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In Gatheral's "The Volatility Surface : A Practitioner's Guide", Equation (1.10) page 13, the following relation linking squared local volatility and squared implied volatility is expressed :

$$\begin{equation} \sigma^2(y, t) = \frac{\partial_T w}{1 - \frac{y}{w} \partial_y w + \frac{1}{4}(-\frac{1}{4} - \frac{1}{w} + \frac{y^2}{w^2})(\partial_y w)^2 + \frac{1}{2} \partial_{y,y}w}\end{equation}$$ where $w(y, t) = \sigma_{BS}^2(y, t)t$ and $y = \log K/F$.

This seem to mean that $1 - \frac{y}{w} \partial_y w + \frac{1}{4}(-\frac{1}{4} - \frac{1}{w} + \frac{y^2}{w^2})(\partial_y w)^2 + \frac{1}{2} \partial_{y,y}w > 0$.

Does this inequality have a known interpretation, and a name ?

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1 Answer 1

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This is actually explained in the article "Arbitrage-free SVI volatility surfaces" by Gatheral : It means that the surface is free of butterfly arbitrage.

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