Proof of non-convergence of the short-maturity expansion for the SABR model

Abstract: 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 $T_c = \frac{1.32}{\omega\sigma_0}$.