In Derman/Kani/Zou paper about local vol they rebuilt a local vol surface from an implied vol surface. Each implied volatility depicted in the surface of the "implied Vol" is the Black-Scholes implied volatility. Bascially the volatility you have to enter into the Black-Scholes formula to have its theoretical option value match the option’s market price.
Now, in the local vol model, they extract the market’s consensus for future local volatilities σ(S,t), as a function of future index level S and time t, from the spectrum of available options prices as quoted by their implied Black-Scholes volatilities. The model fits a consistent implied tree to these quoted option prices, and then allows the calculation of the fair values and exposures of all (standard and exotic) options, consistent with all the initial liquid options prices.
Question: If I compare the graphs in the paper of the implied vol surface and the local vol surface why is it so different? The local vol should be consistent with the liquid option prices. i.e. Term 1.0, 550 level: implied surface 13.5% vol, local vol surface 18% vol. If there is a liquid strike in the market they should have the same vol, am i right?