IDEAS home Printed from https://ideas.repec.org/a/kap/compec/v61y2023i4d10.1007_s10614-022-10263-5.html
   My bibliography  Save this article

Derivation and Application of Some Fractional Black–Scholes Equations Driven by Fractional G-Brownian Motion

Author

Listed:
  • Changhong Guo

    (Guangdong University of Technology)

  • Shaomei Fang

    (South China Agricultural University)

  • Yong He

    (Guangdong University of Technology)

Abstract

In this paper, a new concept for some stochastic process called fractional G-Brownian motion (fGBm) is developed and applied to the financial markets. Compared to the standard Brownian motion, fractional Brownian motion and G-Brownian motion, the fGBm can consider the long-range dependence and uncertain volatility simultaneously. Thus it generalizes the concepts of the former three processes, and can be a better alternative in real applications. Driven by the fGBm, a generalized fractional Black–Scholes equation (FBSE) for some European call option and put option is derived with the help of Taylor’s series of fractional order and the theory of absence of arbitrage. Meanwhile, some explicit option pricing formulas for the derived FBSE are also obtained, which generalize the classical Black–Scholes formulas for the prices of European options given by Black and Scholes in 1973.

Suggested Citation

  • Changhong Guo & Shaomei Fang & Yong He, 2023. "Derivation and Application of Some Fractional Black–Scholes Equations Driven by Fractional G-Brownian Motion," Computational Economics, Springer;Society for Computational Economics, vol. 61(4), pages 1681-1705, April.
    • Handle: RePEc:kap:compec:v:61:y:2023:i:4:d:10.1007_s10614-022-10263-5
    DOI: 10.1007/s10614-022-10263-5
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10614-022-10263-5
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.
    File URL: https://libkey.io/10.1007/s10614-022-10263-5?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. Robert J. Elliott & John Van Der Hoek, 2003. "A General Fractional White Noise Theory And Applications To Finance," Mathematical Finance, Wiley Blackwell, vol. 13(2), pages 301-330, April.
    2. Larry G. Epstein & Shaolin Ji, 2013. "Ambiguous Volatility and Asset Pricing in Continuous Time," The Review of Financial Studies, Society for Financial Studies, vol. 26(7), pages 1740-1786.
    3. Ahmad Golbabai & Omid Nikan, 2020. "A Computational Method Based on the Moving Least-Squares Approach for Pricing Double Barrier Options in a Time-Fractional Black–Scholes Model," Computational Economics, Springer;Society for Computational Economics, vol. 55(1), pages 119-141, January.
    4. L. C. G. Rogers, 1997. "Arbitrage with Fractional Brownian Motion," Mathematical Finance, Wiley Blackwell, vol. 7(1), pages 95-105, January.
    5. Tomas Björk & Henrik Hult, 2005. "A note on Wick products and the fractional Black-Scholes model," Finance and Stochastics, Springer, vol. 9(2), pages 197-209, April.
    6. Cartea, Álvaro & del-Castillo-Negrete, Diego, 2007. "Fractional diffusion models of option prices in markets with jumps," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 374(2), pages 749-763.
    7. Haq, Sirajul & Hussain, Manzoor, 2018. "Selection of shape parameter in radial basis functions for solution of time-fractional Black–Scholes models," Applied Mathematics and Computation, Elsevier, vol. 335(C), pages 248-263.
    8. Andrew W. Lo, A. Craig MacKinlay, 1988. "Stock Market Prices do not Follow Random Walks: Evidence from a Simple Specification Test," The Review of Financial Studies, Society for Financial Studies, vol. 1(1), pages 41-66.
    9. Zengjing Chen & Larry Epstein, 2002. "Ambiguity, Risk, and Asset Returns in Continuous Time," Econometrica, Econometric Society, vol. 70(4), pages 1403-1443, July.
    10. Tommi Sottinen, 2001. "Fractional Brownian motion, random walks and binary market models," Finance and Stochastics, Springer, vol. 5(3), pages 343-355.
    11. Vorbrink, Jörg, 2014. "Financial markets with volatility uncertainty," Journal of Mathematical Economics, Elsevier, vol. 53(C), pages 64-78.
    12. Peng, Shige, 2008. "Multi-dimensional G-Brownian motion and related stochastic calculus under G-expectation," Stochastic Processes and their Applications, Elsevier, vol. 118(12), pages 2223-2253, December.
    13. M. Avellaneda & A. Levy & A. ParAS, 1995. "Pricing and hedging derivative securities in markets with uncertain volatilities," Applied Mathematical Finance, Taylor & Francis Journals, vol. 2(2), pages 73-88.
    14. 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.
    15. Jumarie, Guy, 2008. "Stock exchange fractional dynamics defined as fractional exponential growth driven by (usual) Gaussian white noise. Application to fractional Black-Scholes equations," Insurance: Mathematics and Economics, Elsevier, vol. 42(1), pages 271-287, February.
    16. Jumarie, Guy, 2006. "Fractionalization of the complex-valued Brownian motion of order n using Riemann–Liouville derivative. Applications to mathematical finance and stochastic mechanics," Chaos, Solitons & Fractals, Elsevier, vol. 28(5), pages 1285-1305.
    17. T. J. Lyons, 1995. "Uncertain volatility and the risk-free synthesis of derivatives," Applied Mathematical Finance, Taylor & Francis Journals, vol. 2(2), pages 117-133.
    Full references (including those not matched with items on IDEAS)

    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. Changhong Guo & Shaomei Fang & Yong He, 2023. "A Generalized Stochastic Process: Fractional G-Brownian Motion," Methodology and Computing in Applied Probability, Springer, vol. 25(1), pages 1-34, March.
    2. Stoyan V. Stoyanov & Yong Shin Kim & Svetlozar T. Rachev & Frank J. Fabozzi, 2017. "Option pricing for Informed Traders," Papers 1711.09445, arXiv.org.
    3. Dufera, Tamirat Temesgen, 2024. "Fractional Brownian motion in option pricing and dynamic delta hedging: Experimental simulations," The North American Journal of Economics and Finance, Elsevier, vol. 69(PB).
    4. Wei Chen, 2013. "Fractional G-White Noise Theory, Wavelet Decomposition for Fractional G-Brownian Motion, and Bid-Ask Pricing Application to Finance Under Uncertainty," Papers 1306.4070, arXiv.org.
    5. Stoyan V. Stoyanov & Svetlozar T. Rachev & Stefan Mittnik & Frank J. Fabozzi, 2019. "Pricing Derivatives In Hermite Markets," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 22(06), pages 1-27, September.
    6. Hölzermann, Julian, 2018. "Bond Pricing under Knightian Uncertainty. A Short Rate Model with Drift and Volatility Uncertainty," Center for Mathematical Economics Working Papers 582, Center for Mathematical Economics, Bielefeld University.
    7. Yuhong Xu, 2014. "Robust valuation and risk measurement under model uncertainty," Papers 1407.8024, arXiv.org.
    8. Julian Holzermann, 2019. "Term Structure Modeling under Volatility Uncertainty," Papers 1904.02930, arXiv.org, revised Sep 2021.
    9. Julian Holzermann, 2018. "The Hull-White Model under Volatility Uncertainty," Papers 1808.03463, arXiv.org, revised Jan 2021.
    10. Hölzermann, Julian & Lin, Qian, 2019. "Term Structure Modeling under Volatility Uncertainty: A Forward Rate Model driven by G-Brownian Motion," Center for Mathematical Economics Working Papers 613, Center for Mathematical Economics, Bielefeld University.
    11. Park, Kyunghyun & Wong, Hoi Ying & Yan, Tingjin, 2023. "Robust retirement and life insurance with inflation risk and model ambiguity," Insurance: Mathematics and Economics, Elsevier, vol. 110(C), pages 1-30.
    12. Yuecai Han & Chunyang Liu, 2018. "Asian Option Pricing under Uncertain Volatility Model," Papers 1808.00656, arXiv.org.
    13. Shige Peng & Shuzhen Yang, 2020. "Distributional uncertainty of the financial time series measured by G-expectation," Papers 2011.09226, arXiv.org, revised Jul 2021.
    14. Larry G. Epstein & Shaolin Ji, 2013. "Ambiguous Volatility and Asset Pricing in Continuous Time," The Review of Financial Studies, Society for Financial Studies, vol. 26(7), pages 1740-1786.
    15. Panhong Cheng & Zhihong Xu & Zexing Dai, 2023. "Valuation of vulnerable options with stochastic corporate liabilities in a mixed fractional Brownian motion environment," Mathematics and Financial Economics, Springer, volume 17, number 3, March.
    16. Vorbrink, Jörg, 2014. "Financial markets with volatility uncertainty," Journal of Mathematical Economics, Elsevier, vol. 53(C), pages 64-78.
    17. Ahmad Golbabai & Omid Nikan, 2020. "A Computational Method Based on the Moving Least-Squares Approach for Pricing Double Barrier Options in a Time-Fractional Black–Scholes Model," Computational Economics, Springer;Society for Computational Economics, vol. 55(1), pages 119-141, January.
    18. Xu, Yuhong, 2022. "Optimal growth under model uncertainty," The North American Journal of Economics and Finance, Elsevier, vol. 60(C).
    19. Wang, Xiao-Tian, 2011. "Scaling and long-range dependence in option pricing V: Multiscaling hedging and implied volatility smiles under the fractional Black–Scholes model with transaction costs," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 390(9), pages 1623-1634.
    20. Gapeev Pavel V. & Sottinen Tommi & Valkeila Esko, 2011. "Robust replication in H-self-similar Gaussian market models under uncertainty," Statistics & Risk Modeling, De Gruyter, vol. 28(1), pages 37-50, March.

    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:kap:compec:v:61:y:2023:i:4:d:10.1007_s10614-022-10263-5. 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: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    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.