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Why the Rotation Count Algorithm works

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  • Roger Lord

    ()
    (Erasmus Universiteit Rotterdam, and Rabobank International)

  • Christian Kahl

    ()
    (University of Wuppertal, and ABN AMRO, London)

Abstract

The characteristic functions of many affine jump-diffusion models, such as Heston’s stochastic volatility model and all of its extensions, involve multivalued functions such as the complex logarithm. If we restrict the logarithm to its principal branch, as is done in most software packages, the characteristic function can become discontinuous, leading to completely wrong option prices if options are priced by Fourier inversion. In this paper we prove under non-restrictive conditions on the parameters that the rotation count algorithm of Kahl and Jäckel chooses the correct branch of the complex logarithm. Under the same restrictions we prove that in an alternative formulation of the characteristic function the principal branch is the correct one. Seen as this formulation is easier to implement and numerically more stable than Heston’s formulation, it should be the preferred one. The remainder of this paper shows how complex discontinuities can be avoided in the Schöbel-Zhu model and the exact simulation algorithm of the Heston model, recently proposed by Broadie and Kaya. Finally, we show that Matytsin’s SVJJ model has a closed-form characteristic function, though the complex discontinuities that arise there due to the branch switching of the exponential integral cannot be avoided under all circumstances.

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Bibliographic Info

Paper provided by Tinbergen Institute in its series Tinbergen Institute Discussion Papers with number 06-065/2.

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Date of creation: 27 Jul 2006
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Handle: RePEc:dgr:uvatin:20060065

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Web page: http://www.tinbergen.nl

Related research

Keywords: Complex logarithm; affine jump-diffusion; stochastic volatility; Heston; characteristic function; moment stability; option pricing;

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References

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  1. 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.
  2. Madan, Dilip B & Seneta, Eugene, 1990. "The Variance Gamma (V.G.) Model for Share Market Returns," The Journal of Business, University of Chicago Press, vol. 63(4), pages 511-24, October.
  3. 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.
  4. Bates, David S, 1996. "Jumps and Stochastic Volatility: Exchange Rate Processes Implicit in Deutsche Mark Options," Review of Financial Studies, Society for Financial Studies, vol. 9(1), pages 69-107.
  5. Roger Lord & Christian Kahl, 2006. "Optimal Fourier Inversion in Semi-analytical Option Pricing," Tinbergen Institute Discussion Papers 06-066/2, Tinbergen Institute, revised 05 Jun 2007.
  6. Alan L. Lewis, 2001. "A Simple Option Formula for General Jump-Diffusion and other Exponential Levy Processes," Related articles explevy, Finance Press.
  7. Gaspar, Raquel M., 2004. "General Quadratic Term Structures of Bond, Futures and Forward Prices," Working Paper Series in Economics and Finance 559, Stockholm School of Economics.
  8. Heston, Steven L, 1993. "A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options," Review of Financial Studies, Society for Financial Studies, vol. 6(2), pages 327-43.
  9. Stein, Elias M & Stein, Jeremy C, 1991. "Stock Price Distributions with Stochastic Volatility: An Analytic Approach," Review of Financial Studies, Society for Financial Studies, vol. 4(4), pages 727-52.
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Citations

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Cited by:
  1. Marjon Ruijter & Kees Oosterlee (CWI), 2012. "Two-dimensional Fourier cosine series expansion method for pricing financial options," CPB Discussion Paper 225, CPB Netherlands Bureau for Economic Policy Analysis.
  2. Giacomo Bormetti & Valentina Cazzola & Guido Montagna & Oreste Nicrosini, 2008. "Probability distribution of returns in the exponential Ornstein-Uhlenbeck model," Papers 0805.0540, arXiv.org, revised Oct 2008.
  3. Kilin, Fiodar, 2006. "Accelerating the calibration of stochastic volatility models," MPRA Paper 2975, University Library of Munich, Germany, revised 22 Apr 2007.
  4. Jean-Pierre Fouque & Matthew Lorig, 2010. "A Fast Mean-Reverting Correction to Heston's Stochastic Volatility Model," Papers 1007.4366, arXiv.org, revised Apr 2012.

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