《山东大学学报(理学版)》 ›› 2020, Vol. 55 ›› Issue (5): 105-113.

• •

### 在跳环境和混合高斯过程下的资产定价及模拟

1. 兰州财经大学统计学院, 甘肃 兰州 730020
• 发布日期:2020-05-06
• 作者简介:彭波(1993— ),男,硕士研究生,研究方向为金融统计与风险管理. E-mail:m19993082290@163.com*通信作者简介:郭精军(1976— ),男,博士,教授,博士生导师,研究方向为金融统计与风险管理. E-mail:guojj@lzufe.edu.cn
• 基金资助:
国家自然科学基金资助项目(71561017,71961013);甘肃省飞天学者资助项目

### Asset pricing and simulation under the environment of jumping and mixed Gaussian process

PENG Bo, GUO Jing-jun*

1. School of Statistics, Lanzhou University of Finance and Economics, Lanzhou 730020, Gansu, China
• Published:2020-05-06

Abstract: The European option pricing model is established by based on mixed sub-fractional Brownian motion in jump environment. Firstly, the partial differential equation satisfying European option can be obtained through the Delta hedging principle. Secondly, the call option, the put option pricing formula and the call-put parity formula are respectively obtained by using the quasi-conditional expectation. Then, the asset risk is further quantified by Greeks Δ, Δ, ρ, Θ, Γ, ν and the partial derivative formula forthe Hurst index H. Finally, numerical simulation show that the Hurst index H and jump intensity λ in pricing parameters have a significant impact on the value of option.

• F224.7
 [1] BLACK F, SCHOLES M. The pricing of options and corporate liabilities[J]. Journal of Political Economy, 1973, 81(3):637-654.[2] CHERIDITO P. Mixed fractional Brownian motion[J]. Bernoulli, 2001, 7(6):913-934.[3] SUN L. Pricing currency options in the mixed fractional Brownian motion[J]. Physica A: Statistical Mechanics & its Applications, 2013, 392(16):3441-3458.[4] SHOKROLLAHI F, KıLıçMAN A, MAGDZIARZ M. Pricing european options and currency options by time changed mixed fractional Brownian motion with transaction costs[J]. International Journal of Financial Engineering, 2016, 3(1):637-654.[5] MERTON R C. Option pricing when underlying stock returns are discontinuous[J]. Journal of Financial Economics, 1976, 3(1/2):125-144.[6] LO A W, MACKINLAY A C. Stock market prices do not follow random walks: evidence from a simple specification test[J]. Review of Financial Studies, 1988, 1(1):41-66.[7] SHOKROLLAHI F, KıLıçMAN A. Pricing currency option in a mixed fractional Brownian motion with jumps environment[J/OL]. Mathematical Problems in Engineering, 2014[2020-03-19]. https://doi.org/10.1155/2014/858210.[8] PETERS E E. Fractal structure in the capital markets[J]. Financial Analysts Journal, 1989, 45(4):32-37.[9] NECULA C. Option pricing in a fractional Brownian motion environment[J]. Mathematical Reports, 2002, 2(3):259-273. [10] MURWANINGTYAS C E, KARTIKO S H, GUNARDI, et al. Option pricing by using a mixed fractional Brownian motion with jumps[J]. Journal of Physics: Conference Series, 2019(1180):012011.[11] ROGERS L C G. Arbitrage with fractional Brownian motion[J]. Mathematical Finance, 1997, 7(1):95-105.[12] 郭精军, 张亚芳. 次分数Vasicek随机利率模型下的欧式期权定价[J]. 应用数学, 2017,30(3):503-511. GUO Jingjun, ZHANG Yafang. European option pricing under the sub-fractional Vasicek stochastic interest rate model[J]. Applied Mathematics, 2017, 30(3):503-511.[13] 程志勇,郭精军,张亚芳.次分数布朗运动下支付红利的欧式期权定价[J].应用概率统计,2018,34(1):37-48. CHENG Zhiyong, GUO Jingjun, ZHANG Yafang. European option pricing for paying dividends under sub-fractional Brownian motion[J]. Applied Probability and Statistics, 2018, 34(1):37-48.[14] 肖炜麟,张卫国,徐维军. 次分数布朗运动下带交易费用的备兑权证定价[J]. 中国管理科学,2014,22(5):1-7. XIAO Weilin, ZHANG Weiguo, XU Weijun. Pricing of covered warrants with transaction costs under sub-fractional Brownian motion[J]. Chinese Journal of Management Science, 2014, 22(5):1-7.[15] TUDOR C. Some properties of the sub-fractional Brownian motion[J]. Stochastics an International Journal of Probability and Stochastic Processes, 2007, 79(5):431-448.[16] XU Feng, LI Runze. The pricing formulas of compound option based on the sub-fractional Brownian motion model[J]. Journal of Physics: Conference Series, 2018(1053):012027.
 [1] 陈丽,林玲. 具有时滞效应的股票期权定价[J]. 山东大学学报（理学版）, 2018, 53(4): 36-41. [2] 李国成,王继霞. 交叉熵蝙蝠算法求解期权定价模型参数估计问题[J]. 《山东大学学报(理学版)》, 2018, 53(12): 80-89. [3] 郭尊光1,孔涛2*,李鹏飞2, 张微2. 基于最优实施边界的美式期权定价的数值方法[J]. J4, 2012, 47(3): 110-119. [4] 张慧1,2,孟纹羽1,来翔3. 不确定环境下障碍再装期权的动态定价模型 ——基于BSDE解的期权定价方法[J]. J4, 2011, 46(3): 52-57. [5] 苗杰1,师恪2,蔡华1. 跳扩散模型下的可分离债券的定价[J]. J4, 2010, 45(8): 109-117. [6] 陈祥利. 脆弱期权的公司价值分形定价模型[J]. J4, 2010, 45(11): 109-114. [7] 孙 鹏,张 蕾,赵卫东 . 美式期权定价问题的一类有限体积数值模拟方法[J]. J4, 2007, 42(6): 1-06 . [8] 孙 鹏,赵卫东 . 亚式期权定价问题的交替方向迎风有限体积方法[J]. J4, 2007, 42(6): 16-21 .
Viewed
Full text

Abstract

Cited

Shared
Discussed