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《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

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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

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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
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