IDEAS home Printed from https://ideas.repec.org/a/kap/apfinm/v21y2014i1p1-14.html
   My bibliography  Save this article

Application of Homotopy Analysis Method to Option Pricing Under Lévy Processes

Author

Listed:
  • Takayuki Sakuma
  • Yuji Yamada

Abstract

Option pricing under the Lévy process has been considered an important research direction in the field of financial engineering, where a closed-form expression for the standard European option is available due to the existence of analytically tractable characteristic function according to the Lévy–Khinchin representation. However, this approach cannot be applied to exotic derivatives (such as barrier options) directly, although a large volume of exotic derivatives are actively traded in the current options market. An alternative approach is to solve the corresponding partial integro-differential equation (PIDE) numerically, which is, in fact, time-consuming and is not computationally tractable in general. In this paper, we apply the so-called homotopy analysis method (HAM) to solve the corresponding PIDE in a semi analytic form, being obtained from the following three steps: (1) Apply the Fourier transform to convert the PIDE to an ordinal differential equitation (ODE), and construct a differential system of ODEs. (2) Solve the system of ODEs, where each differential equation is shown to have an analytical solution. (3) Express the option price using the sum of infinite series, where each term may be expressed analytically and derived by applying Steps (1) and (2) recursively. To illustrate our technique more precisely, we take the variance gamma model as an example and provide the semi-analytic form. Numerical examples demonstrate a fast convergence of our proposed method to the prices of European and down-and-out call options with a few number of terms. Note that this method is easy to implement and can be applied to other types of options under general Lévy processes. Copyright Springer Japan 2014

Suggested Citation

  • Takayuki Sakuma & Yuji Yamada, 2014. "Application of Homotopy Analysis Method to Option Pricing Under Lévy Processes," Asia-Pacific Financial Markets, Springer;Japanese Association of Financial Economics and Engineering, vol. 21(1), pages 1-14, March.
    • Handle: RePEc:kap:apfinm:v:21:y:2014:i:1:p:1-14
    DOI: 10.1007/s10690-013-9175-2
    as

    Download full text from publisher

    File URL: http://hdl.handle.net/10.1007/s10690-013-9175-2
    Download Restriction: Access to full text is restricted to subscribers.
    File URL: https://libkey.io/10.1007/s10690-013-9175-2?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. Marc Jeannin & Martijn Pistorius, 2010. "A transform approach to compute prices and Greeks of barrier options driven by a class of Levy processes," Quantitative Finance, Taylor & Francis Journals, vol. 10(6), pages 629-644.
    2. Jing Zhao & Hoi Ying Wong, 2012. "A closed-form solution to American options under general diffusion processes," Quantitative Finance, Taylor & Francis Journals, vol. 12(5), pages 725-737, July.
    3. 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.
    4. Liming Feng & Vadim Linetsky, 2008. "Pricing Discretely Monitored Barrier Options And Defaultable Bonds In Lévy Process Models: A Fast Hilbert Transform Approach," Mathematical Finance, Wiley Blackwell, vol. 18(3), pages 337-384, July.
    5. Song-Ping Zhu, 2006. "An exact and explicit solution for the valuation of American put options," Quantitative Finance, Taylor & Francis Journals, vol. 6(3), pages 229-242.
    6. Fiorani, Filo, 2004. "Option Pricing Under the Variance Gamma Process," MPRA Paper 15395, University Library of Munich, Germany.
    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. Cristina Viegas & José Azevedo-Pereira, 2020. "A Quasi-Closed-Form Solution for the Valuation of American Put Options," IJFS, MDPI, vol. 8(4), pages 1-16, October.
    2. Jun Cheng & Jin Zhang, 2012. "Analytical pricing of American options," Review of Derivatives Research, Springer, vol. 15(2), pages 157-192, July.
    3. Leunglung Chan & Song-Ping Zhu, 2014. "An exact and explicit formula for pricing lookback options with regime switching," Papers 1407.4864, arXiv.org.
    4. Leippold, Markus & Vasiljević, Nikola, 2017. "Pricing and disentanglement of American puts in the hyper-exponential jump-diffusion model," Journal of Banking & Finance, Elsevier, vol. 77(C), pages 78-94.
    5. Wenting Chen & Kai Du & Xinzi Qiu, 2017. "Analytic properties of American option prices under a modified Black-Scholes equation with spatial fractional derivatives," Papers 1701.01515, arXiv.org.
    6. Kontosakos, Vasileios E. & Mendonca, Keegan & Pantelous, Athanasios A. & Zuev, Konstantin M., 2021. "Pricing discretely-monitored double barrier options with small probabilities of execution," European Journal of Operational Research, Elsevier, vol. 290(1), pages 313-330.
    7. In oon Kim & Bong-Gyu Jang & Kyeong Tae Kim, 2013. "A simple iterative method for the valuation of American options," Quantitative Finance, Taylor & Francis Journals, vol. 13(6), pages 885-895, May.
    8. Li, Chenxu & Ye, Yongxin, 2019. "Pricing and Exercising American Options: an Asymptotic Expansion Approach," Journal of Economic Dynamics and Control, Elsevier, vol. 107(C), pages 1-1.
    9. Ricardo Crisóstomo, 2017. "Speed and biases of Fourier-based pricing choices: Analysis of the Bates and Asymmetric Variance Gamma models," CNMV Working Papers CNMV Working Papers no. 6, CNMV- Spanish Securities Markets Commission - Research and Statistics Department.
    10. Qianru Shang & Brian Byrne, 2021. "American option pricing: Optimal Lattice models and multidimensional efficiency tests," Journal of Futures Markets, John Wiley & Sons, Ltd., vol. 41(4), pages 514-535, April.
    11. Leunglung Chan & Song-Ping Zhu, 2014. "An exact and explicit formula for pricing Asian options with regime switching," Papers 1407.5091, arXiv.org.
    12. Xun Li & Ping Lin & Xue-Cheng Tai & Jinghui Zhou, 2015. "Pricing Two-asset Options under Exponential L\'evy Model Using a Finite Element Method," Papers 1511.04950, arXiv.org.
    13. Lu, Xiaoping & Yan, Dong & Zhu, Song-Ping, 2022. "Optimal exercise of American puts with transaction costs under utility maximization," Applied Mathematics and Computation, Elsevier, vol. 415(C).
    14. André Catalão & Rogério Rosenfeld, 2020. "Analytical Path-Integral Pricing Of Deterministic Moving-Barrier Options Under Non-Gaussian Distributions," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 23(01), pages 1-52, February.
    15. Song-Ping Zhu & Guiyuan Ma, 2018. "An analytical solution for the HJB equation arising from the Merton problem," International Journal of Financial Engineering (IJFE), World Scientific Publishing Co. Pte. Ltd., vol. 5(01), pages 1-26, March.
    16. Ravi Kashyap, 2016. "Options as Silver Bullets: Valuation of Term Loans, Inventory Management, Emissions Trading and Insurance Risk Mitigation using Option Theory," Papers 1609.01274, arXiv.org, revised Mar 2022.
    17. Ulze, Markus & Stadler, Johannes & Rathgeber, Andreas W., 2021. "No country for old distributions? On the comparison of implied option parameters between the Brownian motion and variance gamma process," The Quarterly Review of Economics and Finance, Elsevier, vol. 82(C), pages 163-184.
    18. Cristina Viegas & Jos� Azevedo-Pereira, 2012. "Mortgage valuation: a quasi-closed-form solution," Quantitative Finance, Taylor & Francis Journals, vol. 12(7), pages 993-1001, May.
    19. He, Yong & Zhou, Xia & Chen, Peimin & Wang, Xiaoyang, 2022. "An analytical solution for the robust investment-reinsurance strategy with general utilities," The North American Journal of Economics and Finance, Elsevier, vol. 63(C).
    20. Mojtaba Hajipour & Alaeddin Malek, 2015. "Efficient High-Order Numerical Methods for Pricing of Options," Computational Economics, Springer;Society for Computational Economics, vol. 45(1), pages 31-47, January.

    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:apfinm:v:21:y:2014:i:1:p:1-14. 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.