quant research // summarizing the paperLOADING
Jian Sun · 2026-06-22
The paper studies option 'smiles' (implied volatility across strikes) and proves, using only basic no-arbitrage logic, that certain normalized strike coordinates always move in one direction as strike changes. It shows this holds for both the standard Black-Scholes and the normal (Bachelier) volatility conventions, and works directly with real, finitely-quoted option chains rather than idealized continuous smiles. It also derives a model-free formula expressing remaining normal variance as a weighted integral of squared Bachelier implied volatility.
Why it matters: Anyone building or validating volatility surfaces from discrete option quotes may find the monotonicity conditions useful as a lightweight no-arbitrage sanity check that needs no smoothing, differentiation, or density extraction. The results are mathematical structure rather than a trading signal, so their practical use is mainly in data cleaning, smile construction, and consistency checks.
⚠ This is a purely theoretical/mathematical result about smile structure; it offers no trading strategy and its practical value is limited to volatility-surface modeling and arbitrage-checking.
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 \((F-K)/σ_N(K)\) 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.
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AI summary generated from the paper’s public abstract via arXiv; it may miss nuance — read the source before relying on it. Thank you to arXiv for its open-access interoperability; StockTools is not affiliated with arXiv, and all rights remain with the authors. Educational only, not financial advice.