Implied volatility as price transform

1. Implied volatility

The way I understand it, traders often think of implied volatility as a transformed price. So in a way, the Black Scholes model is considered a 'model-free' blackbox that takes a market price and returns an 'implied volatility'. A trader might very well say 'I bought AstraZeneca at 20 vol'. Why is that they prefer implied vol as a price?

2. Implied vol surface

When you devise a new stochastic volatility model, you want it to match the empirical volatility surface as closely as possible (thereby matching the price surface as closely as possible because there is this one to one relationship between implied volatility and price). Why do quants prefer the implied vol surface instead of bespoke price surface?

Thanks!

Answer

It is difficult to gain intuition by just looking at the price surface, and it is also easier to calibrate models on the volatility surface rather than on the price surface because with the later you are dealing with numbers of very different sizes (depending on the moneyness and maturity) which is not good for minimization algorithms. However low and high strikes extrapolation of the volatility surface are often specified in terms of parametric functions for the price.

A useful quantity directly related to the price surface is its second derivative w.r.t strike because it gives you the (discounted) implied density of the underlying asset.