Local Volatility Models
A) Look to Describe Volatility Accurately Locally $S$

Local volatility models

A) Look to describe volatility accurately locally in time (i.e., around the current time) but not necessarily in the longer run.
B) Look to describe volatility accurately locally in space (i.e., around the current stock price $S$ ) but not necessarily when stock prices move away from current levels.
C) Look to identify the form $\sigma \left( S _ { t } , t \right)$ for evolution of volatility that will give rise to the currently observed set of option prices.
D) And stochastic volatility models are exactly the same thing.