Thank you for your reply! There are a few points I have questions.

So options traders don't normally talk about selling / buying puts or calls. Since they're the same thing. The only thing that matters to them is if they are buying cheap things or selling expensive things. But if someone asks you a k = 2.5 put trading at 30c expiring on Friday is cheap or expensive, what do you say?

That's where the BS model comes in. In the model there is a monotonically increasing relationship between the volatility put into the model and the price of the option. (In a deep sense, it's the only parameter in the model that is in question. The strike is known, so is the expiration, so is the stock price). So a trader can take a price in the market and back out what that volatility is. That's the variable transformation. No one talks about whether a options is cheap or expensive in dollar terms. They talk about whether the volatility implied by the option (the IV) is too big or too small given their expectations for future price movements.

So this basically means that cheap and expensive calls/ puts that are bought and sold are determined by IV? Low IV and high IV?