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

Simon Ellersgaard, Martin Jönsson and Rolf Poulsen

Quantitative Finance, 2017, vol. 17, issue 4, 515-529

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.

Date: 2017
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DOI: 10.1080/14697688.2016.1222078

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