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pith:7LJ7LDKI

pith:2026:7LJ7LDKI6U3DDRPP4LWEMJI7MH

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Explicit Rational Formulae for Bachelier (Normal) Implied Volatility

Fabien Le Floc'h

Two rational formulas calculate Bachelier implied volatility directly from option price, forward, strike and expiry without iteration.

arxiv:2605.18343 v1 · 2026-05-18 · q-fin.CP · q-fin.PR

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Claims

C1strongest claim

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.

C2weakest assumption

The chosen rational branch structure and the specific tail approximation for reciprocal absolute standardized moneyness remain accurate enough across all relevant moneyness and time-to-expiry regimes that the overall error stays near machine precision.

C3one line summary

Two new rational approximation formulas compute normal implied volatility to near machine precision without iteration by using a simpler near-the-money variable and tailored tail approximations.

References

8 extracted · 8 resolved · 0 Pith anchors

[1] Théorie de la spéculation.Annales Scientifiques de l’École Normale Supérieure1900,17, 21–86

[2] Fast and Accurate Analytic Basis Point Volatility 2026

[3] Using the Right Implied Volatility Quotes in Times of Low Interest Rates: An Empirical Analysis across Different Currencies.Finance Research Letters2018,25, 196–201 2017 · doi:10.1016/j.frl.2017.10.013

[4] Switch to Bachelier Options Pricing Model—Effective April 22, 2020 2020

[5] A Black–Scholes User’s Guide to the Bachelier Model.Journal of Futures Markets2022,42, 959–980 · doi:10.1002/fut.22315

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Receipt and verification

First computed	2026-05-20T00:05:56.041640Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

fad3f58d48f53631c5efe2ec46251f61fe36de6651b98a74f9a8602f4e574945

Aliases

arxiv: 2605.18343 · arxiv_version: 2605.18343v1 · doi: 10.48550/arxiv.2605.18343 · pith_short_12: 7LJ7LDKI6U3D · pith_short_16: 7LJ7LDKI6U3DDRPP · pith_short_8: 7LJ7LDKI

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Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/7LJ7LDKI6U3DDRPP4LWEMJI7MH \
  | jq -c '.canonical_record' \
  | python3 -c "import sys,json,hashlib; b=json.dumps(json.loads(sys.stdin.read()), sort_keys=True, separators=(',',':'), ensure_ascii=False).encode(); print(hashlib.sha256(b).hexdigest())"
# expect: fad3f58d48f53631c5efe2ec46251f61fe36de6651b98a74f9a8602f4e574945

Canonical record JSON

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    "license": "http://creativecommons.org/licenses/by-sa/4.0/",
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    "submitted_at": "2026-05-18T12:56:35Z",
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