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Fitting Local Volatility:Analytic and Numerical Approaches in Black-Scholes and Local Variance Gamma Models

Author

Listed:
  • Andrey Itkin

    (New York University, USA)

Abstract

The concept of local volatility as well as the local volatility model are one of the classical topics of mathematical finance. Although the existing literature is wide, there still exist various problems that have not drawn sufficient attention so far, for example: a) construction of analytical solutions of the Dupire equation for an arbitrary shape of the local volatility function; b) construction of parametric or non-parametric regression of the local volatility surface suitable for fast calibration; c) no-arbitrage interpolation and extrapolation of the local and implied volatility surfaces; d) extension of the local volatility concept beyond the Black–Scholes model, etc. Also, recent progresses in deep learning and artificial neural networks as applied to financial engineering have made it reasonable to look again at various classical problems of mathematical finance including that of building a no-arbitrage local/implied volatility surface and calibrating it to the option market data.

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Suggested Citation

  • Andrey Itkin, 2020. "Fitting Local Volatility:Analytic and Numerical Approaches in Black-Scholes and Local Variance Gamma Models," World Scientific Books, World Scientific Publishing Co. Pte. Ltd., number 11623, December.
    • Handle: RePEc:wsi:wsbook:11623
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    File URL: https://www.worldscientific.com/worldscibooks/10.1142/11623
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    Citations

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    Cited by:

    1. Kim, Sangkwon & Kim, Junseok, 2021. "Robust and accurate construction of the local volatility surface using the Black–Scholes equation," Chaos, Solitons & Fractals, Elsevier, vol. 150(C).
    2. Carlo Marinelli, 2024. "On certain representations of pricing functionals," Annals of Finance, Springer, vol. 20(1), pages 91-127, March.
    3. A. Itkin & A. Lipton & D. Muravey, 2021. "Multilayer heat equations: application to finance," Papers 2102.08338, arXiv.org.
    4. Carlo Marinelli, 2021. "On certain representations of pricing functionals," Papers 2109.05564, arXiv.org.

    Book Chapters

    The following chapters of this book are listed in IDEAS

    More about this item

    Keywords

    Local Volatility; Stochastic Clock; Geometric Process; Gamma Distribution; Piecewise Linear Volatility; Variance Gamma Process; Closed Form Solution; Fast Calibration; No-Arbitrage;
    All these keywords.

    JEL classification:

    • C6 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling
    • C02 - Mathematical and Quantitative Methods - - General - - - Mathematical Economics
    • C63 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Computational Techniques

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