Explicit Rational Formulae for Bachelier (Normal) Implied Volatility

2026-05-18

AI Summary

This study addresses the inefficiency and limited accuracy of traditional iterative methods for computing Bachelier implied volatility, particularly for extreme strikes. The authors propose two novel non-iterative explicit rational formulas—including the LFK-2026C variant—that employ a piecewise structure to separately handle near-the-money and far-from-the-money regions. A key innovation is the introduction of a normalized moneyness variable based on the absolute forward-strike difference and tail time value, which eliminates the need for logarithmic operations and asymptotic expansions in small parameters. By integrating explicit rational approximations, a tailored piecewise design, and double-precision optimization, the method achieves near-machine precision while significantly accelerating scalar computations, outperforming existing benchmarks in both speed and robustness.

Abstract

We present two explicit rational formulae for Bachelier, or normal, implied volatility. The formulae take the option price, forward, strike, and expiry as inputs and return the implied normal volatility without iteration. They follow the branch structure of LFK-4, but use the simpler near-the-money variable given by the absolute forward-strike difference divided by the tail time value, avoiding a logarithm and a small-argument Taylor branch in that region. LFK-2026 is the accuracy-oriented formula and approximates reciprocal absolute standardized moneyness directly in the far tail. LFK-2026C keeps the same shifted out-of-the-money rational tail approximation, but splits the near-the-money branch into a very small low- \(u\) rational and a mid-range rational. In double precision tests both remain close to machine accuracy, while LFK-2026C is the faster scalar implementation on the current benchmark mix

Problem

• Bachelier implied volatility
• normal volatility
• explicit formula
• rational approximation
• option pricing

Innovation

• Bachelier implied volatility
• explicit rational approximation
• normal volatility
• non-iterative formula
• asymptotic tail approximation