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Explicit Rational Formulae for Bachelier (Normal) Implied Volatility

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  • Fabien Le Floc'h

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 two low degree rationals. In double precision tests both remain close to machine accuracy, while LFK-2026C is the faster scalar implementation on the current benchmark mix.

Suggested Citation

  • Fabien Le Floc'h, 2026. "Explicit Rational Formulae for Bachelier (Normal) Implied Volatility," Papers 2605.18343, arXiv.org, revised Jun 2026.
    • Handle: RePEc:arx:papers:2605.18343
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    References listed on IDEAS

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    1. Jaehyuk Choi & Kwangmoon Kim & Minsuk Kwak, 2009. "Numerical Approximation of the Implied Volatility Under Arithmetic Brownian Motion," Applied Mathematical Finance, Taylor & Francis Journals, vol. 16(3), pages 261-268.
    2. Patel, Jinal & Russo, Vincenzo & Fabozzi, Frank J., 2018. "Using the right implied volatility quotes in times of low interest rates: An empirical analysis across different currencies," Finance Research Letters, Elsevier, vol. 25(C), pages 196-201.
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