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Model-Free Stochastic Collocation for an Arbitrage-Free Implied Volatility, Part II

Author

Listed:
  • Fabien Le Floc’h

    (Delft Institute of Applied Mathematics, TU Delft, 2628 XE Delft, The Netherlands)

  • Cornelis W. Oosterlee

    (Delft Institute of Applied Mathematics, TU Delft, 2628 XE Delft, The Netherlands
    CWI-Centrum Wiskunde & Informatica, 1098 XE Amsterdam, The Netherlands)

Abstract

This paper explores the stochastic collocation technique, applied on a monotonic spline, as an arbitrage-free and model-free interpolation of implied volatilities. We explore various spline formulations, including B-spline representations. We explain how to calibrate the different representations against market option prices, detail how to smooth out the market quotes, and choose a proper initial guess. The technique is then applied to concrete market options and the stability of the different approaches is analyzed. Finally, we consider a challenging example where convex spline interpolations lead to oscillations in the implied volatility and compare the spline collocation results with those obtained through arbitrage-free interpolation technique of Andreasen and Huge.

Suggested Citation

  • Fabien Le Floc’h & Cornelis W. Oosterlee, 2019. "Model-Free Stochastic Collocation for an Arbitrage-Free Implied Volatility, Part II," Risks, MDPI, vol. 7(1), pages 1-21, March.
    • Handle: RePEc:gam:jrisks:v:7:y:2019:i:1:p:30-:d:211431
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    References listed on IDEAS

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    1. Mr. Kevin C Cheng, 2010. "A New Framework to Estimate the Risk-Neutral Probability Density Functions Embedded in Options Prices," IMF Working Papers 2010/181, International Monetary Fund.
    2. 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.
    3. Peter Christoffersen & Steven Heston & Kris Jacobs, 2009. "The Shape and Term Structure of the Index Option Smirk: Why Multifactor Stochastic Volatility Models Work So Well," Management Science, INFORMS, vol. 55(12), pages 1914-1932, December.
    4. Allan M. Malz, 2014. "Simple and reliable way to compute option-based risk-neutral distributions," Staff Reports 677, Federal Reserve Bank of New York.
    5. Heston, Steven L, 1993. "A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options," The Review of Financial Studies, Society for Financial Studies, vol. 6(2), pages 327-343.
    6. Minqiang Li & Kyuseok Lee, 2011. "An adaptive successive over-relaxation method for computing the Black-Scholes implied volatility," Quantitative Finance, Taylor & Francis Journals, vol. 11(8), pages 1245-1269.
    7. Bates, David S, 1996. "Jumps and Stochastic Volatility: Exchange Rate Processes Implicit in Deutsche Mark Options," The Review of Financial Studies, Society for Financial Studies, vol. 9(1), pages 69-107.
    8. Josip Arneric & Zdravka Aljinovic & Tea Poklepovic, 2015. "Extraction of market expectations from risk-neutral density," Zbornik radova Ekonomskog fakulteta u Rijeci/Proceedings of Rijeka Faculty of Economics, University of Rijeka, Faculty of Economics and Business, vol. 33(2), pages 235-256.
    9. 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.
    10. Satchell, Stephen & Knight, John, 2007. "Forecasting Volatility in the Financial Markets," Elsevier Monographs, Elsevier, edition 3, number 9780750669429.
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    Cited by:

    1. Fabien Le Floc’h & Cornelis W. Oosterlee, 2019. "Model-Free Stochastic Collocation for an Arbitrage-Free Implied Volatility, Part II," Risks, MDPI, vol. 7(1), pages 1-21, March.
    2. 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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    More about this item

    Keywords

    stochastic collocation; implied volatility; quantitative finance; arbitrage-free; risk neutral density; B-spline;
    All these keywords.

    JEL classification:

    • C63 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Computational Techniques
    • G17 - Financial Economics - - General Financial Markets - - - Financial Forecasting and Simulation
    • G13 - Financial Economics - - General Financial Markets - - - Contingent Pricing; Futures Pricing

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