I hope someone can clarify my ideas about the constant implied volatility in the classical Black Scholes framework.
As well known, market practitioners quote the prices of vanilla call and put options in terms of implied volatilities. For inputs $K$, $S$, $r$, $T$ and the price of the option $V$, one can determine the implied volatility $σ$ such that
When the market quoted implied volatilities are plotted against different strike prices for a fixed maturity $T$, the graph would tipically exhibit a 'smile' shape and hence the name volatility smile.
Theory says that this implies a deficiency in the Black Scholes model since it assumes a constant volatility parameter, not depending on $K$ nor $T$. Hence the volatility smile would be flat.
Here my ideas get confused. Assuming that $S$, $r$ and $T$ remain constant, for a fixed market price $V$ of a vanilla option the implied volatility will vary depending on the value of strike $K$ under the Black Scholes model (1). Hence, if the implied volatility is plotted against different strikes for a fixed $V$ it will indeed show a smile behaviour, which is in contrast to what theory states.
Furthermore, do the market quoted implied volatilities that form the volatility smile according to the theory correspond to a fixed vanilla option price $V$ and with varying $K$?
I think I am making a mistake in my reasoning but I do not understand where. I would be glad if someone can point me in the right thinking direction.
Thanks in advance.