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A Comparison of Biased Simulation Schemes for Stochastic Volatility Models


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  • Roger Lord

    (Erasmus Universiteit Rotterdam, and Rabobank)

  • Remmert Koekkoek

    (Robeco Alternative Investments)

  • Dick van Dijk

    (Faculty of Economics, Erasmus Universiteit Rotterdam)


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.

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Bibliographic Info

Paper provided by Tinbergen Institute in its series Tinbergen Institute Discussion Papers with number 06-046/4.

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Date of creation: 18 May 2006
Date of revision: 07 Jun 2007
Handle: RePEc:dgr:uvatin:20060046

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Keywords: Stochastic volatility; Heston; square root process; Euler-Maruyama; discretisation; strong convergence; weak convergence; boundary behaviour;

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Cited by:
  1. C. Kaebe & J. Maruhn & E. Sachs, 2009. "Adjoint-based Monte Carlo calibration of financial market models," Finance and Stochastics, Springer, vol. 13(3), pages 351-379, September.
  2. Eckhard Platen & Renata Rendek, 2009. "Exact Scenario Simulation for Selected Multi-dimensional Stochastic Processes," Research Paper Series 259, Quantitative Finance Research Centre, University of Technology, Sydney.
  3. Ewald, Christian-Oliver & Menkens, Olaf & Hung Marten Ting, Sai, 2013. "Asian and Australian options: A common perspective," Journal of Economic Dynamics and Control, Elsevier, vol. 37(5), pages 1001-1018.
  4. Roger Lord & Christian Kahl, 2006. "Why the Rotation Count Algorithm works," Tinbergen Institute Discussion Papers 06-065/2, Tinbergen Institute.
  5. Kilin, Fiodar, 2006. "Accelerating the calibration of stochastic volatility models," MPRA Paper 2975, University Library of Munich, Germany, revised 22 Apr 2007.
  6. Andreas Neuenkirch & Lukasz Szpruch, 2012. "First order strong approximations of scalar SDEs with values in a domain," Papers 1209.0390,
  7. Rodrigue Oeuvray & Pascal Junod, 2013. "On time scaling of semivariance in a jump-diffusion process," Papers 1311.1122,
  8. Carl Chiarella & Chih-Ying Hsiao & Thuy-Duong To, 2011. "Stochastic Correlation and Risk Premia in Term Structure Models," Research Paper Series 298, Quantitative Finance Research Centre, University of Technology, Sydney.
  9. Jessica Wachter, 2008. "Can time-varying risk of rare disasters explain aggregate stock market volatility?," 2008 Meeting Papers 944, Society for Economic Dynamics.
  10. F. Antonelli & A. Ramponi & S. Scarlatti, 2010. "Exchange option pricing under stochastic volatility: a correlation expansion," Review of Derivatives Research, Springer, vol. 13(1), pages 45-73, April.
  11. Xianming Sun & Siqing Gan, 2014. "An Efficient Semi-Analytical Simulation for the Heston Model," Computational Economics, Society for Computational Economics, vol. 43(4), pages 433-445, April.
  12. Dell'Era, Mario, 2010. "Geometrical Considerations on Heston's Market Model," MPRA Paper 21523, University Library of Munich, Germany.
  13. 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.
  14. Dell'Era, Mario, 2010. "Geometrical Approximation method and stochastic volatility market models," MPRA Paper 22568, University Library of Munich, Germany.
  15. Dell'Era, Mario, 2010. "Vanilla Option Pricing on Stochastic Volatility market models," MPRA Paper 25645, University Library of Munich, Germany.
  16. Fahim, Arash & Touzi, Nizar & Warin, Xavier, 2011. "A Probabilistic Numerical Method for Fully Nonlinear Parabolic PDEs," Economics Papers from University Paris Dauphine 123456789/5524, Paris Dauphine University.
  17. repec:hal:wpaper:hal-00409861 is not listed on IDEAS
  18. Medvedev, Alexey & Scaillet, Olivier, 2010. "Pricing American options under stochastic volatility and stochastic interest rates," Journal of Financial Economics, Elsevier, vol. 98(1), pages 145-159, October.
  19. Campillo, Fabien & Joannides, Marc & Larramendy-Valverde, Irène, 2014. "Approximation of the Fokker–Planck equation of the stochastic chemostat," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 99(C), pages 37-53.
  20. Carl Chiarella & Susanne Griebsch & Boda Kang, 2013. "Investigating Time-Efficient Methods to Price Compound Options in the Heston Model," Research Paper Series 328, Quantitative Finance Research Centre, University of Technology, Sydney.
  21. Benjamin Jourdain & Mohamed Sbai, 2013. "High order discretization schemes for stochastic volatility models," Post-Print hal-00409861, HAL.


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