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Model-free stochastic collocation for an arbitrage-free implied volatility: Part I

Author

Listed:
  • Fabien Floc’h

    (TU Delft)

  • Cornelis W. Oosterlee

    (TU Delft
    CWI-Centrum Wiskunde & Informatica)

Abstract

This paper explains how to calibrate a stochastic collocation polynomial against market option prices directly. The method is first applied to the interpolation of short-maturity equity option prices in a fully arbitrage-free manner and then to the joint calibration of the constant maturity swap convexity adjustments with the interest rate swaptions smile. To conclude, we explore some limitations of the stochastic collocation technique.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:decfin:v:42:y:2019:i:2:d:10.1007_s10203-019-00238-x
    DOI: 10.1007/s10203-019-00238-x
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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 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.
    3. Bates, David S, 1996. "Jumps and Stochastic Volatility: Exchange Rate Processes Implicit in Deutsche Mark Options," Review of Financial Studies, Society for Financial Studies, vol. 9(1), pages 69-107.
    4. 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.
    5. 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.
    6. 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.
    7. Satchell, Stephen & Knight, John, 2007. "Forecasting Volatility in the Financial Markets," Elsevier Monographs, Elsevier, edition 3, number 9780750669429.
    8. Robert R. Bliss & Nikolaos Panigirtzoglou, 2004. "Option-Implied Risk Aversion Estimates," Journal of Finance, American Finance Association, vol. 59(1), pages 407-446, February.
    9. 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.
    10. Allan M. Malz, 2014. "Simple and reliable way to compute option-based risk-neutral distributions," Staff Reports 677, Federal Reserve Bank of New York.
    11. Heston, Steven L, 1993. "A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options," Review of Financial Studies, Society for Financial Studies, vol. 6(2), pages 327-343.
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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;
    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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