指数半鞅模型中的均值修正鞅测度

数学物理学报 ›› 2019, Vol. 39 ›› Issue (4): 932-941.

指数半鞅模型中的均值修正鞅测度

姚落根¹(),杨刚^1,*(),杨向群²()

¹ 湖南商学院数学与统计学院长沙 410205
² 湖南财政经济学院数学与统计学院长沙 410205

收稿日期:2017-12-13 出版日期:2019-08-26 发布日期:2019-09-11
通讯作者: 杨刚 E-mail:yaoluogen@sina.com;yanggangmath@sina.com;xqyang@hunnu.edu.cn
作者简介:姚落根, E-mail:yaoluogen@sina.com|杨向群, E-mail:xqyang@hunnu.edu.cn
基金资助:
湖南省自然科学基金(2019JJ40141);湖南省自然科学基金(2017JJ2127);国家社会科学基金(15BJY122);湖南省教育厅优秀青年基金(16B141)

Mean Correcting Martingale Measure for Exponential Semimartingale Market Models

Luogen Yao¹(),Gang Yang^1,*(),Xiangqun Yang²()

¹ School of Mathematics and Statistics, Hunan University of Commerce, Changsha 410205
² College of Mathematics and Statistics, Hunan University of Finance and Economics, Changsha 410205

Received:2017-12-13 Online:2019-08-26 Published:2019-09-11
Contact: Gang Yang E-mail:yaoluogen@sina.com;yanggangmath@sina.com;xqyang@hunnu.edu.cn
Supported by:
the Natural Science Foundation of Hunan Province(2019JJ40141);the Natural Science Foundation of Hunan Province(2017JJ2127);the National Social Science Fund of China(15BJY122);the Excellent Youth Foundation of Hunan Provincial Department of Education(16B141)

摘要/Abstract

摘要：

在半鞅模型中，利用均值修正变换构造了一个鞅测度.在半鞅存在连续的局部鞅时，证明了该鞅测度是均值修正鞅测度.对纯跳半鞅，虽然该鞅测度与市场概率不等价，但在无套利区间能达到平凡边界时，证明了具有凸收益函数的期权在该鞅测度下的价格仍然无套利.

关键词: 半鞅, 均值修正鞅测度, 期权定价

Abstract:

A martingale measure is constructed by using a mean correcting transform in a general semimartingale market model. It is shown that this measure is the mean correcting martingale measure if the semimartingale exists a continuous local martingale part. Although this measure cannot be equivalent to the physical probability for a pure jump semimartingale process, we show that option price of a European option with a convex payoff function under this measure is still arbitrage free if the arbitrage-free interval can reach universal bounds.

Key words: Semimartingale, Mean correcting martingale measure, Option pricing

中图分类号:

O211.6

姚落根,杨刚,杨向群. 指数半鞅模型中的均值修正鞅测度[J]. 数学物理学报, 2019, 39(4): 932-941.

Luogen Yao,Gang Yang,Xiangqun Yang. Mean Correcting Martingale Measure for Exponential Semimartingale Market Models[J]. Acta mathematica scientia,Series A, 2019, 39(4): 932-941.

图/表 1

参考文献 16

1	Schoutens W . Lévy Processes in Finance:Pricing Financial Derivatives. England: John Wiley and Sons, 2003
2	Robert J , Elliott R J , Madan D G . A discrete time equivalent martingale measure. Mathematical Finance, 1998, 8 (2): 127- 152 doi: 10.1111/1467-9965.00048
3	Badescu A , Elliott R , Kulperger R , Miettinen J , Siu T . A comparison of pricing kernels for Garch option pricing with generalized hyperbolic distributions. International Journal of Theoretical and Applied Finance, 2011, 14 (5): 669- 708 doi: 10.1142/S0219024911006401
4	Miyahara Y . Martingale measures for the geometric Lévy process models. Discussion Papers in Economics, Nagoya City University, 2005, 431: 1- 14
5	Jacod J , Shiryaev A N . Limit Theorems for Stochastic Processes. New York: Springer-Verlag, 2002
6	Papapantoleon A. Applications of Semimartingales and Processes in Finance: Duallity and Valuation[D]. Freiburg: University of Freiburg, 2007
7	Kallsen J , Shiryaev A N . The cumulant process and Esscher's change of measure. Finance and Stochastics, 2002, 6 (4): 397- 428 doi: 10.1007/s007800200069
8	Kabanov Yu M , Liptser R S , Shiryaev A N . Absolute continuity and singularity of locally absolutely continuous probability distributions. Mathematics of the USSR-Sbornik, 1980, 36 (1): 31- 58 doi: 10.1070/SM1980v036n01ABEH001760
9	Gushchin A A , Mordecki E . Bounds on option prices for semimartingale market models. Proceedings of the Steklov Institute of Mathematics, 2002, 237: 73- 113
10	Jakubenas P . On option pricing in certain incomplete markets. Proceedings of the Steklov Institute of Mathematics, 2002, 237: 114- 133
11	Madan D B , Carr P , Chang E C . The variance gamma process and option pricing. European Finance Review, 1998, 2: 79- 105 doi: 10.1023/A:1009703431535
12	Barndorff-Nielsen O E . Processes of normal inverse Gaussian type. Finance and Stochastics, 1998, 2: 41- 68
13	Grigelionis B . Processes of Meixner type. Lithuanian Mathematics Journal, 1999, 39: 33- 41 doi: 10.1007/BF02465533
14	Carr P , Geman H , Madan D H , Yor M . The fine structure of asset returns:an empirical investigation. Journal of Business, 2002, 75: 305- 332 doi: 10.1086/338705
15	Barndorff-Nielsen O E . Exponentially decreasing distributions for the logarithm of particle size. Proceedings of the Royal Society of London, Series A, Mathematical and Physical Sciences, 1977, 353: 401- 419 doi: 10.1098/rspa.1977.0041
16	Heston S L . Invisible parameters in option prices. Journal of Finance, 1993, 48: 933- 947 doi: 10.1111/j.1540-6261.1993.tb04025.x

$\gamma$	$-0.3000$	$ -0.3500$	$ -0.4000$	$-0.4500 $	$-0.4577 $	$-0.5000 $
平凡边界	[0, 1]	[0, 1]	[0, 1]	[0, 1]	[0, 1]	[0, 1]
$[J_-, J_+]$	[0, 0.3297]	[0, 0.3624]	[0, 0.3935]	[0, 0.4231]	[0, 0.4263]	[0, 0.4512]
$C^{Q^{\alpha^*}}(K, T)$	0.4263	0.4263	0.4263	0.4263	0.4263	0.4263

[1]	许聪聪,许作良. 跳-扩散模型下期权定价方法及参数校准[J]. 数学物理学报, 2019, 39(3): 649-663.
[2]	肖爽, 蹇明, 陈爱香, 蹇贝. 模糊环境中跳扩散模型下带交易费用的期权定价方法[J]. 数学物理学报, 2015, 35(1): 118-130.
[3]	孙玉东, 师义民, 吴敏. 参数依赖股票价格情形下的障碍期权定价[J]. 数学物理学报, 2013, 33(5): 912-925.
[4]	费为银. 半鞅非Lipschitz系数随机微分方程解的大偏差[J]. 数学物理学报, 2009, 29(4): 1074-1083.
[5]	坚雄飞;易法槐. 分期付款购房模型及其抛物障碍问题[J]. 数学物理学报, 2007, 27(6): 987-995.