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Option pricing in the illiquid markets under the mixed fractional Brownian motion model

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  • Ma, Pengcheng
  • Taghipour, Mehran
  • Cattani, Carlo

Abstract

This paper deals with the option pricing in the illiquid markets under the mixed fractional geometric Brownian motion model with jump process. We propose a general long memory financial model, where its featuring parameters are related to demand and supply by showing also the existence of some restrictions on them. Moreover, by using the delta Hedging strategy and replicating portfolio, we obtain an integro partial differential equation (PIDE) for the option price which is solved by the spectral numerical method with suitable diagonal functions and an infinite series. In particular, by using some operational matrices and Gauss–Hermite quadrature rule, we derive a linear system of algebraic equations solved by a standard collocation method. Moreover, we study the existence and uniqueness of the solution of PIDE and prove the convergence of the numerical scheme. The applicability and efficiency of the collocation method are shown on some nontrivial numerical examples.

Suggested Citation

  • Ma, Pengcheng & Taghipour, Mehran & Cattani, Carlo, 2024. "Option pricing in the illiquid markets under the mixed fractional Brownian motion model," Chaos, Solitons & Fractals, Elsevier, vol. 182(C).
    • Handle: RePEc:eee:chsofr:v:182:y:2024:i:c:s0960077924003588
    DOI: 10.1016/j.chaos.2024.114806
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    References listed on IDEAS

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    1. Wang, Xiao-Tian & Wu, Min & Zhou, Ze-Min & Jing, Wei-Shu, 2012. "Pricing European option with transaction costs under the fractional long memory stochastic volatility model," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 391(4), pages 1469-1480.
    2. Taghipour, M. & Aminikhah, H., 2022. "A spectral collocation method based on fractional Pell functions for solving time–fractional Black–Scholes option pricing model," Chaos, Solitons & Fractals, Elsevier, vol. 163(C).
    3. Li, Zhe & Zhang, Wei-Guo & Liu, Yong-Jun, 2018. "Analytical valuation for geometric Asian options in illiquid markets," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 507(C), pages 175-191.
    4. Kanne, Stefan & Korn, Olaf & Uhrig-Homburg, Marliese, 2023. "Stock illiquidity and option returns," Journal of Financial Markets, Elsevier, vol. 63(C).
    5. Foad Shokrollahi & Adem Kılıçman, 2014. "Pricing Currency Option in a Mixed Fractional Brownian Motion with Jumps Environment," Mathematical Problems in Engineering, Hindawi, vol. 2014, pages 1-13, April.
    6. Tai-Ho Wang, 2015. "Nonlinear Option Pricing," Quantitative Finance, Taylor & Francis Journals, vol. 15(1), pages 19-21, January.
    7. L. C. G. Rogers, 1997. "Arbitrage with Fractional Brownian Motion," Mathematical Finance, Wiley Blackwell, vol. 7(1), pages 95-105, January.
    8. Liu, Hong & Yong, Jiongmin, 2005. "Option pricing with an illiquid underlying asset market," Journal of Economic Dynamics and Control, Elsevier, vol. 29(12), pages 2125-2156, December.
    9. Luo, Ziyang & Zhang, Xingdong & Wang, Shuo & Yao, Lin, 2022. "Numerical approximation of time fractional partial integro-differential equation based on compact finite difference scheme," Chaos, Solitons & Fractals, Elsevier, vol. 161(C).
    10. Wang, Xiao-Tian & Liang, Xiang-Qian & Ren, Fu-Yao & Zhang, Shi-Ying, 2006. "On some generalization of fractional Brownian motions," Chaos, Solitons & Fractals, Elsevier, vol. 28(4), pages 949-957.
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    2. Zhang, Weiting & He, Guitian & Luo, Maokang & Liang, Wenjie, 2025. "Valuation of R&D projects of new energy vehicles based on generalized mixed sub-fractional Brownian motion under fuzzy environment," Applied Mathematics and Computation, Elsevier, vol. 507(C).

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