what's the difference between market implied volatility and implied volatility, how it could be calculated? also what's the quoted implied volatility? thanks.
Where do you find a difference between market implied volatility (IV) and IV? IV is implied from prices. These are usually observed at markets. In my experience this should all be the same.
"quoted implied volatility" is probably an interesting concept. For example with call options comparing prives of calls that are deeper in the money than others and therfore more expensive does not give as more insight than their moneyness.
Quoting implied volatility is a more meaningful measure of richness. A call as well as a put are more expensive the higher the (implied/input) volatility is -everything else being fixed. Summing up: quoting vol instead of prices allows you to compare richness for various levels of moneyness and time to maturity.