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arxiv: 2606.23883 · v1 · pith:TU6COEHZnew · submitted 2026-06-22 · 💱 q-fin.MF

Monotonicity of Normalized Implied-Volatility Coordinates under No-Arbitrage

Jian Sun This is my paper

Pith reviewed 2026-06-26 05:24 UTC · model grok-4.3

classification 💱 q-fin.MF
keywords arbitrage-free optionsimplied volatilitynormalized coordinatesmonotonicityBlack-ScholesBacheliervariance identityno-arbitrage proof
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The pith

Arbitrage-free option prices at finite strikes force the normalized coordinate k/v(k) to be monotonic for fixed maturity.

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

The paper shows that no-arbitrage conditions on quoted option prices induce natural normalized strike coordinates in which the central Black-Scholes quantity k/v(k) is monotonic. The proof uses only finite-strike comparisons, convexity of prices in strike, monotonicity, and put-call parity, so it applies directly to discrete market quotes without requiring a continuous smile or differentiability. The same monotonicity holds for the Bachelier normalized coordinate (F-K)/σ_N(K). A model-free identity is also derived that expresses remaining normal variance as a normal-density weighted integral of squared Bachelier implied volatility.

Core claim

For arbitrage-free option prices at a fixed maturity, the normalized coordinate k/v(k) is monotonic in strike; the same holds for (F-K)/σ_N(K) in the Bachelier case. The proofs rely solely on discrete no-arbitrage properties and do not invoke continuous limits or densities. Remaining normal variance equals the integral of squared Bachelier implied volatility against the normal density in the normalized coordinate, providing the direct analogue of Fukasawa's lognormal variance identity.

What carries the argument

The normalized strike coordinates k/v(k) for Black-Scholes and (F-K)/σ_N(K) for Bachelier implied volatility, shown to be monotonic via convexity and put-call parity on finite strikes.

If this is right

  • Monotonicity holds for any arbitrage-free chain of quoted prices without needing interpolation or density extraction.
  • The argument extends unchanged to the Bachelier implied-volatility coordinate.
  • Remaining normal variance admits an exact integral representation in terms of squared Bachelier implied volatility.
  • The results supply discrete, model-free foundations that complement continuous-strike normalizing transformations.

Where Pith is reading between the lines

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

  • The monotonicity may yield new model-free bounds on the slope of implied-volatility smiles.
  • The normal-variance identity could be used to extract variance-swap-like quantities directly from quoted normal vols.
  • Similar normalizing coordinates might exist for other pricing kernels that admit elementary convexity arguments.
  • The finite-strike proof technique could be adapted to show monotonicity for other implied-volatility parametrizations.

Load-bearing premise

Quoted option prices at finite strikes satisfy convexity in the strike variable together with monotonicity and put-call parity.

What would settle it

A concrete set of finite-strike call and put prices that obey put-call parity and convexity in strike but where the computed sequence k/v(k) increases over some interval of strikes.

read the original abstract

For a fixed maturity, an arbitrage-free option smile induces natural normalized strike coordinates. This paper makes three contributions. First, it gives an elementary discrete no-arbitrage proof of monotonicity for the central Black--Scholes normalized coordinate \(k/v(k)\), using only finite-strike comparisons, convexity, monotonicity, and put--call parity. Thus the argument applies directly to finitely quoted option chains and does not require a continuously quoted smile, differentiability of option prices, differentiability of implied volatility, digital prices, or density extraction. Second, it extends the same monotonicity principle to the normal, or Bachelier, implied volatility formula, proving that the normalized coordinate \((F-K)/\sigma_N(K)\) is decreasing in strike under static no-arbitrage. Third, it proves a model-free normal-variance identity: remaining normal variance can be represented as a normal-density weighted integral of squared Bachelier implied volatility in the normalized coordinate. This third result is the normal/Bachelier analogue of Fukasawa's lognormal variance identity, which expresses variance-type quantities through Black implied variance in normalized coordinates. The paper therefore complements Fukasawa's continuous-strike normalizing transformation theory with a finite-quote no-arbitrage proof and a new normal-variance counterpart, while connecting the results to the volatility-derivatives literature surveyed by Carr and Lee.

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

0 major / 3 minor

Summary. The paper claims three results for arbitrage-free option prices at a fixed maturity. First, an elementary discrete proof shows that the normalized Black-Scholes coordinate k/v(k) is monotonic, relying solely on finite-strike comparisons together with convexity in strike, monotonicity, and put-call parity. Second, the same no-arbitrage conditions imply that the Bachelier normalized coordinate (F-K)/σ_N(K) is decreasing in strike. Third, a model-free identity expresses remaining normal variance as a normal-density weighted integral of squared Bachelier implied volatility in the normalized coordinate, presented as the direct analogue of Fukasawa's lognormal variance identity.

Significance. If the derivations hold, the results are significant because they deliver finite-quote, discrete proofs that apply directly to observed option chains without continuous interpolation, differentiability of prices or implied volatility, or density extraction. The model-free normal-variance identity supplies a new counterpart to Fukasawa's result and connects to the volatility-derivatives literature. The exclusive reliance on standard no-arbitrage axioms (convexity, monotonicity, put-call parity) enhances applicability to market data.

minor comments (3)
  1. [§2] The definition of the normalized coordinate v(k) and the precise statement of the finite-strike comparison lemmas should appear in the first section that introduces the Black-Scholes case, to make the discrete argument self-contained.
  2. [§4] The normal-density weighting in the variance identity is stated without an explicit integral formula; adding the precise expression (analogous to the lognormal case) would improve readability.
  3. [§3] A short remark clarifying that the monotonicity statements remain valid when some strikes are missing (provided the quoted strikes still satisfy the convexity and parity conditions) would strengthen the finite-quote applicability claim.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript and for recommending acceptance. The referee's summary correctly identifies the three main contributions: the discrete no-arbitrage proof of monotonicity for the normalized Black-Scholes coordinate, the extension to the Bachelier normalized coordinate, and the model-free normal-variance identity as the analogue of Fukasawa's result. No major comments require a point-by-point response.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper's core contributions are an elementary proof that standard no-arbitrage properties (convexity in strike, monotonicity, put-call parity) at finite strikes imply monotonicity of the normalized coordinates k/v(k) and (F-K)/σ_N(K), plus a model-free integral identity for normal variance. These properties are external inputs from arbitrage-free option prices; the normalized coordinates are not used to define or fit the inputs, nor does any step reduce by construction to a fitted parameter or self-citation. The argument explicitly avoids continuous interpolation, differentiability, or density extraction. The normal-variance identity is presented as an analogue of Fukasawa's result without importing uniqueness theorems or ansatzes from the author's prior work. The derivation chain is therefore independent of the target monotonicity statements.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on standard domain assumptions from option pricing theory; no free parameters or invented entities are introduced.

axioms (2)
  • domain assumption Option prices satisfy convexity in strike
    Invoked as one of the only properties used in the discrete no-arbitrage proof.
  • domain assumption Put-call parity holds for the quoted strikes
    Invoked as one of the only properties used in the discrete no-arbitrage proof.

pith-pipeline@v0.9.1-grok · 5771 in / 1492 out tokens · 36532 ms · 2026-06-26T05:24:14.600420+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

14 extracted references · 2 canonical work pages

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