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Geometric Local Variance Gamma Model

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

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  • Andrey Itkin

Abstract

As mentioned in Chapter 5, the Local Variance Gamma (LVG) volatility model was first introduced by P. Carr in 2008 and then presented in [Carr and Nadtochiy (2014, 2017)] as an extension of the local volatility model by [Dupire (1994)] and [Derman and Kani (1994a)]. The latter was developed on top of the celebrating Black-Scholes model to take into account the existence of option smile. The main advantage of all local volatility models is that given European options prices or their implied volatilities at points (T, K) where K, T are the option strike and time to maturity, they are able to exactly replicate the local volatility function σ(T, K) at these points. This process is called calibration of the local volatility (or, alternatively, implied volatility) surface, and is one of the main topics of this book…

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  • 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..
    • Handle: RePEc:wsi:wschap:9789811212772_0006
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    References listed on IDEAS

    as
    1. Itkin, Andrey, 2015. "To sigmoid-based functional description of the volatility smile," The North American Journal of Economics and Finance, Elsevier, vol. 31(C), pages 264-291.
    2. 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.
    3. Roger W. Lee, 2004. "The Moment Formula For Implied Volatility At Extreme Strikes," Mathematical Finance, Wiley Blackwell, vol. 14(3), pages 469-480, July.
    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. Andrey Itkin & Alexander Lipton, 2016. "Filling the gaps smoothly," Papers 1608.05145, arXiv.org.
    6. John Hull & Alan White, 2015. "A generalized procedure for building trees for the short rate and its application to determining market implied volatility functions," Quantitative Finance, Taylor & Francis Journals, vol. 15(3), pages 443-454, March.
    7. Alexander Lipton, 2001. "Mathematical Methods for Foreign Exchange:A Financial Engineer's Approach," World Scientific Books, World Scientific Publishing Co. Pte. Ltd., number 4694, February.
    8. Erik Ekström & Johan Tysk, 2012. "Dupire'S Equation For Bubbles," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 15(06), pages 1-12.
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    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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