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Yes, there is a unique time homogeneous local vol model. This is proven in http://www.sciencedirect.com/science/article/pii/S0304414912002487. There is a slight generalization required that if the option-implied density is zero somewhere, the corresponding local vol is infinite in that region, giving a "gap diffusion". No, there is no nice formula for the ...

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No. In practice the local volatility model has a finite number of slices, so a single slice works as well. Now the problem is : how to compute the time derivative ? Well without adding any information you know that $$C(0,K) = (S_0-K)_+$$ so you could try $$C_\tau = \frac{C(\tau,K)-C(0,K)}{\tau}$$ but it is a very crude approximation. What you may want ...

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The local volatility is just a $\mathbb{R}_+\times[0,T]\mapsto \mathbb{R}_+$ function where $T$ is some time horizon. It is the solution of a simple equation so it expression is written as $\sigma(K,t)$ but here $K$ is essentially a notation to denote a strike value as the Dupire equation relates the function $\sigma$ to vanilla market prices at a given ...

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