IDEAS home Printed from https://ideas.repec.org/a/wsi/ijfexx/v05y2018i02ns2424786318500093.html
   My bibliography  Save this article

Exact solutions for time-fractional Fokker–Planck–Kolmogorov equation of Geometric Brownian motion via Lie point symmetries

Author

Listed:
  • Azadeh Naderifard

    (Faculty of Mathematical Sciences, Shahrood University of Technology, Shahrood, Semnan, Iran)

  • Elham Dastranj

    (Faculty of Mathematical Sciences, Shahrood University of Technology, Shahrood, Semnan, Iran)

  • S. Reza Hejazi

    (Faculty of Mathematical Sciences, Shahrood University of Technology, Shahrood, Semnan, Iran)

Abstract

In this paper, the transition joint probability density function of the solution of geometric Brownian motion (GBM) equation is obtained via Lie group theory of differential equations (DEs). Lie symmetry analysis is applied to find new solutions for time-fractional Fokker–Planck–Kolmogorov equation of GBM. This analysis classifies the forms of the solutions for the equation by the similarity variables arising from the symmetry operators. Finally, an analytic method called invariant subspace method is applied in order to find another exact solution.

Suggested Citation

  • Azadeh Naderifard & Elham Dastranj & S. Reza Hejazi, 2018. "Exact solutions for time-fractional Fokker–Planck–Kolmogorov equation of Geometric Brownian motion via Lie point symmetries," International Journal of Financial Engineering (IJFE), World Scientific Publishing Co. Pte. Ltd., vol. 5(02), pages 1-15, June.
    • Handle: RePEc:wsi:ijfexx:v:05:y:2018:i:02:n:s2424786318500093
    DOI: 10.1142/S2424786318500093
    as

    Download full text from publisher

    File URL: http://www.worldscientific.com/doi/abs/10.1142/S2424786318500093
    Download Restriction: Access to full text is restricted to subscribers
    File URL: https://libkey.io/10.1142/S2424786318500093?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. Bjork, Tomas, 2009. "Arbitrage Theory in Continuous Time," OUP Catalogue, Oxford University Press, edition 3, number 9780199574742.
    2. Huang, Qing & Zhdanov, Renat, 2014. "Symmetries and exact solutions of the time fractional Harry-Dym equation with Riemann–Liouville derivative," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 409(C), pages 110-118.
    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. Habibi, Noora & Lashkarian, Elham & Dastranj, Elham & Hejazi, S. Reza, 2019. "Lie symmetry analysis, conservation laws and numerical approximations of time-fractional Fokker–Planck equations for special stochastic process in foreign exchange markets," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 513(C), pages 750-766.
    2. Dastranj, Elham & Sahebi Fard, Hossein & Abdolbaghi, Abdolmajid & Reza Hejazi, S., 2020. "Power option pricing under the unstable conditions (Evidence of power option pricing under fractional Heston model in the Iran gold market)," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 537(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. Will Hicks, 2020. "Pseudo-Hermiticity, Martingale Processes and Non-Arbitrage Pricing," Papers 2009.00360, arXiv.org, revised Apr 2021.
    2. Timothy DeLise, 2021. "Neural Options Pricing," Papers 2105.13320, arXiv.org.
    3. Wang, Jingxing & Chung, Seokhyun & AlShelahi, Abdullah & Kontar, Raed & Byon, Eunshin & Saigal, Romesh, 2021. "Look-ahead decision making for renewable energy: A dynamic “predict and store” approach," Applied Energy, Elsevier, vol. 296(C).
    4. Alexander M. G. Cox & Christoph Hoeggerl, 2013. "Model-independent no-arbitrage conditions on American put options," Papers 1301.5467, arXiv.org.
    5. Battulga Gankhuu, 2021. "Equity-Linked Life Insurances on Maximum of Several Assets," Papers 2111.04038, arXiv.org, revised Sep 2024.
    6. Jihun Han & Hyungbin Park, 2014. "The Intrinsic Bounds on the Risk Premium of Markovian Pricing Kernels," Papers 1411.4606, arXiv.org, revised Sep 2015.
    7. Samim Ghamami & Lisa R. Goldberg, 2014. "Stochastic Intensity Models of Wrong Way Risk: Wrong Way CVA Need Not Exceed Independent CVA," Finance and Economics Discussion Series 2014-54, Board of Governors of the Federal Reserve System (U.S.).
    8. Gauthier Tshiswaka-Kashalala & Steven F Koch, 2018. "The Demand for Reproductive Health Care," Journal of African Economies, Centre for the Study of African Economies, vol. 27(4), pages 405-429.
    9. Battulga Gankhuu, 2022. "Augmented Dynamic Gordon Growth Model," Papers 2201.06012, arXiv.org, revised Sep 2024.
    10. Shah, Anand, 2016. "Pricing and Risk Mitigation Analysis of a Cyber Liability Insurance using Gaussian, t and Gumbel Copulas – A case for Cyber Risk Index," MPRA Paper 111968, University Library of Munich, Germany.
    11. Mordecki, Ernesto & Rodríguez, Andrés Sosa, 2021. "Country risk for emerging economies: a dynamical index proposal with a case study," Brazilian Review of Econometrics, Sociedade Brasileira de Econometria - SBE, vol. 40(2), April.
    12. Mahringer, Steffen & Fuess, Roland & Prokopczuk, Marcel, 2015. "Electricity Market Coupling and the Pricing of Transmission Rights: An Option-based Approach," Working Papers on Finance 1512, University of St. Gallen, School of Finance.
    13. Fabio Antonelli & Alessandro Ramponi & Sergio Scarlatti, 2013. "Option-based risk management of a bond portfolio under regime switching interest rates," Decisions in Economics and Finance, Springer;Associazione per la Matematica, vol. 36(1), pages 47-70, May.
    14. HABIBI, Reza, 2021. "Role Of Early Warning Systems In Predicting The Stock Price Crisis: What We Learnt From Grasshopper And Ants Fable," Studii Financiare (Financial Studies), Centre of Financial and Monetary Research "Victor Slavescu", vol. 25(2), pages 6-20, June.
    15. Zhang, Zhi-Yong & Li, Guo-Fang, 2020. "Lie symmetry analysis and exact solutions of the time-fractional biological population model," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 540(C).
    16. Mondher Bellalah & Detao Zhang & Panpan Zhang, 2020. "Optimal Portfolio Choice Under Shadow Costs with Fixed Assets when Time-Horizon Is Uncertain," Computational Economics, Springer;Society for Computational Economics, vol. 56(1), pages 5-20, June.
    17. Raquel M. Gaspar, 2016. "On Path–dependency of Constant Proportion Portfolio Insurance strategies," EcoMod2016 9381, EcoMod.
    18. Hashemi, M.S., 2015. "Group analysis and exact solutions of the time fractional Fokker–Planck equation," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 417(C), pages 141-149.
    19. Hashemi, M.S., 2018. "Invariant subspaces admitted by fractional differential equations with conformable derivatives," Chaos, Solitons & Fractals, Elsevier, vol. 107(C), pages 161-169.
    20. Leccadito, Arturo & Paletta, Tommaso & Tunaru, Radu, 2016. "Pricing and hedging basket options with exact moment matching," Insurance: Mathematics and Economics, Elsevier, vol. 69(C), pages 59-69.

    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:wsi:ijfexx:v:05:y:2018:i:02:n:s2424786318500093. 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: Tai Tone Lim (email available below). General contact details of provider: http://www.worldscientific.com/worldscinet/ijfe .

    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.