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Pitfalls of the Fourier Transform Method in Affine Models, and Remedies

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  • Sergei Levendorskiĭ

Abstract

We study sources of potentially serious errors of popular numerical realizations of the Fourier method in affine models and explain that, in many cases, a calibration procedure based on such a realization will be able to find a “correct parameter set” only in a rather small region of the parameter space, with a blind spot: an interval of strikes depending on the model and time to maturity, where accurate calculations are extremely time-consuming. We explain how to construct more accurate and faster pricing and calibration procedures. An important ingredient of our method is the study of the analytic continuation of the solution of the associated system of generalized Riccati equations, and contour deformation techniques. As a byproduct, we show that the straightforward application of the Runge–Kutta method may lead to sizable errors, and suggest certain remedies. In the paper, the method is applied to a wide class of stochastic volatility models with stochastic interest rate and interest rate models of An(n) class. The methodology of the paper can be applied to other models (e.g., quadratic term structure models, Wishart dynamics, 3/2-model).

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  • Sergei Levendorskiĭ, 2016. "Pitfalls of the Fourier Transform Method in Affine Models, and Remedies," Applied Mathematical Finance, Taylor & Francis Journals, vol. 23(2), pages 81-134, March.
    • Handle: RePEc:taf:apmtfi:v:23:y:2016:i:2:p:81-134
    DOI: 10.1080/1350486X.2016.1159918
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    Citations

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    Cited by:

    1. Sergei Levendorskiĭ, 2022. "Operators and Boundary Problems in Finance, Economics and Insurance: Peculiarities, Efficient Methods and Outstanding Problems," Mathematics, MDPI, vol. 10(7), pages 1-36, March.
    2. Svetlana Boyarchenko & Sergei Levendorskiä¬ & J. Lars Kyrkby & Zhenyu Cui, 2021. "Sinh-Acceleration For B-Spline Projection With Option Pricing Applications," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 24(08), pages 1-50, December.
    3. Svetlana Boyarchenko & Sergei Levendorskiĭ, 2019. "Sinh-Acceleration: Efficient Evaluation Of Probability Distributions, Option Pricing, And Monte Carlo Simulations," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 22(03), pages 1-49, May.
    4. Weinan Zhang & Pingping Zeng, 2023. "A transform-based method for pricing Asian options under general two-dimensional models," Quantitative Finance, Taylor & Francis Journals, vol. 23(11), pages 1677-1697, November.
    5. Giorgia Callegaro & Lucio Fiorin & Martino Grasselli, 2019. "Quantization meets Fourier: a new technology for pricing options," Annals of Operations Research, Springer, vol. 282(1), pages 59-86, November.
    6. Svetlana Boyarchenko & Sergei Levendorskiĭ, 2020. "Static and semistatic hedging as contrarian or conformist bets," Mathematical Finance, Wiley Blackwell, vol. 30(3), pages 921-960, July.
    7. Svetlana Boyarchenko & Sergei Levendorskiu{i}, 2022. "L\'evy models amenable to efficient calculations," Papers 2207.02359, arXiv.org.
    8. Svetlana Boyarchenko & Sergei Levendorskiu{i}, 2024. "Efficient inverse $Z$-transform and Wiener-Hopf factorization," Papers 2404.19290, arXiv.org, revised May 2024.
    9. Svetlana Boyarchenko & Sergei Levendorskiu{i}, 2019. "Gauge transformations in the dual space, and pricing and estimation in the long run in affine jump-diffusion models," Papers 1912.06948, arXiv.org, revised Dec 2019.

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