者相关文章　　
张苗
刘晖
张飞龙

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.

链接本文:

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

[1] BLACK F, SCHOLE M. The pricing of options and corporate liabilities[J]. Journal of Political Economy,1973,81(3):133-155.
[2] HESTON S L. A closed-form solution for options with stochastic volatility with applications to bond and currency options[J]. The Review of Financial Studies,1993,6(2):327-343.
[3] HULL J,WHITE A. The pricing of options on assets with Stochastic volatility[J].Journal of Finance,1987,42(2):281-300.
[4] STEIN E, STEIN C. Stock price distributions with stochastic volatility: An analytic approach[J].Review of Financial Studies,1991(4):727-752.
[5] TALAMANCA G F. Testing volatility autocorrelation in the constant elasticity of variance stochastic volatility model[J]. Computational Statistics and Data Analysis,2009,53:2201-2218.
[6] MARIO L. First passage problems for asymmetric wiener processes[J]. Ecolepoly Technique de Montreal,2006,43(4):175-184.
[7] CARLSUND A. Cover Times, Sign-Dependent Random Walks, and Maxima[M]. Stockholm: Matematik,2003.
[8] BOROVKOV K, NOVIKOV A. On exit times of Levy-driven Ornstein-Uhlenbeck processes[J]. Statistics & Probability Letters,2007,78(12):1517-1525.
[9] PATIE P. On a martingale associated to generalized Ornstein-Uhlenbeck processes and an application to finance[J]. Stoch Process,2005,115:593-607.
[10] LOEFFEN R L, PATIE P. Absolute ruin in the Ornstein-Uhlenbeck type risk model[J]. Quantitative Finance,2010,arXiv:1006.2712.
[11] ALILI L, PATIE P, PEDERSEN J L. Representations of the first hitting time density of an Ornstein-Uhlenbeck process[J]. Stock Models,2005,21:967-980.
[12] BO L J, HAO C. First passage times of constant-elasticity-of-variance processes with two-sided reflecting barriers[J]. Journal of Applied Probability,2012,49(4):1119-1133.
[13] GERBER H U, YANG H L. Absolute ruin probabilities in a jump diffusion risk model with investment[J]. NAAJ,2007,1l(3):159-168.
[14] BO L J, REN G, WANG Y J. First passage times of reflected generalized Ornstein-Uhlenbeck processes [J]. Stochastics and Dynamics,2013,13(1):1250014.
[15] BO L J, WANG Y J. Some integral functionals of reflected SDEs and their applications in finance[J]. Quantitative Finance,2011,11(11):343-348.
[16] BO L J, TANG D, WANG Y J, et al. On the conditional default probability in a regulated market: A structural approach[J]. Quantitative Finance,2010,11(12):1695-1702.
[17] 冯海林,薄立军.随机过程——计算与应用[M].西安:西安电子科技大学出版社,2012. FENG H L,BO L J. Stochastic Process-Calculation and Application[M]. Xi'an:Xidian University Press,2012.
[18] 孙健.金融衍生品定价模型-数理金融引论[M].北京:中国经济出版社,2008. SUN J. Financial Derivatives Pricing Model-An Introduction to Mathematical Finance[M]. Beijing: China Economy Press,2008.
[19] 王高雄,周之铭.常微分方程[M].北京:高等教育出版社,2006. WANG G X, ZHOU Z M. Ordinary Differential Equations[M]. Beijing: Higher Education Press,2006.
No related articles found!