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Extreme values for solution to uncertain fractional differential equation and application to American option pricing model

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  • Jin, Ting
  • Sun, Yun
  • Zhu, Yuanguo

Abstract

Uncertain fractional differential equation plays an important role of describing uncertain dynamic process. This paper focuses on extreme values (including supremum and infimum) for solution to an uncertain fractional differential equation for the Caputo type. Theorems for the inverse uncertain distributions of the extreme values are given based on the definition of α-path. And then, numerical algorithms for them are designed, numerical examples are shown for validating the availability about algorithms. The absolute errors between the numerical and analytical results are also presented to demonstrate the accuracy of the algorithms. Finally, as an application of the extreme values, an uncertain stock model is proposed on the basis of uncertain fractional differential equation of the Caputo type. The American option pricing formulas of such stock model are studied by using the proposed extreme theorems. Besides, numerical calculations are also illustrated with respect to different parameters p.

Suggested Citation

  • Jin, Ting & Sun, Yun & Zhu, Yuanguo, 2019. "Extreme values for solution to uncertain fractional differential equation and application to American option pricing model," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 534(C).
    • Handle: RePEc:eee:phsmap:v:534:y:2019:i:c:s0378437119313573
    DOI: 10.1016/j.physa.2019.122357
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    References listed on IDEAS

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    1. Black, Fischer & Scholes, Myron S, 1973. "The Pricing of Options and Corporate Liabilities," Journal of Political Economy, University of Chicago Press, vol. 81(3), pages 637-654, May-June.
    2. Lu, Ziqiang & Zhu, Yuanguo, 2019. "Numerical approach for solution to an uncertain fractional differential equation," Applied Mathematics and Computation, Elsevier, vol. 343(C), pages 137-148.
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    Cited by:

    1. Jin, Ting & Ding, Hui & Xia, Hongxuan & Bao, Jinfeng, 2021. "Reliability index and Asian barrier option pricing formulas of the uncertain fractional first-hitting time model with Caputo type," Chaos, Solitons & Fractals, Elsevier, vol. 142(C).
    2. Yan, Hongyan & Jin, Ting & Sun, Yun, 2020. "Uncertain bang–bang control problem for multi-stage switched systems," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 551(C).
    3. Jin, Ting & Zhu, Yuanguo, 2020. "First hitting time about solution for an uncertain fractional differential equation and application to an uncertain risk index model," Chaos, Solitons & Fractals, Elsevier, vol. 137(C).
    4. Weiwei Wang & Dan A. Ralescu, 2021. "Option pricing formulas based on uncertain fractional differential equation," Fuzzy Optimization and Decision Making, Springer, vol. 20(4), pages 471-495, December.
    5. Jin, Ting & Yang, Xiangfeng, 2021. "Monotonicity theorem for the uncertain fractional differential equation and application to uncertain financial market," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 190(C), pages 203-221.

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