I am a beginner in Finance and I get confused by the statement "Black-Scholes model implies flat implied volatility plots"
Here is one form of this statement: (Dan Stefanica, 150 most frequently asked questions on Quant Interviews, 3.3.12)
"On the same asset, prices for options with multiple strikes and maturities are quoted and implied volatilities can be computed for each of these options. If the price of the asset had a lognormal distribution - as assumed in the Black-Scholes model - then the resulting plots of implied volatility vs Strike would be flat"
But surely the volatility is assumed constant in the derivation of BS in the first place? It is an assumption, not a consequence of "the price of the asset having a lognormal distribution" ?
Here is another example of my confusion , this time from this paper http://www.columbia.edu%2F~mh2078%2FBlackScholesCtsTime.pdf
As i understand it, in the Black-Scholes model, we fix K,T, we assume $\exists \sigma=\sigma(K,T)$ for those K,T and we come up with the BS formula above
There is nothing in the BS derivation that assumes $\exists \sigma, \forall K,T ....$ ?