IDEAS home Printed from https://ideas.repec.org/a/taf/quantf/v5y2005i2p181-198.html
   My bibliography  Save this article

On accurate and provably efficient GARCH option pricing algorithms

Author

Listed:
  • Yuh-Dauh Lyuu
  • Chi-Ning Wu

Abstract

The GARCH model has been very successful in capturing the serial correlation of asset return volatilities. As a result, applying the model to options pricing attracts a lot of attention. However, previous tree-based GARCH option pricing algorithms suffer from exponential running time, a cut-off maturity, inaccuracy, or some combination thereof. Specifically, this paper proves that the popular trinomial-tree option pricing algorithms of Ritchken and Trevor (Ritchken, P. and Trevor, R., Pricing options under generalized GARCH and stochastic volatility processes. J. Finance, 1999, 54(1), 377-402.) and Cakici and Topyan (Cakici, N. and Topyan, K., The GARCH option pricing model: a lattice approach. J. Comput. Finance, 2000, 3(4), 71-85.) explode exponentially when the number of partitions per day, n, exceeds a threshold determined by the GARCH parameters. Furthermore, when explosion happens, the tree cannot grow beyond a certain maturity date, making it unable to price derivatives with a longer maturity. As a result, the algorithms must be limited to using small n, which may have accuracy problems. The paper presents an alternative trinomial-tree GARCH option pricing algorithm. This algorithm provably does not have the short-maturity problem. Furthermore, the tree-size growth is guaranteed to be quadratic if n is less than a threshold easily determined by the model parameters. This level of efficiency makes the proposed algorithm practical. The surprising finding for the first time places a tree-based GARCH option pricing algorithm in the same complexity class as binomial trees under the Black-Scholes model. Extensive numerical evaluation is conducted to confirm the analytical results and the numerical accuracy of the proposed algorithm. Of independent interest is a simple and efficient technique to calculate the transition probabilities of a multinomial tree using generating functions.

Suggested Citation

  • 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.
  • Handle: RePEc:taf:quantf:v:5:y:2005:i:2:p:181-198
    DOI: 10.1080/14697680500040157
    as

    Download full text from publisher

    File URL: http://www.tandfonline.com/doi/abs/10.1080/14697680500040157
    Download Restriction: Access to full text is restricted to subscribers.

    File URL: https://libkey.io/10.1080/14697680500040157?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. Duan, Jin-Chuan, 1997. "Augmented GARCH (p,q) process and its diffusion limit," Journal of Econometrics, Elsevier, vol. 79(1), pages 97-127, July.
    2. 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.
    3. 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.
    4. R. F. Engle & A. J. Patton, 2001. "What good is a volatility model?," Quantitative Finance, Taylor & Francis Journals, vol. 1(2), pages 237-245.
    5. Bollerslev, Tim, 1986. "Generalized autoregressive conditional heteroskedasticity," Journal of Econometrics, Elsevier, vol. 31(3), pages 307-327, April.
    6. Nelson, Daniel B & Ramaswamy, Krishna, 1990. "Simple Binomial Processes as Diffusion Approximations in Financial Models," The Review of Financial Studies, Society for Financial Studies, vol. 3(3), pages 393-430.
    7. 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.
    8. Turan G. Bali, 1999. "An empirical comparison of continuous time models of the short term interest rate," Journal of Futures Markets, John Wiley & Sons, Ltd., vol. 19(7), pages 777-797, October.
    9. Hull, John & White, Alan, 1993. "One-Factor Interest-Rate Models and the Valuation of Interest-Rate Derivative Securities," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 28(2), pages 235-254, June.
    10. 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.
    11. Peter Ritchken & L. Sankarasubramanian & Anand M. Vijh, 1993. "The Valuation of Path Dependent Contracts on the Average," Management Science, INFORMS, vol. 39(10), pages 1202-1213, October.
    12. Li, Anlong & Ritchken, Peter & Sankarasubramanian, L, 1995. "Lattice Models for Pricing American Interest Rate Claims," Journal of Finance, American Finance Association, vol. 50(2), pages 719-737, June.
    13. Jin‐Chuan Duan & Geneviève Gauthier & Caroline Sasseville & Jean‐Guy Simonato, 2003. "Approximating American option prices in the GARCH framework," Journal of Futures Markets, John Wiley & Sons, Ltd., vol. 23(10), pages 915-929, October.
    14. Cox, John C. & Ross, Stephen A. & Rubinstein, Mark, 1979. "Option pricing: A simplified approach," Journal of Financial Economics, Elsevier, vol. 7(3), pages 229-263, September.
    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 de Frutos & Victor Gaton, 2016. "A spectral method for an Optimal Investment problem with Transaction Costs under Potential Utility," Papers 1612.09469, arXiv.org.
    2. 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.
    3. Javier de Frutos & Victor Gaton, 2017. "Chebyshev Reduced Basis Function applied to Option Valuation," Papers 1701.01429, arXiv.org, revised Jun 2017.
    4. Huang, Hung-Hsi & Lin, Shin-Hung & Wang, Chiu-Ping, 2019. "Reasonable evaluation of VIX options for the Taiwan stock index," The North American Journal of Economics and Finance, Elsevier, vol. 48(C), pages 111-130.
    5. Javier Frutos & Víctor Gatón, 2017. "Chebyshev reduced basis function applied to option valuation," Computational Management Science, Springer, vol. 14(4), pages 465-491, October.
    6. 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.
    7. Yuh‐Dauh Lyuu & Yu‐Quan Zhang, 2023. "Pricing multiasset time‐varying double‐barrier options with time‐dependent parameters," Journal of Futures Markets, John Wiley & Sons, Ltd., vol. 43(3), pages 404-434, March.

    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. 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.
    2. 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.
    3. Papantonis, Ioannis, 2016. "Volatility risk premium implications of GARCH option pricing models," Economic Modelling, Elsevier, vol. 58(C), pages 104-115.
    4. 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.
    5. 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.
    6. 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.
    7. Jean-Guy Simonato, 2011. "Johnson binomial trees," Quantitative Finance, Taylor & Francis Journals, vol. 11(8), pages 1165-1176.
    8. 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.
    9. 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.
    10. 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.
    11. 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.
    12. Tim Bollerslev, 2008. "Glossary to ARCH (GARCH)," CREATES Research Papers 2008-49, Department of Economics and Business Economics, Aarhus University.
    13. 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.
    14. Katarzyna Toporek, 2012. "Simple is better. Empirical comparison of American option valuation methods," Ekonomia journal, Faculty of Economic Sciences, University of Warsaw, vol. 29.
    15. Peter Christoffersen & Kris Jacobs, 2002. "Which Volatility Model for Option Valuation?," CIRANO Working Papers 2002s-33, CIRANO.
    16. Jin-Chuan Duan & Kris Jacobs, 2001. "Short and Long Memory in Equilibrium Interest Rate Dynamics," CIRANO Working Papers 2001s-22, CIRANO.
    17. repec:dau:papers:123456789/2138 is not listed on IDEAS
    18. Peter Christoffersen & Redouane Elkamhi & Bruno Feunou & Kris Jacobs, 2010. "Option Valuation with Conditional Heteroskedasticity and Nonnormality," The Review of Financial Studies, Society for Financial Studies, vol. 23(5), pages 2139-2183.
    19. Asger Lunde & Peter R. Hansen, 2005. "A forecast comparison of volatility models: does anything beat a GARCH(1,1)?," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 20(7), pages 873-889.
    20. Alex Backwell & Thomas A. McWalter & Peter H. Ritchken, 2022. "On buybacks, dilutions, dividends, and the pricing of stock‐based claims," Mathematical Finance, Wiley Blackwell, vol. 32(1), pages 273-308, January.
    21. Gondzio, Jacek & Kouwenberg, Roy & Vorst, Ton, 2003. "Hedging options under transaction costs and stochastic volatility," Journal of Economic Dynamics and Control, Elsevier, vol. 27(6), pages 1045-1068, April.

    More about this item

    Statistics

    Access and download statistics

    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:taf:quantf:v:5:y:2005:i:2:p:181-198. 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 Longhurst (email available below). General contact details of provider: http://www.tandfonline.com/RQUF20 .

    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.