IDEAS home Printed from https://ideas.repec.org/a/eee/matcom/v190y2021icp203-221.html
   My bibliography  Save this article

Monotonicity theorem for the uncertain fractional differential equation and application to uncertain financial market

Author

Listed:
  • Jin, Ting
  • Yang, Xiangfeng

Abstract

Uncertain fractional differential equations (UFDEs) have non-locality features to reflect memory and hereditary characteristics for the asset price changes, thus are more suitable to model the real financial market. Based on this characteristic, this paper primarily investigates the monotonicity theorem for uncertain fractional differential equations in Caputo sense and its application. Firstly, monotonicity theorems for solutions of UFDEs are presented by using the α-path method. Secondly, as the application of the monotone function theorem, a novel uncertain fractional mean-reverting model with a floating interest rate is presented. Lastly, the pricing formulas of the European and American options are derived for the proposed model based on the monotone function and present extreme values and time integral theorems, respectively. In addition, numerical schemes are designed, and numerical calculations are illustrated concerning different parameters through the predictor–corrector method.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:matcom:v:190:y:2021:i:c:p:203-221
    DOI: 10.1016/j.matcom.2021.05.018
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0378475421001890
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.matcom.2021.05.018?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. 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).
    2. Daniel Kahneman & Amos Tversky, 2013. "Prospect Theory: An Analysis of Decision Under Risk," World Scientific Book Chapters, in: Leonard C MacLean & William T Ziemba (ed.), HANDBOOK OF THE FUNDAMENTALS OF FINANCIAL DECISION MAKING Part I, chapter 6, pages 99-127, World Scientific Publishing Co. Pte. Ltd..
    3. Ziqiang Lu & Hongyan Yan & Yuanguo Zhu, 2019. "European option pricing model based on uncertain fractional differential equation," Fuzzy Optimization and Decision Making, Springer, vol. 18(2), pages 199-217, June.
    4. Tian, Miao & Yang, Xiangfeng & Zhang, Yi, 2019. "Barrier option pricing of mean-reverting stock model in uncertain environment," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 166(C), pages 126-143.
    5. 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).
    6. 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.
    7. Alghalith, Moawia, 2018. "Pricing the American options using the Black–Scholes pricing formula," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 507(C), pages 443-445.
    8. Yang, Xiangfeng & Zhang, Zhiqiang & Gao, Xin, 2019. "Asian-barrier option pricing formulas of uncertain financial market," Chaos, Solitons & Fractals, Elsevier, vol. 123(C), pages 79-86.
    9. Jin, Ting & Sun, Yun & Zhu, Yuanguo, 2020. "Time integral about solution of an uncertain fractional order differential equation and application to zero-coupon bond model," Applied Mathematics and Computation, Elsevier, vol. 372(C).
    10. Yang, Xiangfeng & Liu, Yuhan & Park, Gyei-Kark, 2020. "Parameter estimation of uncertain differential equation with application to financial market," Chaos, Solitons & Fractals, Elsevier, vol. 139(C).
    11. Schwartz, Eduardo S., 1977. "The valuation of warrants: Implementing a new approach," Journal of Financial Economics, Elsevier, vol. 4(1), pages 79-93, January.
    12. Cox, John C. & Ross, Stephen A. & Rubinstein, Mark, 1979. "Option pricing: A simplified approach," Journal of Financial Economics, Elsevier, vol. 7(3), pages 229-263, September.
    13. Kai Yao & Baoding Liu, 2020. "Parameter estimation in uncertain differential equations," Fuzzy Optimization and Decision Making, Springer, vol. 19(1), pages 1-12, March.
    14. Tian, Miao & Yang, Xiangfeng & Kar, Samarjit, 2019. "Equity warrants pricing problem of mean-reverting model in uncertain environment," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 531(C).
    15. Brennan, Michael J. & Schwartz, Eduardo S., 1978. "Finite Difference Methods and Jump Processes Arising in the Pricing of Contingent Claims: A Synthesis," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 13(3), pages 461-474, September.
    16. Lars Stentoft, 2019. "Efficient Numerical Pricing of American Call Options Using Symmetry Arguments," JRFM, MDPI, vol. 12(2), pages 1-26, April.
    17. Yiyao Sun & Taoyong Su, 2017. "Mean-reverting stock model with floating interest rate in uncertain environment," Fuzzy Optimization and Decision Making, Springer, vol. 16(2), pages 235-255, June.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Xiang Wang & Chang-Franw Lee & Jiabei Jiang & Xiaoyang Zhu, 2023. "Discussion on Sustainable Development Strategy of China’s Rehabilitation Assistive Device Industry Based on Diamond Model," Sustainability, MDPI, vol. 15(3), pages 1-24, January.
    2. Du, Wentong & Xiao, Min & Ding, Jie & Yao, Yi & Wang, Zhengxin & Yang, Xinsong, 2023. "Fractional-order PD control at Hopf bifurcation in a delayed predator–prey system with trans-species infectious diseases," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 205(C), pages 414-438.
    3. Hu, Chaoming & Wan, Zhao Man & Zhu, Saihua & Wan, Zhong, 2022. "An integrated stochastic model and algorithm for constrained multi-item newsvendor problems by two-stage decision-making approach," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 193(C), pages 280-300.
    4. Fang Xu & Yifan Ma & Chang Liu & Ying Ji, 2024. "Emergency Logistics Facilities Location Dual-Objective Modeling in Uncertain Environments," Sustainability, MDPI, vol. 16(4), pages 1-34, February.
    5. Pan, Zeyu & Gao, Yin & Yuan, Lin, 2021. "Bermudan options pricing formulas in uncertain financial markets," Chaos, Solitons & Fractals, Elsevier, vol. 152(C).

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    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. 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).
    3. Jian Zhou & Yujiao Jiang & Athanasios A. Pantelous & Weiwen Dai, 2023. "A systematic review of uncertainty theory with the use of scientometrical method," Fuzzy Optimization and Decision Making, Springer, vol. 22(3), pages 463-518, September.
    4. Liu He & Yuanguo Zhu & Ziqiang Lu, 2023. "Parameter estimation for uncertain fractional differential equations," Fuzzy Optimization and Decision Making, Springer, vol. 22(1), pages 103-122, March.
    5. Lu, Jing & Yang, Xiangfeng & Tian, Miao, 2022. "Barrier swaption pricing formulae of mean-reverting model in uncertain environment," Chaos, Solitons & Fractals, Elsevier, vol. 160(C).
    6. 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.
    7. Pan, Zeyu & Gao, Yin & Yuan, Lin, 2021. "Bermudan options pricing formulas in uncertain financial markets," Chaos, Solitons & Fractals, Elsevier, vol. 152(C).
    8. Jia, Lifen & Chen, Wei, 2020. "Knock-in options of an uncertain stock model with floating interest rate," Chaos, Solitons & Fractals, Elsevier, vol. 141(C).
    9. Sheng, Yuhong & Yao, Kai & Qin, Zhongfeng, 2020. "Continuity and variation analysis of fractional uncertain processes," Chaos, Solitons & Fractals, Elsevier, vol. 140(C).
    10. Shi, Gang & Gao, Jinwu, 2021. "European Option Pricing Problems with Fractional Uncertain Processes," Chaos, Solitons & Fractals, Elsevier, vol. 143(C).
    11. Yu, Yongjiu & Yang, Xiangfeng & Lei, Qing, 2022. "Pricing of equity swaps in uncertain financial market," Chaos, Solitons & Fractals, Elsevier, vol. 154(C).
    12. He, Liu & Zhu, Yuanguo, 2024. "Nonparametric estimation for uncertain fractional differential equations," Chaos, Solitons & Fractals, Elsevier, vol. 178(C).
    13. Song-Ping Zhu & Xin-Jiang He, 2018. "A hybrid computational approach for option pricing," International Journal of Financial Engineering (IJFE), World Scientific Publishing Co. Pte. Ltd., vol. 5(03), pages 1-16, September.
    14. Yang, Xiangfeng & Liu, Yuhan & Park, Gyei-Kark, 2020. "Parameter estimation of uncertain differential equation with application to financial market," Chaos, Solitons & Fractals, Elsevier, vol. 139(C).
    15. Jie, Ke-Wei & Liu, San-Yang & Sun, Xiao-Jun & Xu, Yun-Cheng, 2023. "A dynamic ripple-spreading algorithm for solving mean–variance of shortest path model in uncertain random networks," Chaos, Solitons & Fractals, Elsevier, vol. 167(C).
    16. Suresh M. Sundaresan, 2000. "Continuous‐Time Methods in Finance: A Review and an Assessment," Journal of Finance, American Finance Association, vol. 55(4), pages 1569-1622, August.
    17. Manuel Moreno & Javier Navas, 2003. "On the Robustness of Least-Squares Monte Carlo (LSM) for Pricing American Derivatives," Review of Derivatives Research, Springer, vol. 6(2), pages 107-128, May.
    18. Mark Broadie & Jérôme Detemple, 1996. "Recent Advances in Numerical Methods for Pricing Derivative Securities," CIRANO Working Papers 96s-17, CIRANO.
    19. Phelim Boyle & Yisong Tian, 1998. "An explicit finite difference approach to the pricing of barrier options," Applied Mathematical Finance, Taylor & Francis Journals, vol. 5(1), pages 17-43.
    20. Wang, Weiwei & Ralescu, Dan A., 2021. "Valuation of lookback option under uncertain volatility model," Chaos, Solitons & Fractals, Elsevier, vol. 153(P1).

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:matcom:v:190:y:2021:i:c:p:203-221. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Catherine Liu (email available below). General contact details of provider: http://www.journals.elsevier.com/mathematics-and-computers-in-simulation/ .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.