Proof of non-convergence of the short-maturity expansion for the SABR model
研究了无相关对数正态SABR模型中期权价格短到期展开的收敛性，证明该展开是渐近的（对任何T>0不收敛），并给出了隐含波动率展开收敛半径的条件。
We study the convergence properties of the short maturity expansion of option prices in the uncorrelated log-normal (β=1) SABR model. In this model, the option time-value can be represented as an integral of the form V(T)=∫0∞e−u22Tg(u)du with g(u) a ‘payoff function’ which is given by an integral over the McKean kernel G(t,s). We study the analyticity properties of the function g(u) in the complex u-plane and show that it is holomorphic in the strip |ℑ(u)|<π. Using this result, we show that the T-series expansion of V(T) 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 σ0→∞ at fixed ω=1, or the small vol-of-vol limit ω→0 limit at fixed ωσ0, the short maturity T-expansion for the implied volatility has a finite convergence radius Tc=1.32ωσ0.