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

The SABR model has been around for a relatively long time and it’s 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. This new research preprint is posted to the arXiv here

Update: The final published version (in the journal Quantitative Finance) has limited (1st 50 clicks) free availability: 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) = {\text{erf}}(2^{-3/2} \sqrt{T}),

using the error function. After division by a \sqrt{T} , this is an entire function in the complex T-plane.

In principle, similar “nice” behavior could occur in the SABR model. That is, the value function or some simple normalized version of it, could have had a power series that converged for
|T| \lt R in the complex T-plane. Here \{ R: 0 \lt R \le \infty\} denotes a convergence radius. In fact, we prove this doesn’t happen.

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