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To sigmoid-based functional description of the volatility smile

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  • Andrey Itkin

Abstract

We propose a new static parameterization of the implied volatility surface which is constructed by using polynomials of sigmoid functions combined with some other terms. This parameterization is flexible enough to fit market implied volatilities which demonstrate smile or skew. An arbitrage-free calibration algorithm is considered that constructs the implied volatility surface as a grid in the strike-expiration space and guarantees a lack of arbitrage at every node of this grid. We also demonstrate how to construct an arbitrage-free interpolation and extrapolation in time, as well as build a local volatility and implied pdf surfaces. Asymptotic behavior of this parameterization is discussed, as well as results on stability of the calibrated parameters are presented. Numerical examples show robustness of the proposed approach in building all these surfaces as well as demonstrate a better quality of the fit as compared with some known models.

Suggested Citation

  • Andrey Itkin, 2014. "To sigmoid-based functional description of the volatility smile," Papers 1407.0256, arXiv.org, revised Dec 2014.
    • Handle: RePEc:arx:papers:1407.0256
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    Cited by:

    1. Andrey Itkin, 2020. "Geometric Local Variance Gamma Model," World Scientific Book Chapters, in: Fitting Local Volatility Analytic and Numerical Approaches in Black-Scholes and Local Variance Gamma Models, chapter 6, pages 137-173, World Scientific Publishing Co. Pte. Ltd..
    2. Christian Bayer & Blanka Horvath & Aitor Muguruza & Benjamin Stemper & Mehdi Tomas, 2019. "On deep calibration of (rough) stochastic volatility models," Papers 1908.08806, arXiv.org.
    3. P. Carr & A. Itkin, 2021. "An Expanded Local Variance Gamma Model," Computational Economics, Springer;Society for Computational Economics, vol. 57(4), pages 949-987, April.
    4. Andrey Itkin & Alexander Lipton, 2016. "Filling the gaps smoothly," Papers 1608.05145, arXiv.org.

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    JEL classification:

    • C6 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling
    • C61 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Optimization Techniques; Programming Models; Dynamic Analysis
    • G17 - Financial Economics - - General Financial Markets - - - Financial Forecasting and Simulation

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