This might be a simple question, but I couldn't find the answer anywhere: is there a separate Volatility smile (and surface) based on Calls and a separate Volatility smile (surface) based on Puts? Or is there simply one Volatility smile (and one surface) based on the most liquid options, mixing calls and puts?
I would have thought that if you plot strike on the x-axis and IV on the y-axis, then:
(i) To the left of ATM strike, you'd use OTM puts
(ii) To the right of ATM strike, you'd use OTM calls
But various pictures I have found online simply show the smile as if it can be constructed just from calls or just from puts, i.e.:
The problem I see with constructing the surface just based on calls, or just based on puts, is that ITM options may not be liquid enough or traded at all.
Last but not least: say that you'd use OTM puts to the left of ATM, and OTM calls to the right of ATM: what about the ATM point? What if the ATM call IV is different to the ATM put IV?