🤖 AI Summary
This study addresses the monotonicity of normalized implied volatility coordinates within a finite quoted option chain under no-arbitrage conditions and derives model-independent variance identities. Relying solely on static no-arbitrage assumptions—including discrete strike comparisons, convexity, monotonicity, and call-put parity—the authors provide the first purely discrete proof of monotonicity for normalized coordinates in both the Black–Scholes and Bachelier frameworks, without requiring continuous quotes or differentiability. The main contributions are twofold: (1) a rigorous discrete verification of monotonicity in these two canonical implied volatility models, and (2) the introduction of a normal variance identity that serves as the natural counterpart to Fukasawa’s log-normal result, thereby establishing a model-independent theoretical foundation for volatility derivatives.
📝 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 \((F-K)/σ_N(K)\) 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.