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Fast computation of vanilla prices in time-changed models and implied volatilities using rational approximations

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  • Martijn Pistorius
  • Johannes Stolte

Abstract

We present a new numerical method to price vanilla options quickly in time-changed Brownian motion models. The method is based on rational function approximations of the Black-Scholes formula. Detailed numerical results are given for a number of widely used models. In particular, we use the variance-gamma model, the CGMY model and the Heston model without correlation to illustrate our results. Comparison to the standard fast Fourier transform method with respect to accuracy and speed appears to favour the newly developed method in the cases considered. We present error estimates for the option prices. Additionally, we use this method to derive a procedure to compute, for a given set of arbitrage-free European call option prices, the corresponding Black-Scholes implied volatility surface. To achieve this, rational function approximations of the inverse of the Black-Scholes formula are used. We are thus able to work out implied volatilities more efficiently than one can by the use of other common methods. Error estimates are presented for a wide range of parameters.

Suggested Citation

  • Martijn Pistorius & Johannes Stolte, 2012. "Fast computation of vanilla prices in time-changed models and implied volatilities using rational approximations," Papers 1203.6899, arXiv.org.
  • Handle: RePEc:arx:papers:1203.6899
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    File URL: http://arxiv.org/pdf/1203.6899
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    References listed on IDEAS

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    1. Clark, Peter K, 1973. "A Subordinated Stochastic Process Model with Finite Variance for Speculative Prices," Econometrica, Econometric Society, vol. 41(1), pages 135-155, January.
    2. Fang, Fang & Oosterlee, Kees, 2008. "A Novel Pricing Method For European Options Based On Fourier-Cosine Series Expansions," MPRA Paper 9319, University Library of Munich, Germany.
    3. Leippold, Markus & Wu, Liuren, 2002. "Asset Pricing under the Quadratic Class," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 37(02), pages 271-295, June.
    4. Li, Minqiang, 2008. "Approximate inversion of the Black-Scholes formula using rational functions," European Journal of Operational Research, Elsevier, vol. 185(2), pages 743-759, March.
    5. Corrado, Charles J. & Miller, Thomas Jr., 1996. "A note on a simple, accurate formula to compute implied standard deviations," Journal of Banking & Finance, Elsevier, vol. 20(3), pages 595-603, April.
    6. Merton, Robert C., 1976. "Option pricing when underlying stock returns are discontinuous," Journal of Financial Economics, Elsevier, vol. 3(1-2), pages 125-144.
    7. Eberlein, Ernst & Keller, Ulrich & Prause, Karsten, 1998. "New Insights into Smile, Mispricing, and Value at Risk: The Hyperbolic Model," The Journal of Business, University of Chicago Press, vol. 71(3), pages 371-405, July.
    8. repec:dau:papers:123456789/1392 is not listed on IDEAS
    9. 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 511-524, October.
    10. Helyette Geman & P. Carr & D. Madan & M. Yor, 2003. "Stochastic Volatility for Levy Processes," Post-Print halshs-00144385, HAL.
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    Cited by:

    1. Tat Lung Chan, 2017. "Singular Fourier-Pad\'e Series Expansion of European Option Prices," Papers 1706.06709, arXiv.org, revised Nov 2017.
    2. Kathrin Glau & Paul Herold & Dilip B. Madan & Christian Potz, 2017. "The Chebyshev method for the implied volatility," Papers 1710.01797, arXiv.org.

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