What Is the Relationship between the Volatility Surface and the Concept of ‘Local Volatility’?

The local volatility model is a refinement used to construct the volatility surface. It assumes that the volatility of the underlying asset is a deterministic function of both the current asset price and time.

The local volatility surface is a theoretical construct that, when plugged into a pricing model, is guaranteed to reproduce the market-observed option prices (the implied volatility surface).