IDEAS home Printed from https://ideas.repec.org/a/wsi/ijtafx/v10y2007i02ns0219024907004202.html
   My bibliography  Save this article

Componentwise Splitting Methods For Pricing American Options Under Stochastic Volatility

Author

Listed:
  • SAMULI IKONEN

    (Nordea Markets, FIN-00020 Nordea, Finland)

  • JARI TOIVANEN

    (Department of Mathematical Information Technology, University of Jyväskylä, Agora FIN-40014, Finland)

Abstract

Efficient numerical methods for pricing American options using Heston's stochastic volatility model are proposed. Based on this model the price of a European option can be obtained by solving a two-dimensional parabolic partial differential equation. For an American option the early exercise possibility leads to a lower bound for the price of the option. This price can be computed by solving a linear complementarity problem. The idea of operator splitting methods is to divide each time step into fractional time steps with simpler operators. This paper proposes componentwise splitting methods for solving the linear complementarity problem. The basic componentwise splitting decomposes the discretized problem into three linear complementarity problems with tridiagonal matrices. These problems can be efficiently solved using the Brennan and Schwartz algorithm, which was originally introduced for American options under the Black and Scholes model. The accuracy of the componentwise splitting method is increased by applying the Strang symmetrization. The good accuracy and the computational efficiency of the proposed symmetrized splitting method are demonstrated by numerical experiments.

Suggested Citation

  • Samuli Ikonen & Jari Toivanen, 2007. "Componentwise Splitting Methods For Pricing American Options Under Stochastic Volatility," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 10(02), pages 331-361.
  • Handle: RePEc:wsi:ijtafx:v:10:y:2007:i:02:n:s0219024907004202
    DOI: 10.1142/S0219024907004202
    as

    Download full text from publisher

    File URL: http://www.worldscientific.com/doi/abs/10.1142/S0219024907004202
    Download Restriction: Access to full text is restricted to subscribers

    File URL: https://libkey.io/10.1142/S0219024907004202?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. Wilmott,Paul & Howison,Sam & Dewynne,Jeff, 1995. "The Mathematics of Financial Derivatives," Cambridge Books, Cambridge University Press, number 9780521497893.
    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. Carl Chiarella & Boda Kang & Gunter H. Meyer & Andrew Ziogas, 2009. "The Evaluation Of American Option Prices Under Stochastic Volatility And Jump-Diffusion Dynamics Using The Method Of Lines," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 12(03), pages 393-425.
    2. Oleksandr Zhylyevskyy, 2010. "A fast Fourier transform technique for pricing American options under stochastic volatility," Review of Derivatives Research, Springer, vol. 13(1), pages 1-24, April.
    3. Maryam Safaei & Abodolsadeh Neisy & Nader Nematollahi, 2018. "New Splitting Scheme for Pricing American Options Under the Heston Model," Computational Economics, Springer;Society for Computational Economics, vol. 52(2), pages 405-420, August.
    4. Andrey Itkin, 2015. "HIGH ORDER SPLITTING METHODS FOR FORWARD PDEs AND PIDEs," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 18(05), pages 1-24.
    5. Andrey Itkin, 2015. "LSV models with stochastic interest rates and correlated jumps," Papers 1511.01460, arXiv.org, revised Nov 2016.
    6. Chan, Tat Lung (Ron), 2019. "Efficient computation of european option prices and their sensitivities with the complex fourier series method," The North American Journal of Economics and Finance, Elsevier, vol. 50(C).
    7. Karel in 't Hout & Jari Toivanen, 2015. "Application of Operator Splitting Methods in Finance," Papers 1504.01022, arXiv.org.
    8. Andrey Itkin & Dmitry Muravey, 2023. "American options in time-dependent one-factor models: Semi-analytic pricing, numerical methods and ML support," Papers 2307.13870, arXiv.org.
    9. Tinne Haentjens & Karel in 't Hout, 2013. "ADI schemes for pricing American options under the Heston model," Papers 1309.0110, arXiv.org.

    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. Yonggu Kim & Keeyoung Shin & Joseph Ahn & Eul-Bum Lee, 2017. "Probabilistic Cash Flow-Based Optimal Investment Timing Using Two-Color Rainbow Options Valuation for Economic Sustainability Appraisement," Sustainability, MDPI, vol. 9(10), pages 1-16, October.
    2. Weaver, Robert D. & Moon, Yongma, 2010. "Private Labels: A Mechanism For Fulfilling Consumer Demand For Healthy Food?," 115th Joint EAAE/AAEA Seminar, September 15-17, 2010, Freising-Weihenstephan, Germany 116397, European Association of Agricultural Economists.
    3. Peter Buchen & Otto Konstandatos, 2005. "A New Method Of Pricing Lookback Options," Mathematical Finance, Wiley Blackwell, vol. 15(2), pages 245-259, April.
    4. George Chang, 2018. "Examining the Efficiency of American Put Option Pricing by Monte Carlo Methods with Variance Reduction," International Journal of Economics and Finance, Canadian Center of Science and Education, vol. 10(2), pages 10-13, February.
    5. Jiao Li, 2016. "Trading VIX Futures under Mean Reversion with Regime Switching," Papers 1605.07945, arXiv.org, revised Jun 2016.
    6. Moon, Yongma & Baran, Mesut, 2018. "Economic analysis of a residential PV system from the timing perspective: A real option model," Renewable Energy, Elsevier, vol. 125(C), pages 783-795.
    7. Zaheer Imdad & Tusheng Zhang, 2014. "Pricing European options in a delay model with jumps," Journal of Financial Engineering (JFE), World Scientific Publishing Co. Pte. Ltd., vol. 1(04), pages 1-13.
    8. San-Lin Chung & Mark Shackleton, 2005. "On the use and improvement of Hull and White's control variate technique," Applied Financial Economics, Taylor & Francis Journals, vol. 15(16), pages 1171-1179.
    9. Jiao Li, 2016. "Trading VIX futures under mean reversion with regime switching," International Journal of Financial Engineering (IJFE), World Scientific Publishing Co. Pte. Ltd., vol. 3(03), pages 1-20, September.
    10. Tarn Driffield & Peter C. Smith, 2007. "A Real Options Approach to Watchful Waiting: Theory and an Illustration," Medical Decision Making, , vol. 27(2), pages 178-188, March.
    11. Xueping Wu & Jin Zhang, 1999. "Options on the minimum or the maximum of two average prices," Review of Derivatives Research, Springer, vol. 3(2), pages 183-204, May.
    12. Peter Buchen & Otto Konstandatos, 2009. "A New Approach to Pricing Double-Barrier Options with Arbitrary Payoffs and Exponential Boundaries," Applied Mathematical Finance, Taylor & Francis Journals, vol. 16(6), pages 497-515.
    13. Kenji Hamatani & Masao Fukushima, 2011. "Pricing American options with uncertain volatility through stochastic linear complementarity models," Computational Optimization and Applications, Springer, vol. 50(2), pages 263-286, October.
    14. Moriconi, L., 2007. "Delta hedged option valuation with underlying non-Gaussian returns," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 380(C), pages 343-350.
    15. Erhan Bayraktar & Hao Xing, 2009. "Pricing American options for jump diffusions by iterating optimal stopping problems for diffusions," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 70(3), pages 505-525, December.
    16. Tim Leung & Michael Ludkovski, 2010. "Optimal Timing to Purchase Options," Papers 1008.3650, arXiv.org, revised Apr 2011.
    17. Tobias Lipp & Grégoire Loeper & Olivier Pironneau, 2013. "Mixing Monte-Carlo and Partial Differential Equations for Pricing Options," Post-Print hal-01558826, HAL.
    18. Masatoshi Miyake & Hiroshi Inoue & Satoru Takahashi, 2011. "Option Pricing For Weighted Average Of Asset Prices," Asia-Pacific Journal of Operational Research (APJOR), World Scientific Publishing Co. Pte. Ltd., vol. 28(05), pages 651-672.
    19. Conrad, Jon M., 1997. "On the option value of old-growth forest," Ecological Economics, Elsevier, vol. 22(2), pages 97-102, August.
    20. Park, Hojeong & Lim, Jaekyu, 2009. "Valuation of marginal CO2 abatement options for electric power plants in Korea," Energy Policy, Elsevier, vol. 37(5), pages 1834-1841, May.

    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:ijtafx:v:10:y:2007:i:02:n:s0219024907004202. 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.worldscinet.com/ijtaf/ijtaf.shtml .

    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.