JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE) ›› 2022, Vol. 57 ›› Issue (4): 100-110.doi: 10.6040/j.issn.1671-9352.0.2021.622

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Pricing and simulation of lookback options under the mixed sub-fractional jump-diffusion model

AN Xiang, GUO Jing-jun*   

  1. School of Statistics, Lanzhou University of Finance and Economics, Lanzhou 730020, Gansu, China
  • Published:2022-03-29

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Abstract: The pricing model of European lookback options with transaction costs is established based on the mixed sub-fractional Brownian motion and Poisson process. Firstly, the nonlinear partial differential equation satisfied by the price of the option is obtained using the Delta hedging principle, and its numerical solution is obtained by constructing a Crank-Nicolson format. Secondly, the validity of the numerical method is verified, and the effects of transaction costs, volatility and risk-free interest rate on the value of the option are respectively discussed. Finally, the daily closing price of Shanghai Pudong Development Bank is selected for the simulation, and the results show that the simulated price based on the mixed sub-fractional jump-diffusion model is closer to the real value of the stock, and can better reflect the overall stock trend.

Key words: mixed sub-fractional Brownian motion, jump-diffusion model, European lookback options, transaction costs, Crank-Nicolson format

CLC Number: 

  • O211.6
[1] GOLDMAN M B, SOSIN H B, GATTO M A. Path dependent options: “buy at the low, sell at the high”[J]. The Journal of Finance, 1979, 34(5):1111-1127.
[2] BROADIE M, GLASSERMAN P, KOU S G. Connecting discrete and continuous path-dependent options[J]. Finance and Stochastics, 1999, 3(1):55-82.
[3] BUCHEN P, KONSTANDATOS O. A new method of pricing lookback options[J]. Mathematical Finance, 2005, 15(2):245-259.
[4] SUN L. Pricing currency options in the mixed fractional Brownian motion [J]. Physica A:Statistical Mechanics and its Applications, 2013, 392(16):3441-3458.
[5] ROSTEK S, SCHÖBEL R. A note on the use of fractional Brownian motion for financial modeling[J]. Economic Modelling, 2013, 30:30-35.
[6] SHOKROLLAHI F. Mixed fractional Merton model to evaluate European options with transaction costs[J]. Journal of Mathematical Finance, 2018, 8(4):623-639.
[7] YANG Z Q. Default probability of American lookback option in a mixed jump-diffusion model[J]. Physica A: Statistical Mechanics and its Applications, 2020, 540:1-12.
[8] CHEN Q, ZHANG Q, LIU C. The pricing and numerical analysis of lookback options for mixed fractional Brownian motion [J]. Chaos, Solitons & Fractals, 2019, 128:123242.
[9] MERTON R C. Option pricing when underlying stock returns are discontinuous [J]. Journal of Financial Economics, 1976, 3(1/2):125-144.
[10] XIAO W L, ZHANG W G, ZHANG X L, et al. Pricing currency options in a fractional Brownian motion with jumps[J]. Economic Modelling, 2010, 27(5):935-942.
[11] SHOKROLLAHI F, KılıçMAN A. Actuarial approach in a mixed fractional Brownian motion with jumps environment for pricing currency option[J]. Advances in Difference Equations, 2015, 2015(1):257.
[12] 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. PENG Bo, GUO Jingjun. Asset pricing and simulation under the environment of jumping and mixed Gaussian process[J]. Journal of Shandong University(Natural Science), 2020, 55(5):105-113.
[13] KIM K I, PARK H S, QIAN X S. A mathematical modeling for the lookback option with jump-diffusion using binomial tree method[J]. Journal of Computational and Applied Mathematics, 2011, 235(17):5140-5154.
[14] 杨朝强.一类特殊混合跳扩散Black-Scholes模型的欧式回望期权定价[J].数学物理学报, 2019, 39A(6): 1514-1531. YANG Zhaoqiang. Pricing European lookback option in a special kind of mixed jump-diffusion Black-Scholes model[J]. Acta Mathematica Scientia, 2019, 39A(6):1514-1531.
[15] LELAND H E. Option pricing and replication with transactions costs[J]. The Journal of Finance, 1985, 40(5):1283-1301.
[16] WANG X T. Scaling and long-range dependence in option pricing I: pricing European option with transaction costs under the fractional Black-Scholes model[J]. Physica A: Statistical Mechanics and its Applications, 2010, 389(3):438-444.
[17] WANG X T, ZHU E H, TANG M M, et al. Scaling and long-range dependence in option pricing II: Pricing European option with transaction costs under the mixed Brownian fractional-Brownian model[J]. Physica A: Statistical Mechanics and its Applications, 2010, 389(3):445-451.
[18] LESMANA D C, WANG S. An upwind finite difference method for a nonlinear Black-Scholes equation governing European option valuation under transaction costs[J]. Applied Mathematics and Computation, 2013, 219(16):8811-8828.
[19] 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 covered warrants in a sub-fractional Brownian motion with transaction costs[J]. Chinese Journal of Management Science, 2014, 22(5):1-7.
[20] SUN J J, ZHOU S, ZHANG Y, et al. Lookback option pricing with fixed proportional transaction costs under fractional Brownian motion[J]. International Scholarly Research Notices, 2014, 2014:1-7.
[21] 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.
[22] CHARLES E N, MOUNIR Z. On the sub-mixed fractional Brownian motion[J]. Applied Mathematics-A Journal of Chinese Universities, 2015, 30(1):27-43.
[1] LI Guo-cheng, WANG Ji-xia. Calibrating option pricing models with cross entropy bat algorithm [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2018, 53(12): 80-89.
[2] MIAO Jie 1, SHI Ke 2, CAI Hua 1. The pricing of bond with attached warrant under the jump-diffusion model [J]. J4, 2010, 45(8): 109-117.
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