"Black-Scholes model implies flat implied volatility plots"?

Question

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 $\exists \sigma, \forall K,T ....$ ?

Answer 1

Regarding your second question: Remember that Black/Scholes start by postulating a stochastic model for the dynamics of the underlying asset - a geometric Brownian motion with a constant diffusion coefficient $\sigma$. This asset price process should be the same no matter what option you want to value based on it. Saying that you allow for different values for $\sigma$ for different strikes or maturities essentially means that you use a different model for each vanilla option, which is inconsistent and for sure not what they initially had in mind.

Regarding your first question: The implied volatility $\hat{\sigma}(T, K)$ is the value for the constant diffusion coefficient that results in a Black/Scholes price equal to some observed market price. So obviously, if market prices for all $(T, K)$ were computed using a Black/Scholes model with a common volatility $\sigma^*$, then you would recover this same value as the implied volatility $\hat{\sigma}(T, K) = \sigma^*$ and thus a flat volatility surface. Similarly, this fixed $\sigma^*$ results in log-normally distributed prices. So if implied densities for the terminal spot price of a given maturity were log-normally distributed, then the corresponding implied volatility smile would be flat.