Explicit Rational Formulae for Bachelier (Normal) Implied Volatility

TL;DR

This paper introduces two explicit rational formulae for calculating Bachelier (normal) implied volatility directly from option parameters, avoiding iterative procedures and enhancing computational efficiency.

Contribution

The authors develop simplified, accurate rational formulae for Bachelier implied volatility that improve speed and precision over existing methods.

Findings

Both formulae achieve near-machine accuracy in double precision tests.

LFK-2026C is faster and maintains high accuracy on current benchmarks.

The methods avoid complex logarithmic and Taylor branch calculations.

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