  We study the convergence properties of the short maturity expansion of option
prices in the uncorrelated log-normal ($\beta=1$) SABR model. In this model the
option time-value can be represented as an integral of the form $V(T) =
\int_{0}^\infty e^{-\frac{u^2}{2T}} g(u) du$ with $g(u)$ a "payoff function"
which is given by an integral over the McKean kernel $G(s,t)$. We study the
analyticity properties of the function $g(u)$ in the complex $u$-plane and show
that it is holomorphic in the strip $|\Im(u) |< \pi$. Using this result we show
that the $T$-series expansion of $V(T)$ and implied volatility are asymptotic
(non-convergent for any $T>0$). In a certain limit which can be defined either
as the large volatility limit $\sigma_0\to \infty$ at fixed $\omega=1$, or the
small vol-of-vol limit $\omega\to 0$ limit at fixed $\omega\sigma_0$, the short
maturity $T$-expansion for the implied volatility has a finite convergence
radius $T_c = \frac{1.32}{\omega\sigma_0}$.
