Monotonicity of Normalized Implied-Volatility Coordinates under No-Arbitrage (opens in new tab)

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 requir...