A Fast Implied Volatility Method with Expansions

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Implied volatility inversion is a fundamental numerical primitive in options trading, volatility surface construction, calibration, and risk management. The method is explicitly designed for production-style implementations where IV may be evaluated millions of times. The 1.7-1.9x speedup over the industry-standard Jackel implementation, combined with near machine-precision accuracy and robustness across extreme-delta cases, makes it directly relevant for high-frequency options pricing, real-time risk systems, and large-scale Monte Carlo calibration engines. The Python/Numba portability further broadens applicability.

The method involves multiple regime branches (ATM, mild-OTM, transition, deep-OTM) with different seed formulas, but each seed is algebraic or nearly algebraic (polynomial evaluations, square roots, exponentials). The HH-4 update is a single formula. The pseudo-algorithm is clearly specified. The main complexity lies in correctly implementing the regime boundaries and the polynomial seeds (P1, P3, P7) with their shared-term optimizations. The fallback logic (N<=0, denominator guards) adds some branching. Overall moderate complexity - more than a simple Newton solver but less than a full PDE solver.

Source code and benchmark scripts supplied as anonymized supplementary material. A public repository will be provided upon acceptance. The paper includes detailed pseudo-algorithm (Algorithm 1), all mathematical derivations in appendix, and benchmark results across multiple CPU/compiler environments (MSYS2 UCRT64 GCC, Linux GCC 13.3, Clang on Intel Xeon). Python/Numba implementation confirms portability.