A lower bound for the volatility swap in the lognormal SABR model

TL;DR

This paper proves that in the short time limit, the zero vanna implied volatility provides a lower bound for the volatility swap strike in the lognormal SABR model, improving upon previous bounds especially for non-positive correlation.

Contribution

It establishes a new lower bound for volatility swaps in the lognormal SABR model valid for all correlation values, sharpening previous bounds for certain cases.

Findings

Zero vanna implied volatility is a lower bound for volatility swap strike.

The bound is valid for all correlation parameters.

It is sharper than the at-the-money implied volatility for correlation ≤ 0.

Abstract

In the short time to maturity limit it is proved that for the conditionally lognormal SABR model the zero vanna implied volatility is a lower bound for the volatility swap strike. The result is valid for all values of the correlation parameter and is a sharper lower bound than the at-the-money implied volatility for correlation less than or equal to zero.