Proof of non-convergence of the short-maturity expansion for the SABR model. Issue 9 (2nd September 2022) - Lewis, Alan L.; Pirjol, Dan
Abstract : 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 − u 2 2 T g ( u ) d u 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 σ 0 → ∞ at fixed ω = 1, or the small vol-of-vol limit ω → 0 limit at fixed ω σ 0, the short maturity T -expansion for the implied volatility has a finite convergence radius T c = 1.32 ω σ 0 .