IDEAS home Printed from https://ideas.repec.org/p/tin/wpaper/20060046.html
   My bibliography  Save this paper

A Comparison of Biased Simulation Schemes for Stochastic Volatility Models

Author

Listed:
  • Roger Lord

    (Erasmus Universiteit Rotterdam, and Rabobank)

  • Remmert Koekkoek

    (Robeco Alternative Investments)

  • Dick van Dijk

    (Faculty of Economics, Erasmus Universiteit Rotterdam)

Abstract

This discussion paper resulted in a publication in 'Quantitative Finance' , 2010, 10, 177-194. When using an Euler discretisation to simulate a mean-reverting square root process, one runs into the problem that while the process itself is guaranteed to be nonnegative, the discretisation is not. Although an exact and efficient simulation algorithm exists for this process, at present this is not the case for the Heston stochastic volatility model, where the variance is modelled as a square root process. Consequently, when using an Euler discretisation, one must carefully think about how to fix negative variances. Our contribution is threefold. Firstly, we unify all Euler fixes into a single general framework. Secondly, we introduce the new full truncation scheme, tailored to minimise the upward bias found when pricing European options. Thirdly and finally, we numerically compare all Euler fixes to a recent quasi-second order scheme of Kahl and Jäckel and the exact scheme of Broadie and Kaya. The choice of fix is found to be extremely important. The full truncation scheme by far outperforms all biased schemes in terms of bias, root-mean-squared error, and hence should be the preferred discretisation method for simulation of the Heston model and extensions thereof.

Suggested Citation

  • Roger Lord & Remmert Koekkoek & Dick van Dijk, 2006. "A Comparison of Biased Simulation Schemes for Stochastic Volatility Models," Tinbergen Institute Discussion Papers 06-046/4, Tinbergen Institute, revised 07 Jun 2007.
    • Handle: RePEc:tin:wpaper:20060046
    as

    Download full text from publisher

    File URL: https://papers.tinbergen.nl/06046.pdf
    Download Restriction: no
    ---><---

    Other versions of this item:

    References listed on IDEAS

    as
    1. John C. Cox & Jonathan E. Ingersoll Jr. & Stephen A. Ross, 2005. "A Theory Of The Term Structure Of Interest Rates," World Scientific Book Chapters, in: Sudipto Bhattacharya & George M Constantinides (ed.), Theory Of Valuation, chapter 5, pages 129-164, World Scientific Publishing Co. Pte. Ltd..
    2. Bates, David S, 1996. "Jumps and Stochastic Volatility: Exchange Rate Processes Implicit in Deutsche Mark Options," The Review of Financial Studies, Society for Financial Studies, vol. 9(1), pages 69-107.
    3. Griselda Deelstra & Freddy Delbaen, 1998. "Convergence of discretised stochastic interest rate: processes with stochastic drift term," ULB Institutional Repository 2013/7584, ULB -- Universite Libre de Bruxelles.
    4. Peter Christoffersen & Kris Jacobs, 2004. "Which GARCH Model for Option Valuation?," Management Science, INFORMS, vol. 50(9), pages 1204-1221, September.
    5. G. Deelstra & F. Delbaen, 1998. "Convergence of discretized stochastic (interest rate) processes with stochastic drift term," Applied Stochastic Models and Data Analysis, John Wiley & Sons, vol. 14(1), pages 77-84, March.
    6. Alexander Lipton, 2001. "Mathematical Methods for Foreign Exchange:A Financial Engineer's Approach," World Scientific Books, World Scientific Publishing Co. Pte. Ltd., number 4694, February.
    7. Christian Kahl & Peter Jackel, 2006. "Fast strong approximation Monte Carlo schemes for stochastic volatility models," Quantitative Finance, Taylor & Francis Journals, vol. 6(6), pages 513-536.
    8. Mark Broadie & Özgür Kaya, 2006. "Exact Simulation of Stochastic Volatility and Other Affine Jump Diffusion Processes," Operations Research, INFORMS, vol. 54(2), pages 217-231, April.
    9. Darrell Duffie & Jun Pan & Kenneth Singleton, 2000. "Transform Analysis and Asset Pricing for Affine Jump-Diffusions," Econometrica, Econometric Society, vol. 68(6), pages 1343-1376, November.
    10. Heston, Steven L, 1993. "A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options," The Review of Financial Studies, Society for Financial Studies, vol. 6(2), pages 327-343.
    11. Leif Andersen & Vladimir Piterbarg, 2007. "Moment explosions in stochastic volatility models," Finance and Stochastics, Springer, vol. 11(1), pages 29-50, January.
    12. Alfonsi Aurélien, 2005. "On the discretization schemes for the CIR (and Bessel squared) processes," Monte Carlo Methods and Applications, De Gruyter, vol. 11(4), pages 355-384, December.
    13. 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.
    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. Paul Glasserman & Kyoung-Kuk Kim, 2011. "Gamma expansion of the Heston stochastic volatility model," Finance and Stochastics, Springer, vol. 15(2), pages 267-296, June.
    2. Andrei Cozma & Matthieu Mariapragassam & Christoph Reisinger, 2015. "Convergence of an Euler scheme for a hybrid stochastic-local volatility model with stochastic rates in foreign exchange markets," Papers 1501.06084, arXiv.org, revised Oct 2016.
    3. Corsaro, Stefania & Kyriakou, Ioannis & Marazzina, Daniele & Marino, Zelda, 2019. "A general framework for pricing Asian options under stochastic volatility on parallel architectures," European Journal of Operational Research, Elsevier, vol. 272(3), pages 1082-1095.
    4. Chantal Labb'e & Bruno R'emillard & Jean-Franc{c}ois Renaud, 2010. "A simple discretization scheme for nonnegative diffusion processes, with applications to option pricing," Papers 1011.3247, arXiv.org.
    5. Nan Chen & Zhengyu Huang, 2013. "Localization and Exact Simulation of Brownian Motion-Driven Stochastic Differential Equations," Mathematics of Operations Research, INFORMS, vol. 38(3), pages 591-616, August.
    6. Biffis, Enrico, 2005. "Affine processes for dynamic mortality and actuarial valuations," Insurance: Mathematics and Economics, Elsevier, vol. 37(3), pages 443-468, December.
    7. Kozarski, R., 2013. "Pricing and hedging in the VIX derivative market," Other publications TiSEM 221fefe0-241e-4914-b6bd-c, Tilburg University, School of Economics and Management.
    8. Benjamin Jourdain & Mohamed Sbai, 2013. "High order discretization schemes for stochastic volatility models," Post-Print hal-00409861, HAL.
    9. Alexander Lipton, 2024. "Hydrodynamics of Markets:Hidden Links Between Physics and Finance," Papers 2403.09761, arXiv.org.
    10. Andrei Cozma & Christoph Reisinger, 2017. "Strong order 1/2 convergence of full truncation Euler approximations to the Cox-Ingersoll-Ross process," Papers 1704.07321, arXiv.org, revised Oct 2018.
    11. Gonçalo Faria & João Correia-da-Silva, 2014. "A closed-form solution for options with ambiguity about stochastic volatility," Review of Derivatives Research, Springer, vol. 17(2), pages 125-159, July.
    12. Andrei Cozma & Christoph Reisinger, 2015. "Exponential integrability properties of Euler discretization schemes for the Cox-Ingersoll-Ross process," Papers 1601.00919, arXiv.org.
    13. Alexander Lipton & Andrey Gal & Andris Lasis, 2014. "Pricing of vanilla and first-generation exotic options in the local stochastic volatility framework: survey and new results," Quantitative Finance, Taylor & Francis Journals, vol. 14(11), pages 1899-1922, November.
    14. Xianming Sun & Siqing Gan, 2014. "An Efficient Semi-Analytical Simulation for the Heston Model," Computational Economics, Springer;Society for Computational Economics, vol. 43(4), pages 433-445, April.
    15. Gurdip Bakshi & Charles Cao & Zhaodong (Ken) Zhong, 2021. "Assessing models of individual equity option prices," Review of Quantitative Finance and Accounting, Springer, vol. 57(1), pages 1-28, July.
    16. Nicola Bruti-Liberati, 2007. "Numerical Solution of Stochastic Differential Equations with Jumps in Finance," PhD Thesis, Finance Discipline Group, UTS Business School, University of Technology, Sydney, number 1, July-Dece.
    17. Maya Briani & Lucia Caramellino & Giulia Terenzi & Antonino Zanette, 2019. "Numerical Stability Of A Hybrid Method For Pricing Options," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 22(07), pages 1-46, November.
    18. Mascagni Michael & Hin Lin-Yee, 2013. "Parallel pseudo-random number generators: A derivative pricing perspective with the Heston stochastic volatility model," Monte Carlo Methods and Applications, De Gruyter, vol. 19(2), pages 77-105, July.
    19. Nicola Bruti-Liberati, 2007. "Numerical Solution of Stochastic Differential Equations with Jumps in Finance," PhD Thesis, Finance Discipline Group, UTS Business School, University of Technology, Sydney, number 1-2007, January-A.
    20. Papantonis, Ioannis, 2016. "Volatility risk premium implications of GARCH option pricing models," Economic Modelling, Elsevier, vol. 58(C), pages 104-115.

    More about this item

    Keywords

    Stochastic volatility; Heston; square root process; Euler-Maruyama; discretisation; strong convergence; weak convergence; boundary behaviour;
    All these keywords.

    JEL classification:

    • C63 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Computational Techniques
    • G13 - Financial Economics - - General Financial Markets - - - Contingent Pricing; Futures Pricing

    NEP fields

    This paper has been announced in the following NEP Reports:

    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:tin:wpaper:20060046. 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: Tinbergen Office +31 (0)10-4088900 (email available below). General contact details of provider: https://edirc.repec.org/data/tinbenl.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.