Local Volatility Models

Would you consider Heston, SABR, Black, or LMM as examples? If not, what are some examples of these?

I’m not knowledgeable enough to be sure I’m correct, but I’ll give it a shot. Given this definition: http://en.wikipedia.org/wiki/Local_volatility All of the models, like Heston, SABR, CIR, LMM, that have the ability to scale the dW depending on the levels could be considered to fully or possibly incorporate local volatility. Basically, when the interest rate falls from 10% to 1%, some of these models might multiply dW by square root of S or work with log interest rates so that the volatility scales lower. Similarly, when working with equities if you take the log price changes and fit a Garch model, then when you convert back to arithmetic returns, you will likely see a lower variance as the stock price falls that you wouldn’t see if you only worked with the log numbers.

My understanding is that ‘local volatility models’ are not separate models in and of themselves (as jmh530 points out) and they were used to price exotics while incorporating the skew using a simpler method than was available at the time. What I think you might be getting at is understanding the differences/similarities between local, implied, and stochastic volatilities. HTH.