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 ((Formula presented.)) SABR model. In this model, the option time-value can be represented as an integral of the form (Formula presented.) with (Formula presented.) a ‘payoff function’ which is given by an integral over the McKean kernel (Formula presented.). We study the analyticity properties of the function (Formula presented.) in the complex u-plane and show that it is holomorphic in the strip (Formula presented.). Using this result, we show that the T-series expansion of (Formula presented.) 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 (Formula presented.) at fixed (Formula presented.), or the small vol-of-vol limit (Formula presented.) limit at fixed (Formula presented.), the short maturity T-expansion for the implied volatility has a finite convergence radius (Formula presented.).