您的位置：山东大学 -> 科技期刊社 -> 《山东大学学报（理学版）》

《山东大学学报(理学版)》 ›› 2024, Vol. 59 ›› Issue (4): 98-107.doi: 10.6040/j.issn.1671-9352.0.2022.629

•   • 上一篇    下一篇

混合双分数布朗运动下的永久美式回望期权定价

张亚茹1(),夏莉1,2,*(),张典秋1   

  1. 1. 广东财经大学统计与数学学院，广东 广州 510320
    2. 广东财经大学大数据与教育应用统计实验室，广东 广州 510320
  • 收稿日期:2022-11-28 出版日期:2024-04-20 发布日期:2024-04-12
  • 通讯作者: 夏莉 E-mail:1075214382@qq.com;xaleysherry@163.com
  • 作者简介:张亚茹(1993—)，女，硕士研究生，研究方向为偏微分方程理论及其应用、数理金融. E-mail: 1075214382@qq.com
  • 基金资助:
    国家自然科学基金资助项目(11971200);广东省教育厅科研项目(2018KCXTD013);广东省教育厅科研项目[特色创新项目](2018KTSCX069);广东省教育厅委托项目(0835-210Z33606691)

Perpetual American lookback option pricing under mixed bi-fractional Brownian motion

Yaru ZHANG1(),Li XIA1,2,*(),Dianqiu ZHANG1   

  1. 1. School of Statistics and Mathematics, Guangdong University of Finance and Economics, Guangzhou 510320, Guangdong, China
    2. Big Data and Educational Statistics Application Laboratory, Guangdong University of Finance and Economics, Guangzhou 510320, Guangdong, China
  • Received:2022-11-28 Online:2024-04-20 Published:2024-04-12
  • Contact: Li XIA E-mail:1075214382@qq.com;xaleysherry@163.com

RichHTML

4

PDF (PC)

171

摘要：

构建混合双分数布朗运动驱动下的带有红利的永久美式回望期权定价模型。首先，通过Δ-对冲原理，给出混合双分数布朗运动下的永久美式回望看涨和看跌期权所满足的偏微分方程组；其次，通过变量代换法、特征方程法对建立的偏微分方程组进行求解，给出混合双分数布朗运动下的永久美式回望看涨和看跌期权定价公式及其最佳实施边界；最后，通过数值实验，验证该解的线性等比例缩放性质，并讨论混合双分数布朗运动参数H、K及波动率对期权价格的影响。

关键词: 混合双分数布朗运动, 永久美式回望期权, 期权定价

Abstract:

A pricing model for a perpetual American lookback option with dividends driven by mixed bi-fractional Brownian motion is constructed in this paper. First, the partial differential equations of mixed bi-fractional Brownian motion for perpetual American lookback call and put options are given by the Δ-hedging principle. Then, the established partial differential equations are solved by variable substitution method and characteristic equation method. Finally, numerical experiments are adopted to verify the linear proportional scaling properties of the solution, and the effects of mixed bi-fractional Brownian motion parameters H, K and volatility on the option prices are further discussed.

Key words: mixed bi-fractional Brownian motion, perpetual American lookback option, option pricing

中图分类号: 

  • O211.6

图1

双分数布朗运动模拟资产价格"

图2

资产价格路径模拟图"

图3

线性等比例缩放性质"

表1

不同H、K下永久美式回望期权定价结果"

HKλ1λ2zcC
0.750.707.691 8-0.095 30.087 07.406 3
0.750.757.237 1-0.094 30.086 27.425 0
0.750.806.821 3-0.093 30.085 37.445 0
0.850.607.885 7-0.095 70.087 37.399 0
0.850.706.874 6-0.093 40.085 57.442 0
0.850.806.046 0-0.091 10.083 57.489 0
0.700.757.691 8-0.095 30.087 07.406 3
0.750.757.237 1-0.094 30.086 27.425 0
0.850.756.440 3-0.092 30.084 57.465 0
0.650.857.354 3-0.094 60.086 47.420 0
0.750.856.440 3-0.092 30.084 57.465 0
0.850.855.687 2-0.089 80.082 47.514 0

图4

永久美式回望看涨期权随波动率的变化"

表2

不同σ1、σ2下部分永久美式回望期权定价结果"

σ1σ2λ1λ2zcC
0.200.306.821 3-0.093 30.085 37.444 6
0.250.355.143 2-0.087 60.080 67.559 1
0.300.404.099 7-0.081 80.075 67.681 6
0.350.453.407 5-0.076 00.070 67.807 0
0.400.502.924 8-0.070 40.658 07.931 2
1 SHREVES E.Stochastic calculus for finance Ⅱ: continuous-time models[M].New York:Springer,2004.
2 GOLDMANM B,SOSINH B,GATTOM A.Path dependentoptions: "buy at the low, sell at the high"[J].The Journal of Finance,1979,34(5):1111-1127.
3 EMANUELD C,MACBETHJ D.Further results on the constant elasticity of variance call option pricing model[J].Journal of Financial and Quantitative Analysis,1982,17(4):533-554.
doi: 10.2307/2330906
4 DAIMin.A closed-form solution for perpetual American floating strike lookback options[J].Journal of Computational Finance,2000,4(2):63-68.
doi: 10.21314/JCF.2001.074
5 姜礼尚.期权定价的数学模型和方法[M].2版北京:高等教育出版社,2008.
JIANGLishang.Mathematical modeling and methods of option pricing[M].2nd edBeijing:Higher Education Press,2008.
6 张艳秋,杜雪樵.随机利率下的回望期权的定价[J].合肥工业大学学报(自然科学版),2007,30(4):515-517.
doi: 10.3969/j.issn.1003-5060.2007.04.031
ZHANGYanqiu,DUXueqiao.Pricing of lookback options under stochastic interest rates[J].Journal of Hefei University of Technology (Natural Science),2007,30(4):515-517.
doi: 10.3969/j.issn.1003-5060.2007.04.031
7 袁国军,肖庆宪.CEV过程下有交易费的回望期权定价研究[J].系统工程学报,2011,26(5):642-648.
YUANGuojun,XIAOQingxian.Study of pricing lookback options in CEV process with transaction cost[J].Journal of Systems Engineering,2011,26(5):642-648.
8 TINGS H M,EWALDC O,WANGW K.On the investment-uncertainty relationship in a real option model with stochastic volatility[J].Mathematical Social Sciences,2013,66(1):22-32.
doi: 10.1016/j.mathsocsci.2013.01.005
9 桑利恒.标的资产带有红利支付的回望期权定价[J].长春大学学报,2014,24(12):1671-1674.
SANGLiheng.Valuation of lookback option of underlying asset with dividend payment[J].Journal of Changchun University,2014,24(12):1671-1674.
10 陈海珍,周圣武,孙祥艳.混合分数布朗运动下的回望期权定价[J].华东师范大学学报(自然科学版),2018,(4):47-58.
CHENHaizhen,ZHOUShengwu,SUNXiangyan.Pricing of lookback options under a mixed fractional Brownian movement[J].Journal of East China Normal University (Natural Science),2018,(4):47-58.
11 DENGGuohe.Pricing perpetual American floating strike lookback option under multiscale stochastic volatility model[J].Chaos, Solitons & Fractals,2020,141,110411.
12 顾哲煜. 几类混合双分数布朗运动模型下回望期权的定价研究[D]. 南京: 南京财经大学, 2020.
GU Zheyu. Research on the pricing of lookback options under several mixed bi-fractional Brownian motion models[D]. Nanjing: Nanjing University of Finance and Economics, 2020.
13 安翔,郭精军.混合次分数布朗运动下永久美式回望期权的定价[J].应用数学,2021,34(4):820-828.
ANXiang,GUOJingjun.Pricing of perpetual American lookback option under the sub-mixed fractional Brownian motion[J].Mathematica Applicate,2021,34(4):820-828.
14 安翔,郭精军.混合次分数跳扩散模型下回望期权的定价及模拟[J].山东大学学报(理学版),2022,57(4):100-110.
ANXiang,GUOJingjun.Pricing and simulation of lookback options under the mixed sub-fractional jump-diffusion model[J].Journal of Shandong University (Natural Science),2022,57(4):100-110.
15 张萌,温鲜,霍海峰.基于次分数布朗运动的欧式回望期权定价[J].桂林航天工业学院学报,2022,27(1):110-115.
ZHANGMeng,WENXian,HUOHaifeng.European lookback option pricing based on fractional Brownian motion[J].Journal of Guilin University of Aerospace Technology,2022,27(1):110-115.
16 杨朝强,田有功.一类杠杆公司的破产概率: 回望期权定价方法[J].应用概率统计,2022,38(1):1-23.
YANGZhaoqiang,TIANYougong.Bankruptcy probability of a lever company: lookback option pricing method[J].Chinese Journal of Applied Probability and Statistics,2022,38(1):1-23.
17 ZHANGXili,XIAOWeilin.Arbitrage with fractional Gaussian processes[J].Physica A: Statistical Mechanics and its Applications,2017,471,620-628.
18 RUSSOF,TUDORC A.On bifractional Brownian motion[J].Stochastic Processes and their Applications,2006,116(5):830-856.
[1] 彭波,郭精军. 在跳环境和混合高斯过程下的资产定价及模拟[J]. 《山东大学学报(理学版)》, 2020, 55(5): 105-113.
[2] 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.
[3] 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.
[4] 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.
[5] 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.
[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   
[1] 程智1,2,孙翠芳2,王宁1,杜先能1. 关于Zn的拉回及其性质[J]. J4, 2013, 48(2): 15 -19 .
[2] 张爱平,李刚 . LR拟正规Ehresmann半群[J]. J4, 2006, 41(5): 44 -47 .
[3] 赵同欣1,刘林德1*,张莉1,潘成臣2,贾兴军1. 紫藤传粉昆虫与花粉多型性研究[J]. 山东大学学报（理学版）, 2014, 49(03): 1 -5 .
[4] 马建玲 . 菱体型消色差相位延迟器的光谱特性分析[J]. J4, 2007, 42(7): 27 -29 .
[5] 边斐,何海伦,陈秀兰*,张玉忠 . 离子对深海适冷菌Pseudoalteromonas sp. SM9913胞外蛋白酶分泌的影响[J]. J4, 2006, 41(5): 166 -172 .
[6] 王 震,张 琎 . NLS方程的守恒数值格式及其收敛性分析[J]. J4, 2007, 42(3): 13 -17 .
[7] 王康 李华. 化学计量学方法用于蛤青注射色谱数据重叠峰的分辨[J]. J4, 2009, 44(11): 16 -20 .
[8] 陈 莉, . 非方广义系统带干扰抑制的奇异LQ次优控制问题[J]. J4, 2006, 41(2): 74 -77 .
[9] 李永明1, 丁立旺2. PA误差下半参数回归模型估计的r-阶矩相合[J]. J4, 2013, 48(1): 83 -88 .
[10] 董丽红1,2,郭双建1. Yetter-Drinfeld模范畴上的弱Hopf模基本定理[J]. J4, 2013, 48(2): 20 -22 .
版权所有 © 2018 《山东大学学报(理学版)》编辑部　
地址: 山东省济南市山大南路27号山东大学中心校区（250100）　电话: +86-0531-88366917　E-mail: xblxb@sdu.edu.cn
本系统由北京玛格泰克科技发展有限公司设计开发