The Local Volatility Denominator as a Butterfly Arbitrage Condition

Summary

The document asks how to interpret a condition that arises in Gatheral’s relationship between local variance and the implied total variance surface. In that formula, the local variance is proportional to the maturity derivative of implied total variance divided by a denominator involving its strike-log-moneyness slope and curvature. The question focuses on the requirement that this denominator remain positive.

The answer identifies the condition with freedom from butterfly arbitrage, pointing to Gatheral’s treatment of arbitrage-free SVI volatility surfaces. This connects the shape of the implied volatility surface to the absence of static arbitrage across strikes. The exchange is brief: it names the interpretation but does not derive the condition, spell out regularity assumptions, or discuss how to enforce it in a fitted surface.

Key ideas

• The local variance formula relates implied total variance to its maturity and moneyness derivatives.
• The formula’s denominator must be positive for the stated local variance relationship to make sense.
• The positivity condition is interpreted as freedom from butterfly arbitrage.
• The exchange identifies the condition but does not provide a derivation or implementation method.

Tags

# Does Gatheral formula for local volatility translate to a constraint on implied volatility


# Does Gatheral formula for local volatility translate to a constraint on implied volatility












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 ?

## Answer by servabat (score 5)

https://quant.stackexchange.com/a/78000

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

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.