A Fast Implied Volatility Method with Expansions
We present a regime-split Black--Scholes implied volatility solver in which every initial seed is a fully closed-form analytical expression, derived from the asymptotic structure of the Black--Scholes price in its natural domain. At the money, series reversion of an exact Gaussian identity yields a fourth-order seed with error $\mathcal{O}(s^8)$. In the moderate out-of-the-money region, successive Gaussian CDF approximations of increasing order produce explicit initial seed formulas whose accuracy is proved numerically, with no iteration or numerical inversion at the seed stage. In the deep out-of-the-money region, a Gaussian tail cancellation identity -- the Mills ratio -- reveals the asymptotic structure of the Black--Scholes price and motivates a ratio-corrected seed that achieves near-machine-precision initialisation for large moneyness. All regime boundaries are derived analytically from CDF truncation tolerances and numerical solver theoretical error bounds, with no empirically tuned constants. A universal fourth-order Householder polisher then drives all regimes to machine precision, with mean update iterations strictly below two on both standard and granular benchmark grids -- meeting and surpassing the two-iteration target established by the highest-accuracy reference implementation in the literature (J\"ackel, 2015). The resulting C implementation achieves a $1.6$--$1.89\times$ throughput gain over the state-of-the-art benchmark (J\"ackel, 2015) under identical hardware and compiler conditions, with maximum absolute error $\mathcal{O}(10^{-14})$, stable across grid configurations. A Python/Numba implementation confirms portability. All source code is publicly available.