The Fundamental Theorem of Derivative Trading - Exposition, Extensions, & Experiments

When estimated volatilities are not in perfect agreement with reality, delta hedged option portfolios will incur a non-zero profit-and-loss over time. There is, however, a surprisingly simple formula for the resulting hedge error, which has been known since the late 90s. We call this The Fundamental Theorem of Derivative Trading. This is a survey with twists of that result. We prove a more general version and discuss various extensions (including jumps) and applications (including deriving the Dupire-Gyo ̈ngy-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.