I am trying to get a good understanding of the different models out there, and thus be able to study hedging errors, and strengths and weaknesses. My understanding of the Local Volatility model in layman's terms is the following:
It requires market prices, from which you will derive a deterministic function for volatility. Volatility is dependent on underlying price and time. You basically derive a surface of implied volatilities by fitting to the market with whatever available option prices there are. Then, in order to price options with strikes/expirations for which the market does not have, you use the surface, and plug that implied volatility into the Black Scholes equation to derive an option price. Is this correct?
If the above is correct, I was wondering if there was an even simpler model where volatility is a deterministic function of underlying price only. Such a model can only be used to price a single expiration of options. It would take current market option prices, fit an implied volatility curve to those prices (for example fit using a cubic spline). For strikes without market prices, it would use the implied volatility interpolated/extrapolated by the curve and return an option price.
If so, what is this model called, and where can I find more literature on what curves should be used to fit option prices (cubic spline, parabola, etc...), hedging issues, etc...