Skip to main content
SpringerLink
Log in

Option pricing formulas based on uncertain fractional differential equation

  • Published:
Fuzzy Optimization and Decision Making Aims and scope Submit manuscript

Abstract

Uncertain fractional differential equations have been playing an important role in modelling complex dynamic systems. Early researchers have presented the extreme value theorems and time integral theorem on uncertain fractional differential equation. As applications of these theorems, this paper investigates the pricing problems of American option and Asian option under uncertain financial markets based on uncertain fractional differential equations. Then the analytical solutions and numerical solutions of these option prices are derived, respectively. Finally, some numerical experiments are performed to verify the effectiveness of our results.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  • Chen, X., & Liu, B. (2010). Existence and uniqueness theorem for uncertain differential equations. Fuzzy Optimization and Decision Making, 9(1), 69–81.

    Article  MathSciNet  Google Scholar 

  • Chen, X. (2011). American option pricing formula for uncertain financial market. International Journal of Operations Research, 8(2), 32–37.

    MathSciNet  MATH  Google Scholar 

  • Chen, X., & Gao, J. (2013). Uncertain term structure model of interest rate. Soft Computing, 17(4), 597–604.

    Article  Google Scholar 

  • Gao, Y. (2012). Existence and uniqueness theorem on uncertain differential equations with local Lipschitz condition. Journal of Uncertain Systems, 6(3), 223–232.

    Google Scholar 

  • Gao, R. (2016). Milne method for solving uncertain differential equations. Applied Mathematics and Computation, 274, 774–785.

    Article  MathSciNet  Google Scholar 

  • Jin, T., Sun, Y., & Zhu, Y. (2019). Extreme values for solution to uncertain fractional differential equation and application to American option pricing model. Physica A: Statistical Mechanics and its Applications, 534, 1–13.

    Article  MathSciNet  Google Scholar 

  • Jin, T., Sun, Y., & Zhu, Y. (2020). Time integral about solution of an uncertain fractional order differential equation and application to zero-coupon bond model. Applied Mathematics and Computation, 372, 124991.

    Article  MathSciNet  Google Scholar 

  • Jin, T., & Zhu, Y. (2020). First hitting time about solution for an uncertain fractional differential equation and application to an uncertain risk index model. Chaos, Solitons and Fractals, 137, 109836.

    Article  MathSciNet  Google Scholar 

  • Kahneman, D., & Tversky, A. (1979). Prospect theory: an analysis of decision making under risk. Econometrica, 47, 263–291.

    Article  MathSciNet  Google Scholar 

  • Liu, B. (2007). Uncertainty theory (2nd ed.). Berlin: Springer.

    MATH  Google Scholar 

  • Liu, B. (2008). Fuzzy process, hybrid process and uncertain process. Journal of Uncertain Systems, 2(1), 3–16.

    Google Scholar 

  • Liu, B. (2009). Some research problems in uncertainty theory. Journal of Uncertain Systems, 3(1), 3–10.

    Google Scholar 

  • Liu, B. (2013). Toward uncertain finance theory. Journal of Uncertainty Analysis and Applications, 1(1), 1–15.

    Article  Google Scholar 

  • Liu, Y., Chen, X., & Ralescu, D. (2015). Uncertain currency model and currency option pricing. International Journal of Intelligent Systems, 30(1), 40–51.

    Article  Google Scholar 

  • Lu, Z., & Zhu, Y. (2019). Numerical approach for solution to an uncertain fractional differential equation. Applied Mathematics and Computation, 343, 137–148.

    Article  MathSciNet  Google Scholar 

  • Lu, Z., Yan, H., & Zhu, Y. (2019). European option pricing model based on uncertain fractional differential equation. Fuzzy Optimization and Decision Making, 18, 199–217.

    Article  MathSciNet  Google Scholar 

  • Lu, Z., Zhu, Y., & Li, B. (2019). Critical value-based Asian option pricing model for uncertain financial markets. Physica A: Statistical Mechanics and its Applications, 525, 694–703.

    Article  MathSciNet  Google Scholar 

  • Peng, J., & Yao, K. (2011). A new option pricing model for stocks in uncertainty markets. International Journal of Operations Research, 8(2), 18–26.

    MathSciNet  MATH  Google Scholar 

  • Sun, J., & Chen, X. (2015). Asian option pricing formula for uncertain financial market. Journal of Uncertainty Analysis and Applications, 3(11), 1–11.

    Google Scholar 

  • Yao, K., & Chen, X. (2013). A numerical method for solving uncertain differential equations. Journal of Intelligent and Fuzzy Systems, 25(3), 825–832.

    Article  MathSciNet  Google Scholar 

  • Yao, K. (2013). Extreme values and integral of solution of uncertain differential equation. Journal of Uncertainty Analysis and Applications, 1(2), 1–21.

    MathSciNet  Google Scholar 

  • Yang, X., & Gao, J. (2016). Linear-quadratic uncertain differential games with application to resource extraction problem. IEEE Transactions on Fuzzy Systems, 24(6), 1578–1585.

    Article  MathSciNet  Google Scholar 

  • Zhang, Y., Gao, J., & Huang, Z. (2017). Hamming method for solving uncertain differential equations. Applied Mathematics and Computation, 313, 331–341.

    Article  MathSciNet  Google Scholar 

  • Zhu, Y. (2010). Uncertain optimal control with application to a portfolio selection model. Cybernetics and Systems, 41(7), 535–547.

    Article  Google Scholar 

  • Zhu, Y. (2015). Uncertain fractional differential equations and an interest rate model. Mathematical Methods in the Applied Sciences, 38(15), 3359–3368.

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

This work was supported by the National Natural Science Foundation of China (Grant No. 71771147) and the Shanghai Jiao Tong University.

Author information

Authors and Affiliations

  1. Antai College of Economics and Management, Shanghai Jiao Tong University, Shanghai, 200000, China

    Weiwei Wang

  2. Department of Mathematical Sciences, University of Cincinnati, Cincinnati, OH, 45221-0025, USA

    Dan A. Ralescu

Authors
  1. Weiwei Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Dan A. Ralescu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Weiwei Wang.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Wang, W., Ralescu, D.A. Option pricing formulas based on uncertain fractional differential equation. Fuzzy Optim Decis Making 20, 471–495 (2021). https://doi.org/10.1007/s10700-021-09354-z

Download citation

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10700-021-09354-z

Keywords

Search

Navigation