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Monotonicity of Normalized Implied-Volatility Coordinates under No-Arbitrage

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  • Jian Sun

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.

Suggested Citation

  • Jian Sun, 2026. "Monotonicity of Normalized Implied-Volatility Coordinates under No-Arbitrage," Papers 2606.23883, arXiv.org.
    • Handle: RePEc:arx:papers:2606.23883
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    References listed on IDEAS

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    1. Breeden, Douglas T & Litzenberger, Robert H, 1978. "Prices of State-contingent Claims Implicit in Option Prices," The Journal of Business, University of Chicago Press, vol. 51(4), pages 621-651, October.
    2. Jim Gatheral & Antoine Jacquier, 2014. "Arbitrage-free SVI volatility surfaces," Quantitative Finance, Taylor & Francis Journals, vol. 14(1), pages 59-71, January.
    3. Roger W. Lee, 2004. "The Moment Formula For Implied Volatility At Extreme Strikes," Mathematical Finance, Wiley Blackwell, vol. 14(3), pages 469-480, July.
    4. Black, Fischer & Scholes, Myron S, 1973. "The Pricing of Options and Corporate Liabilities," Journal of Political Economy, University of Chicago Press, vol. 81(3), pages 637-654, May-June.
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