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arxiv: 2605.18343 · v1 · pith:7LJ7LDKInew · submitted 2026-05-18 · 💱 q-fin.CP · q-fin.PR

Explicit Rational Formulae for Bachelier (Normal) Implied Volatility

Fabien Le Floc'h This is my paper

Pith reviewed 2026-05-19 23:35 UTC · model grok-4.3

classification 💱 q-fin.CP q-fin.PR
keywords Bachelier modelnormal implied volatilityrational approximationexplicit formulaoption pricingimplied volatility inversion
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The pith

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

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops two explicit rational formulas that invert the Bachelier normal option pricing equation to recover implied volatility. The formulas accept the market price, forward level, strike and time to expiry as inputs and return the normal volatility parameter directly. They follow a branch structure that switches approximations according to distance from at-the-money, using the absolute forward-strike difference divided by tail time value as the central near-the-money variable. This choice removes the need for a logarithm or a small-argument Taylor expansion in that region. One formula approximates reciprocal absolute standardized moneyness in the far tail for accuracy while the other splits the near-money branch further for speed; both reach errors near machine precision in double-precision tests across wide ranges of moneyness and expiry.

Core claim

The paper claims that the two formulas LFK-2026 and LFK-2026C recover the normal volatility parameter from the Bachelier price formula by means of rational functions alone, employing the absolute forward-strike difference over tail time value near the money and a direct rational fit to reciprocal absolute standardized moneyness in the tails, thereby eliminating any iterative root search.

What carries the argument

Branch-structured rational approximation that replaces the logarithm with the absolute forward-strike difference divided by tail time value near the money and approximates reciprocal absolute standardized moneyness directly in the far tail.

If this is right

  • Pricing and risk systems can obtain normal implied volatility in constant time without convergence failures from iterative solvers.
  • The formulas remain accurate for deep out-of-the-money options and very short expiries where iterative methods can struggle.
  • LFK-2026C delivers faster scalar execution on typical hardware while preserving the same accuracy level as the accuracy-oriented version.
  • Direct embedding into calibration loops becomes possible without added numerical overhead or safeguards.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The branch-rational approach may adapt to implied-volatility inversion for other one-dimensional diffusion models.
  • Machine-precision accuracy supports repeated volatility extraction inside high-frequency risk engines without accumulation of round-off error.
  • Elimination of special-function calls in the near-money region could reduce latency in embedded or low-precision financial hardware.

Load-bearing premise

The chosen rational branch structure and 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.

What would settle it

A numerical test in which the Bachelier pricing formula evaluated at the volatility returned by either rational formula produces an option price that differs from the input price by more than a few units in the last place for some strike, forward and expiry combination.

Figures

Figures reproduced from arXiv: 2605.18343 by Fabien Le Floc'h.

Figure 1
Figure 1. Figure 1: Direct BigFloat relative implied-volatility error for LFK-4, LFK-2026, and LFK-2026C. The left panel focuses on the central region |d| ≤ 8, while the right panel shows the full direct grid |d| ≤ 30. The dotted vertical lines mark the route boundary g = α and the two OTM zone boundaries. move when CPU frequency or scheduling changes during the run. Absolute timings are machine- and power-state-dependent; th… view at source ↗
read the original 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

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper presents two explicit rational formulae (LFK-2026 and LFK-2026C) for Bachelier/normal implied volatility. The inputs are option price, forward, strike and expiry; the output is the implied normal volatility without iteration. The constructions follow the LFK-4 branch structure but replace the near-the-money variable with the simpler |F-K| divided by tail time value, avoiding a logarithm and small-argument Taylor branch. LFK-2026 approximates reciprocal absolute standardized moneyness directly in the far tail; LFK-2026C retains the same tail form but splits the near-the-money region into a very-small-u rational and a mid-range rational. Double-precision tests are reported to reach near machine accuracy, with LFK-2026C faster on the benchmark mix.

Significance. If the accuracy claims are substantiated, the formulae would supply fast, non-iterative, rational-function implementations of normal implied volatility. Such explicit inversions are useful in computational finance for repeated pricing, calibration and risk calculations under the normal model, where iterative solvers are undesirable.

major comments (2)
  1. Abstract: the claim that 'in double precision tests both remain close to machine accuracy' is unsupported by any error tables, coefficient values, test-grid description, or exclusion rules. Without these the central accuracy claim cannot be verified and is load-bearing for the paper's contribution.
  2. Tail approximation section (implied by the description of LFK-2026): the rational fit to 1/|d| for reciprocal absolute standardized moneyness lacks an independent a-priori error bound or exhaustive regime coverage (T ≪ 1, |F-K|/(σ√T) ≫ 10, or exact branch-transition points). Because the underlying Bachelier map is transcendental, any untested corner can produce relative errors exceeding 1e-15 even if average-case results appear good.
minor comments (2)
  1. The abstract introduces 'tail time value' without an explicit definition or equation reference; a one-line definition would improve readability.
  2. The paper should state whether the rational coefficients were obtained by fitting on the same data later used for validation, to address potential circularity concerns.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thorough review and valuable comments on our manuscript. We address each of the major comments below and indicate the revisions we will make to strengthen the paper.

read point-by-point responses

  1. Referee: Abstract: the claim that 'in double precision tests both remain close to machine accuracy' is unsupported by any error tables, coefficient values, test-grid description, or exclusion rules. Without these the central accuracy claim cannot be verified and is load-bearing for the paper's contribution.

    Authors: We agree that the abstract's accuracy claim needs to be supported by explicit documentation to allow independent verification. In the revised manuscript, we will expand the numerical results section to include detailed error tables reporting the maximum, mean, and median relative errors for both formulae across the test cases. We will also list the rational coefficients explicitly, provide a full description of the test grid (including ranges for T from 10^{-8} to 100, standardized moneyness from 0 to 100, and other parameters), and clarify any exclusion criteria for degenerate cases such as zero time value. These additions will substantiate the near-machine-accuracy performance in double precision. revision: yes

  2. Referee: Tail approximation section (implied by the description of LFK-2026): the rational fit to 1/|d| for reciprocal absolute standardized moneyness lacks an independent a-priori error bound or exhaustive regime coverage (T ≪ 1, |F-K|/(σ√T) ≫ 10, or exact branch-transition points). Because the underlying Bachelier map is transcendental, any untested corner can produce relative errors exceeding 1e-15 even if average-case results appear good.

    Authors: We appreciate the referee's point regarding the need for more rigorous validation of the tail approximation. While our current tests include regimes with small T and large |F-K|/(σ√T), we acknowledge that a more exhaustive coverage and explicit branch points would improve the manuscript. In the revision, we will add a new subsection detailing the maximum observed relative errors in the far-tail regime for T ≪ 1 and |F-K|/(σ√T) ≫ 10, specify the exact transition points between the near-the-money and tail branches, and include plots or tables demonstrating coverage. An independent a-priori error bound is challenging to derive for this transcendental inversion without additional analysis; however, we will provide a conservative empirical bound based on the tested regimes and the properties of the rational approximant. revision: partial

Circularity Check

1 steps flagged

Minor self-citation to prior branch structure; central explicit formulae remain independent

specific steps
  1. self citation load bearing [Abstract]
    "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."

    The foundational branch structure is imported from LFK-4 (prior work sharing the LFK author prefix), so the organizational skeleton of the derivation is self-referential even though the concrete variable choice and rational coefficients are novel.

full rationale

The paper constructs and presents two new explicit rational approximations (LFK-2026 and LFK-2026C) for Bachelier implied volatility using a chosen near-the-money variable and tail rational fit. It references LFK-4 only for the overall branch structure while introducing a simpler variable and specific rational forms. This self-citation is not load-bearing for the central result, which consists of the new formulae and their reported numerical accuracy. No step reduces the output formulae to a tautological fit or self-definition by construction; the approximations are offered as practical explicit alternatives with independent validation tests.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Because only the abstract is available, the ledger records the minimal set of modeling choices that must be true for the central claim to hold; no explicit free parameters, axioms or invented entities are stated in the provided text.

pith-pipeline@v0.9.0 · 5683 in / 1190 out tokens · 26539 ms · 2026-05-19T23:35:07.108761+00:00 · methodology

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Reference graph

Works this paper leans on

8 extracted references · 8 canonical work pages

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