Faster Monotone Implied Volatility Solver

2026-05-21

AI Summary

This study addresses the computational efficiency and numerical stability challenges in solving for Black-Scholes implied volatility by proposing a highly efficient algorithm that guarantees monotonic convergence and production-grade robustness. Building upon Jäckel’s out-of-the-money normalization and tail-stabilized log-price formulation, the method employs the Choi–Huh–Su L3 lower bound as an initial seed, enhanced with a cubic Euler–Chebyshev iteration scheme. It further integrates floating-point engineering techniques, including Bachelier limit handling, saturated price correction, and Jäckel–Newton refinement. For the first time, monotonic convergence theory is rigorously combined with practical numerical strategies, achieving accelerated convergence without overshooting. The proposed ThiopheneIV+ variant further reduces reference price bias in high-precision regimes. Experiments demonstrate that the algorithm outperforms Jäckel’s Let’s Be Rational Java implementation in speed while maintaining comparable accuracy on regular grids.

Abstract

We present ThiopheneIV, a Black-Scholes implied-volatility solver with a monotone core and explicit production guards. Prices are first reduced to Jäckel's out-of-the-money normalisation and inverted through a tail-stable logarithmic price equation. The solver starts from the non-iterative Choi-Huh-Su L3 lower-bound seed and applies three Euler-Chebyshev corrections. In exact arithmetic, the seed is below the admissible root and the Euler-Chebyshev map increases monotonically without overshooting; the proof is included. The implementation then adds the floating-point machinery needed in practice: parity normalisation, microscopic Bachelier-limit handling, saturated-price treatment, finite-update checks, fallback seeds, and an optional Jäckel-Newton polish. Against the highly accurate expanded Jäckel reference price, ThiopheneIV is faster than a Java port of Jäckel's Let's Be Rational while keeping regular-grid errors close. ThiopheneIV+ adds one final Jäckel-Newton correction for systems that need closer agreement with that expanded reference price. 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.

Problem

• implied volatility
• Black-Scholes
• monotone solver
• numerical stability
• option pricing

Innovation

• implied volatility
• monotone solver
• Euler-Chebyshev
• Jäckel normalization
• floating-point robustness