In the context of Option pricing model.
Is there a difference between the Dupire Model and the Local volatility model ?
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Dupire model is just one way of generating a local volatility surface from an implied volatility surface. There are many other ways to generate a local volatility surface. One critical aspect of Dupire model is that the input implied volatility (IV) surface should be arbitrage free. If not, you will negative instantaneous variance when generating the local volatility surface. Consequently your local volatility (square root of variance) would be invalid.
There are many ways of smoothing the implied volatility surface. Cubic spline is a good, but does not guarantee an aribtrage free IV surface. Fengler model does. http://www.econbiz.de/archiv1/2008/58072_arbitrage-free_smoothing.pdf Also remember to smooth the IV surface before generating a local volatility surface.
No, if you are referring to the famous Dupire Model (there are others), then they are the same. It is usually referred to as the Local Volatility Model and the Dupire Equation. I would disentagle those with the concept of Local Volatility, which is model independent and a fairly deep result.