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The Fundamental Theorem of Derivative Trading - exposition, extensions and experiments

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  • Simon Ellersgaard
  • Martin Jönsson
  • Rolf Poulsen

Abstract

When estimated volatilities are not in perfect agreement with reality, delta-hedged option portfolios will incur a non-zero profit-and-loss over time. However, there is a surprisingly simple formula for the resulting hedge error, which has been known since the late 1990s. We call this The Fundamental Theorem of Derivative Trading. This paper is a survey with twists on that result. We prove a more general version of it and discuss various extensions and applications, from incorporating a multi-dimensional jump framework to deriving the Dupire–Gyöngy–Derman–Kani formula. We also consider its practical consequences, both in simulation experiments and on empirical data, thus demonstrating the benefits of hedging with implied volatility.

Suggested Citation

  • Simon Ellersgaard & Martin Jönsson & Rolf Poulsen, 2017. "The Fundamental Theorem of Derivative Trading - exposition, extensions and experiments," Quantitative Finance, Taylor & Francis Journals, vol. 17(4), pages 515-529, April.
    • Handle: RePEc:taf:quantf:v:17:y:2017:i:4:p:515-529
    DOI: 10.1080/14697688.2016.1222078
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    Cited by:

    1. John Armstrong & Claudio Bellani & Damiano Brigo & Thomas Cass, 2021. "Option pricing models without probability: a rough paths approach," Mathematical Finance, Wiley Blackwell, vol. 31(4), pages 1494-1521, October.
    2. John Armstrong & Andrei Ionescu, 2023. "Gamma Hedging and Rough Paths," Papers 2309.05054, arXiv.org, revised Mar 2024.

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