# Arbitrage free volatility smile and delta

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?