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Robust Calibration For SVI Model Arbitrage Free

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  • Tahar Ferhati

    (IMJ - Institut de Mathématiques de Jussieu - UPMC - Université Pierre et Marie Curie - Paris 6 - UPD7 - Université Paris Diderot - Paris 7 - CNRS - Centre National de la Recherche Scientifique)

Abstract

The purpose of this paper is to study the Stochastic Volatility Inspired model (SVI) as implied volatility model: we study the analytic part of the SVI with the arbitrage conditions, we establish the initial guess and the parameter's boundaries. Until recently it was not possible to find sufficient conditions that would guarantee an SVI model calibration arbitrage-free. The main contribution in this paper is that we provided a numerical method to resolve the arbitrage problem (butterfly and calendar spread) using the Sequential Least-Squares Quadratic Programming (SLSQP) method. Our method guarantee to get SVI calibration with butterfly and calendar spread arbitrage-free, We provide many numerical examples with arbitrage such as Vogt Axel example and we show how to fix them. The calibration method is tested on 23 equity indexes with 14 maturities each and we get 322 slices fits using the same initial guess and the SVI parameters boundaries for all indexes. This new calibration method is very important and it meets practical need: resolving this arbitrage problem will pave the way to the surface calibration and the transition from implied volatility to local volatility using Dupire's formula, therefore, it allows price different kind of path-dependent options such as barrier options, and American options. The SVI model could also be applied to price interest rate derivatives such as swaptions, interest rate caps, and floors.

Suggested Citation

  • Tahar Ferhati, 2020. "Robust Calibration For SVI Model Arbitrage Free," Working Papers hal-02490029, HAL.
    • Handle: RePEc:hal:wpaper:hal-02490029
    Note: View the original document on HAL open archive server: https://hal.science/hal-02490029v2
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    References listed on IDEAS

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    1. Judith Glaser & Pascal Heider, 2012. "Arbitrage-free approximation of call price surfaces and input data risk," Quantitative Finance, Taylor & Francis Journals, vol. 12(1), pages 61-73, August.
    2. Carr, Peter & Madan, Dilip B., 2005. "A note on sufficient conditions for no arbitrage," Finance Research Letters, Elsevier, vol. 2(3), pages 125-130, September.
    3. Matthias Fengler, 2009. "Arbitrage-free smoothing of the implied volatility surface," Quantitative Finance, Taylor & Francis Journals, vol. 9(4), pages 417-428.
    4. Jim Gatheral & Antoine Jacquier, 2011. "Convergence of Heston to SVI," Quantitative Finance, Taylor & Francis Journals, vol. 11(8), pages 1129-1132.
    5. M. Benko & M. Fengler & W. Härdle & M. Kopa, 2007. "On extracting information implied in options," Computational Statistics, Springer, vol. 22(4), pages 543-553, December.
    6. Roger W. Lee, 2004. "The Moment Formula For Implied Volatility At Extreme Strikes," Mathematical Finance, Wiley Blackwell, vol. 14(3), pages 469-480, July.
    7. 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.
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    Cited by:

    1. Claude Martini & Arianna Mingone, 2020. "No arbitrage SVI," Papers 2005.03340, arXiv.org, revised May 2021.
    2. Navratil, Robert & Taylor, Stephen & Vecer, Jan, 2022. "On the utility maximization of the discrepancy between a perceived and market implied risk neutral distribution," European Journal of Operational Research, Elsevier, vol. 302(3), pages 1215-1229.

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