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Local Variance Gamma and Explicit Calibration to Option Prices

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  • Peter Carr
  • Sergey Nadtochiy

Abstract

In some options markets (e.g. commodities), options are listed with only a single maturity for each underlying. In others, (e.g. equities, currencies), options are listed with multiple maturities. In this paper, we provide an algorithm for calibrating a pure jump Markov martingale model to match the market prices of European options of multiple strikes and maturities. This algorithm only requires solutions of several one-dimensional root-search problems, as well as application of elementary functions. We show how to construct a time-homogeneous process which meets a single smile, and a piecewise time-homogeneous process which can meet multiple smiles.

Suggested Citation

  • Peter Carr & Sergey Nadtochiy, 2013. "Local Variance Gamma and Explicit Calibration to Option Prices," Papers 1308.2326, arXiv.org, revised Jan 2014.
    • Handle: RePEc:arx:papers:1308.2326
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    References listed on IDEAS

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    1. Noble, John M., 2013. "Time homogeneous diffusions with a given marginal at a deterministic time," Stochastic Processes and their Applications, Elsevier, vol. 123(3), pages 675-718.
    2. 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.
    3. Mark Rubinstein., 1994. "Implied Binomial Trees," Research Program in Finance Working Papers RPF-232, University of California at Berkeley.
    4. Rubinstein, Mark, 1994. "Implied Binomial Trees," Journal of Finance, American Finance Association, vol. 49(3), pages 771-818, July.
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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. Rene Carmona & Yi Ma & Sergey Nadtochiy, 2015. "Simulation of Implied Volatility Surfaces via Tangent Levy Models," Papers 1504.00334, arXiv.org.
    3. A. Itkin & A. Lipton & D. Muravey, 2021. "Multilayer heat equations: application to finance," Papers 2102.08338, arXiv.org.
    4. 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.
    5. Dilip B. Madan & Wim Schoutens, 2019. "Arbitrage Free Approximations to Candidate Volatility Surface Quotations," JRFM, MDPI, vol. 12(2), pages 1-21, April.
    6. Sergey Nadtochiy & Jan Obłój, 2017. "Robust Trading Of Implied Skew," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 20(02), pages 1-41, March.
    7. Sergey Nadtochiy & Jan Obloj, 2016. "Robust Trading of Implied Skew," Papers 1611.05518, arXiv.org.
    8. Julien Guyon, 2020. "Inversion of convex ordering in the VIX market," Quantitative Finance, Taylor & Francis Journals, vol. 20(10), pages 1597-1623, October.
    9. Noble, John M., 2015. "Time homogeneous diffusion with drift and killing to meet a given marginal," Stochastic Processes and their Applications, Elsevier, vol. 125(4), pages 1500-1540.
    10. Fabien Le Floc'h, 2020. "An arbitrage-free interpolation of class $C^2$ for option prices," Papers 2004.08650, arXiv.org, revised May 2020.

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