We study the probability mass at the origin in the  SABR stochastic volatility model, and derive several tractable expressions for it , in particular when time becomes small or large. In the uncorrelated case, tedious saddlepoint expansions allow for (semi) closed-form asymptotic formulae. As an application--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 currently used implied volatility expansions. In particular we discuss how much those expansions become erroneous .
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 .
D
probability
A
uncorrelated
D
for it
R
In the uncorrelated case, tedious saddlepoint expansions allow for (semi) closed-form asymptotic formulae. As an application--the
As an application--in fact the
R
arbitrage-free
arbitrage free
R
currently used implied volatility expansions. In particular we discuss how much those expansions become erroneous
existing implied volatility approximations, and in particular how correct/erroneous these approximations become