JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE) ›› 2024, Vol. 59 ›› Issue (4): 98-107.doi: 10.6040/j.issn.1671-9352.0.2022.629

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

ZHANG Yaru1, XIA Li1,2*, ZHANG Dianqiu1   

  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)
  • Published:2024-04-12

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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

CLC Number: 

  • O211.6
[1] SHREVE S E. Stochastic calculus for finance II: continuous-time models[M]. New York: Springer, 2004.
[2] GOLDMAN M B, SOSIN H B, GATTO M A. Path dependentoptions: “buy at the low, sell at the high”[J]. The Journal of Finance, 1979, 34(5):1111-1127.
[3] EMANUEL D C, MACBETH J 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.
[4] DAI Min. A closed-form solution for perpetual American floating strike lookback options[J]. Journal of Computational Finance, 2000, 4(2):63-68.
[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. JIANG Lishang. Mathematical modeling and methods of option pricing[M]. 2nd ed. Beijing: 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. ZHANG Yanqiu, DU Xueqiao. Pricing of lookback options under stochastic interest rates[J]. Journal of Hefei University of Technology(Natural Science), 2007, 30(4):515-517.
[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. YUAN Guojun, XIAO Qingxian. Study of pricing lookback options in CEV process with transaction cost[J]. Journal of Systems Engineering, 2011, 26(5):642-648.
[8] TING S H M, EWALD C O, WANG W K. On the investment-uncertainty relationship in a real option model with stochastic volatility[J]. Mathematical Social Sciences, 2013, 66(1):22-32.
[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. SANG Liheng. 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. CHEN Haizhen, ZHOU Shengwu, SUN Xiangyan. Pricing of lookback options under a mixed fractional Brownian movement[J]. Journal of East China Normal University(Natural Science), 2018(4):47-58.
[11] DENG Guohe. 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. AN Xiang, GUO Jingjun. 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. AN Xiang, GUO Jingjun. 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. ZHANG Meng, WEN Xian, HUO Haifeng. 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. YANG Zhaoqiang, TIAN Yougong. Bankruptcy probability of a lever company: lookback option pricing method[J]. Chinese Journal of Applied Probability and Statistics, 2022, 38(1):1-23.
[17] ZHANG Xili, XIAO Weilin. Arbitrage with fractional Gaussian processes[J]. Physica A: Statistical Mechanics and its Applications, 2017, 471:620-628.
[18] RUSSO F, TUDOR C A. On bifractional Brownian motion[J]. Stochastic Processes and their Applications, 2006, 116(5):830-856.
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