IDEAS home Printed from https://ideas.repec.org/a/inm/oropre/v58y2010i4-part-2p1148-1157.html
   My bibliography  Save this article

Option Pricing Under GARCH Processes Using PDE Methods

Author

Listed:
  • Michèle Breton

    (GERAD and HEC Montréal, Quebec H3T 2A7, Canada)

  • Javier de Frutos

    (GERAD and Departamento de Matemática Aplicada, Universidad de Valladolid, Valladolid, Spain)

Abstract

In this paper, we propose a partial differential equation formulation for the value of an option when the underlying asset price is described by a discrete-time GARCH process. Our numerical approach involves a spectral Fourier-Chebyshev interpolation. Numerical illustrations are provided, and the results are compared with other available valuation methods. Our numerical procedure converges exponentially fast and allows for the efficient computation of option prices, achieving a high level of precision in a few seconds of computing time.

Suggested Citation

  • Michèle Breton & Javier de Frutos, 2010. "Option Pricing Under GARCH Processes Using PDE Methods," Operations Research, INFORMS, vol. 58(4-part-2), pages 1148-1157, August.
    • Handle: RePEc:inm:oropre:v:58:y:2010:i:4-part-2:p:1148-1157
    DOI: 10.1287/opre.1100.0822
    as

    Download full text from publisher

    File URL: http://dx.doi.org/10.1287/opre.1100.0822
    Download Restriction: no
    File URL: https://libkey.io/10.1287/opre.1100.0822?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
    ---><---

    References listed on IDEAS

    as
    1. Longstaff, Francis A & Schwartz, Eduardo S, 2001. "Valuing American Options by Simulation: A Simple Least-Squares Approach," The Review of Financial Studies, Society for Financial Studies, vol. 14(1), pages 113-147.
    2. Stentoft, Lars, 2005. "Pricing American options when the underlying asset follows GARCH processes," Journal of Empirical Finance, Elsevier, vol. 12(4), pages 576-611, September.
    3. Bollerslev, Tim & Ole Mikkelsen, Hans, 1996. "Modeling and pricing long memory in stock market volatility," Journal of Econometrics, Elsevier, vol. 73(1), pages 151-184, July.
    4. Robert J. Myers & Steven D. Hanson, 1993. "Pricing Commodity Options when the Underlying Futures Price Exhibits Time-Varying Volatility," American Journal of Agricultural Economics, Agricultural and Applied Economics Association, vol. 75(1), pages 121-130.
    5. Jan Kallsen & Murad S. Taqqu, 1998. "Option Pricing in ARCH‐type Models," Mathematical Finance, Wiley Blackwell, vol. 8(1), pages 13-26, January.
    6. Jin-Chuan Duan & Jean-Guy Simonato, 1995. "Empirical Martingale Simulation for Asset Prices," CIRANO Working Papers 95s-43, CIRANO.
    7. Hatem Ben-Ameur & Michèle Breton & Juan-Manuel Martinez, 2009. "Dynamic Programming Approach for Valuing Options in the GARCH Model," Management Science, INFORMS, vol. 55(2), pages 252-266, February.
    8. Jin-Chuan Duan & Technology & Jean-Guy Simonato, "undated". "American GARCH Option Pricing by a Markov Chain Approximation," Computing in Economics and Finance 1997 131, Society for Computational Economics.
    9. Bollerslev, Tim & Ole Mikkelsen, Hans, 1999. "Long-term equity anticipation securities and stock market volatility dynamics," Journal of Econometrics, Elsevier, vol. 92(1), pages 75-99, September.
    10. Heston, Steven L & Nandi, Saikat, 2000. "A Closed-Form GARCH Option Valuation Model," The Review of Financial Studies, Society for Financial Studies, vol. 13(3), pages 585-625.
    11. Engle, Robert F & Ng, Victor K, 1993. "Measuring and Testing the Impact of News on Volatility," Journal of Finance, American Finance Association, vol. 48(5), pages 1749-1778, December.
    12. R. F. Engle & A. J. Patton, 2001. "What good is a volatility model?," Quantitative Finance, Taylor & Francis Journals, vol. 1(2), pages 237-245.
    13. Bollerslev, Tim, 1986. "Generalized autoregressive conditional heteroskedasticity," Journal of Econometrics, Elsevier, vol. 31(3), pages 307-327, April.
    14. Peter Christoffersen & Kris Jacobs, 2004. "Which GARCH Model for Option Valuation?," Management Science, INFORMS, vol. 50(9), pages 1204-1221, September.
    15. Duan, Jin-Chuan & Zhang, Hua, 2001. "Pricing Hang Seng Index options around the Asian financial crisis - A GARCH approach," Journal of Banking & Finance, Elsevier, vol. 25(11), pages 1989-2014, November.
    16. Duan, Jin-Chuan & Simonato, Jean-Guy, 2001. "American option pricing under GARCH by a Markov chain approximation," Journal of Economic Dynamics and Control, Elsevier, vol. 25(11), pages 1689-1718, November.
    17. K. Hsieh & P. Ritchken, 2005. "An empirical comparison of GARCH option pricing models," Review of Derivatives Research, Springer, vol. 8(3), pages 129-150, December.
    18. 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.
    19. Jin‐Chuan Duan, 1995. "The Garch Option Pricing Model," Mathematical Finance, Wiley Blackwell, vol. 5(1), pages 13-32, January.
    20. Yuh-Dauh Lyuu & Chi-Ning Wu, 2005. "On accurate and provably efficient GARCH option pricing algorithms," Quantitative Finance, Taylor & Francis Journals, vol. 5(2), pages 181-198.
    21. Bollerslev, Tim & Chou, Ray Y. & Kroner, Kenneth F., 1992. "ARCH modeling in finance : A review of the theory and empirical evidence," Journal of Econometrics, Elsevier, vol. 52(1-2), pages 5-59.
    22. Jin-Chuan Duan & Jean-Guy Simonato, 1998. "Empirical Martingale Simulation for Asset Prices," Management Science, INFORMS, vol. 44(9), pages 1218-1233, September.
    23. Peter Ritchken & Rob Trevor, 1999. "Pricing Options under Generalized GARCH and Stochastic Volatility Processes," Journal of Finance, American Finance Association, vol. 54(1), pages 377-402, February.
    24. Longstaff, Francis A & Schwartz, Eduardo S, 2001. "Valuing American Options by Simulation: A Simple Least-Squares Approach," University of California at Los Angeles, Anderson Graduate School of Management qt43n1k4jb, Anderson Graduate School of Management, UCLA.
    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. Javier Frutos & Víctor Gatón, 2017. "Chebyshev reduced basis function applied to option valuation," Computational Management Science, Springer, vol. 14(4), pages 465-491, October.
    2. Zhang, Yuanyuan & Zhang, Qian & Wang, Zerong & Wang, Qi, 2024. "Option valuation via nonaffine dynamics with realized volatility," Journal of Empirical Finance, Elsevier, vol. 77(C).
    3. Kenneth Judd & Garrett van Ryzin, 2010. "Preface to the Special Issue on Computational Economics," Operations Research, INFORMS, vol. 58(4-part-2), pages 1035-1036, August.
    4. Javier de Frutos & Victor Gaton, 2017. "Chebyshev Reduced Basis Function applied to Option Valuation," Papers 1701.01429, arXiv.org, revised Jun 2017.
    5. Ali Hamzenejad & Saeid Jafarzadeh Ghoushchi & Vahid Baradaran & Abbas Mardani, 2020. "A Robust Algorithm for Classification and Diagnosis of Brain Disease Using Local Linear Approximation and Generalized Autoregressive Conditional Heteroscedasticity Model," Mathematics, MDPI, vol. 8(8), pages 1-19, August.
    6. Sesana, Debora & Marazzina, Daniele & Fusai, Gianluca, 2014. "Pricing exotic derivatives exploiting structure," European Journal of Operational Research, Elsevier, vol. 236(1), pages 369-381.

    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. Hatem Ben-Ameur & Michèle Breton & Juan-Manuel Martinez, 2009. "Dynamic Programming Approach for Valuing Options in the GARCH Model," Management Science, INFORMS, vol. 55(2), pages 252-266, February.
    2. Lars Stentoft, 2013. "American option pricing using simulation with an application to the GARCH model," Chapters, in: Adrian R. Bell & Chris Brooks & Marcel Prokopczuk (ed.), Handbook of Research Methods and Applications in Empirical Finance, chapter 5, pages 114-147, Edward Elgar Publishing.
    3. Stentoft, Lars, 2005. "Pricing American options when the underlying asset follows GARCH processes," Journal of Empirical Finance, Elsevier, vol. 12(4), pages 576-611, September.
    4. Lars Stentoft, 2008. "American Option Pricing Using GARCH Models and the Normal Inverse Gaussian Distribution," Journal of Financial Econometrics, Oxford University Press, vol. 6(4), pages 540-582, Fall.
    5. Stentoft, Lars, 2011. "American option pricing with discrete and continuous time models: An empirical comparison," Journal of Empirical Finance, Elsevier, vol. 18(5), pages 880-902.
    6. Len, Angel & Vaello-Sebasti, Antoni, 2009. "American GARCH employee stock option valuation," Journal of Banking & Finance, Elsevier, vol. 33(6), pages 1129-1143, June.
    7. Peter Christoffersen & Kris Jacobs & Chayawat Ornthanalai, 2012. "GARCH Option Valuation: Theory and Evidence," CREATES Research Papers 2012-50, Department of Economics and Business Economics, Aarhus University.
    8. repec:dau:papers:123456789/2138 is not listed on IDEAS
    9. Christoffersen, Peter & Jacobs, Kris & Ornthanalai, Chayawat & Wang, Yintian, 2008. "Option valuation with long-run and short-run volatility components," Journal of Financial Economics, Elsevier, vol. 90(3), pages 272-297, December.
    10. Javier Frutos & Víctor Gatón, 2017. "Chebyshev reduced basis function applied to option valuation," Computational Management Science, Springer, vol. 14(4), pages 465-491, October.
    11. Duan, Jin-Chuan & Zhang, Hua, 2001. "Pricing Hang Seng Index options around the Asian financial crisis - A GARCH approach," Journal of Banking & Finance, Elsevier, vol. 25(11), pages 1989-2014, November.
    12. Rombouts, Jeroen V.K. & Stentoft, Lars & Violante, Francesco, 2020. "Pricing individual stock options using both stock and market index information," Journal of Banking & Finance, Elsevier, vol. 111(C).
    13. Peter Christoffersen & Kris Jacobs, 2004. "Which GARCH Model for Option Valuation?," Management Science, INFORMS, vol. 50(9), pages 1204-1221, September.
    14. Peter Christoffersen & Kris Jacobs, 2002. "Which Volatility Model for Option Valuation?," CIRANO Working Papers 2002s-33, CIRANO.
    15. Torben G. Andersen & Tim Bollerslev & Peter F. Christoffersen & Francis X. Diebold, 2005. "Volatility Forecasting," PIER Working Paper Archive 05-011, Penn Institute for Economic Research, Department of Economics, University of Pennsylvania.
    16. Papantonis, Ioannis, 2016. "Volatility risk premium implications of GARCH option pricing models," Economic Modelling, Elsevier, vol. 58(C), pages 104-115.
    17. Jin-Chuan Duan & Genevieve Gauthier & Caroline Sasseville & Jean-Guy Simonato, 2002. "Seize the Moments: Approximating American Option Prices in the GARCH Framework," Finance 0206005, University Library of Munich, Germany.
    18. Andersen, Torben G. & Bollerslev, Tim & Christoffersen, Peter F. & Diebold, Francis X., 2006. "Volatility and Correlation Forecasting," Handbook of Economic Forecasting, in: G. Elliott & C. Granger & A. Timmermann (ed.), Handbook of Economic Forecasting, edition 1, volume 1, chapter 15, pages 777-878, Elsevier.
    19. Lin, Shin-Hung & Huang, Hung-Hsi & Li, Sheng-Han, 2015. "Option pricing under truncated Gram–Charlier expansion," The North American Journal of Economics and Finance, Elsevier, vol. 32(C), pages 77-97.
    20. Duan, Jin-Chuan & Simonato, Jean-Guy, 2001. "American option pricing under GARCH by a Markov chain approximation," Journal of Economic Dynamics and Control, Elsevier, vol. 25(11), pages 1689-1718, November.
    21. K. Hsieh & P. Ritchken, 2005. "An empirical comparison of GARCH option pricing models," Review of Derivatives Research, Springer, vol. 8(3), pages 129-150, December.

    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:inm:oropre:v:58:y:2010:i:4-part-2:p:1148-1157. 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: Chris Asher (email available below). General contact details of provider: https://edirc.repec.org/data/inforea.html .

    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.