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

We study the convergence properties of the short maturity expansion of option prices in the uncorrelated log-normal ( $ \beta =1 $ β=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)\,{\rm d}u $ V(T)=∫0∞e−u22Tg(u)du with $ g(u) $ g(u) a ‘payoff function’ which is given by an integral over the McKean kernel $ \mathcal {G}(t,s) $ G(t,s). We study the analyticity properties of the function $ g(u) $ g(u) in the complex u-plane and show that it is holomorphic in the strip $ |\Im (u) | 0). In a certain limit which can be defined either as the large volatility limit $ \sigma _0\to \infty $ σ0→∞ at fixed $ \omega =1 $ ω=1, or the small vol-of-vol limit $ \omega \to 0 $ ω→0 limit at fixed $ \omega \sigma _0 $ ωσ0, the short maturity T-expansion for the implied volatility has a finite convergence radius $ T_c = \frac {1.32}{\omega \sigma _0} $ Tc=1.32ωσ0.