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An arbitrage-free interpolation of class $C^2$ for option prices

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  • Fabien Le Floc'h

Abstract

This paper presents simple formulae for the local variance gamma model of Carr and Nadtochiy, extended with a piecewise-linear local variance function. The new formulae allow to calibrate the model efficiently to market option quotes. On a small set of quotes, exact calibration is achieved under one millisecond. This effectively results in an arbitrage-free interpolation of class $C^2$. The paper proposes a good regularization when the quotes are noisy. Finally, it puts in evidence an issue of the model at-the-money, which is also present in the related one-step finite difference technique of Andreasen and Huge, and gives two solutions for it.

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  • Fabien Le Floc'h, 2020. "An arbitrage-free interpolation of class $C^2$ for option prices," Papers 2004.08650, arXiv.org, revised May 2020.
    • Handle: RePEc:arx:papers:2004.08650
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    References listed on IDEAS

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    1. Peter Carr & Sergey Nadtochiy, 2017. "Local Variance Gamma And Explicit Calibration To Option Prices," Mathematical Finance, Wiley Blackwell, vol. 27(1), pages 151-193, January.
    2. 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.
    3. Breeden, Douglas T & Litzenberger, Robert H, 1978. "Prices of State-contingent Claims Implicit in Option Prices," The Journal of Business, University of Chicago Press, vol. 51(4), pages 621-651, October.
    4. Fabien Floc’h & Cornelis W. Oosterlee, 2019. "Model-free stochastic collocation for an arbitrage-free implied volatility: Part I," Decisions in Economics and Finance, Springer;Associazione per la Matematica, vol. 42(2), pages 679-714, December.
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