The SABR model has been around for a long-time and it often studied via a small-maturity power series. In fact, all stochastic volatility models have a double power series expansion, for the Black-Scholes implied volatility, in the time-to-maturity T and the log-moneyness. With my coauthor Dan Pirjol, we prove that this series (taking the lognormal SABR model at the at-the-money point) is strictly asymptotic: non-convergent for any [$]T>0[$]. Paper is here

For some background, the non-analyticity of the diffusion kernel factor, [$]e^{-d^2/T}[$] at [$]T=0[$] leads many to believe strict asymptotics in [$]T[$] is the only possibility for option values and related solutions for diffusion models. As we explain in the paper, that's not correct. For example, a simple case is the at-the-money Black-Scholes option value [$]V_{BS}(T) = \mbox{Erf}(2^{-3/2} \sqrt{T})[$]. If you divide the right-hand-side by a [$]\sqrt{T}[$], you get an entire function of [$]T[$], and so an associated power series in [$]T[$] that converges for any [$]|T| < \infty[$]. In principle, that or analyticity for [$]|T| < R[$], where [$]R>0[$] is a convergence radius, could hold in the SABR model. We prove it doesn't.

As always, any comments on the paper are appreciated.