Mass at zero in the uncorrelated SABR model and implied volatility asymptotics

TL;DR

This paper analyzes the probability mass at zero in the uncorrelated SABR model and derives asymptotic formulas for implied volatility at different maturities, improving understanding of small-strike behavior.

Contribution

It provides new tractable expressions for the mass at zero and develops small- and large-maturity implied volatility expansions in the uncorrelated SABR model.

Findings

Derived explicit formulas for the mass at zero.

Provided small- and large-maturity implied volatility asymptotics.

Quantified the impact of mass at zero on implied volatility approximations.

Abstract

We study the mass at the origin in the uncorrelated SABR stochastic volatility model, and derive several tractable expressions, in particular when time becomes small or large. As an application--in fact the original motivation for this paper--we derive small-strike expansions for the implied volatility when the maturity becomes short or large. These formulae, by definition arbitrage free, allow us to quantify the impact of the mass at zero on existing implied volatility approximations, and in particular how correct/erroneous these approximations become.