If we have a (parametric) volatility surface which has arbitrage, then consider the delta of the options, i.e. $N(d_1)$, where

$d_1 = \frac{1}{\sqrt{t}\sigma(K)}\log(F/K) + \frac{1}{2}\sigma(K)\sqrt{t}$

Are the following conditions equivalent:

  1. For a fixed $F$, $d_1$ is a strictly decreasing function of $K$ (hence injective)

  2. The volatility surface is arbitrage free

It is quite clear to me that: if there are two strikes with the same value of delta, then cannot be arbitrage free (except perhaps when we have a region where the probability density is 0), is the converse true?


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