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