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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

    (Department of Mathematics, Imperial College London, South Kensington Campus, London SW7 2AZ, UK)

  • JOHANNES STOLTE

    (Department of Mathematics, Imperial College London, South Kensington Campus, London SW7 2AZ, UK)

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," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 15(04), pages 1-34.
    • Handle: RePEc:wsi:ijtafx:v:15:y:2012:i:04:n:s0219024912500318
    DOI: 10.1142/S0219024912500318
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    References listed on IDEAS

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    1. 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.
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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. Maximilian Gaß & Kathrin Glau & Mirco Mahlstedt & Maximilian Mair, 2018. "Chebyshev interpolation for parametric option pricing," Finance and Stochastics, Springer, vol. 22(3), pages 701-731, July.
    3. 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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