Faster Monotone Implied Volatility Solver

Abstract

We present ThiopheneIV, a Black-Scholes implied-volatility solver with a monotone core and explicit production guards. The solver starts from the simple Choi-Huh-Su L3 lower-bound seed and applies three Euler-Chebyshev steps on a lower branch and three Halley steps on the remaining upper branch. We prove that, in exact arithmetic, the seed lies below the root and both maps increase monotonically without overshooting. We also detail the practical challenges encountered for a double-precision implementation: parity normalisation, microscopic Bachelier-limit handling, saturated price treatment, and an optional J\"ackel-Newton polish. Across standard grids, market-like data, high-volatility cases, and adversarial corners, ThiopheneIV agrees closely with multiprecision Black reference prices at low latency. We provide detailed comparisons with recent solvers, including J\"ackel's Let's Be Rational. The broader lesson is that a convergence proof gives a clean core, but robust production inversion still depends on boundary handling and on the pricing objective one chooses to match.