Three questions regarding local volatility implementation (based on the Andreasen, Huge article "Volatility interpolation")

Ok, I did some investigations, asked around and got some answers to most of my questions. Since it might be of general interest for other people I present my findings here.

1. How to transfer market data (option prices) from the real world to the simplified zero rate economy (used in the article by J. Andreasen, B. Huge) back and forth. We have an underlying asset $S(t)$ which satisfies $$ frac{dS(t)}{S(t)} = (r(t) - d(t)) , dt + sigma(t, S) , dW(t), quad S(0) = S_0, $$ where $W(t)$ is a Brownian motion, $r(t)$ is the short rate, $d(t)$ is the dividend yield and $sigma(t, S)$ is the local volatility surface (unknown to us so far). Introduce the notation $$ mu(t) = r(t) - d(t), quad F(0,t) = S_0 cdot exp(int_0^t mu(s) , ds)$$ for the drift and the forward value at time $t$ seen from time $0$. Now introduce the variable $$ X(t) = frac{S(t)}{F(0,t)}. $$ Using Ito's lemma we get (I skip those calculations. It is just a standard application of Ito's lemma for a quotient) $$ dX(t) = X(t) cdot sigma(t,S) , dW(t), $$ where $W(t)$ is the same Brownian motion as the one used for $S$. We write this as $$ dX(t) = X(t) cdot hat{sigma}(t,X(t)) , dW(t), $$ where $$ hat{sigma}(t,X(t)) = sigma(t, X(t) cdot F(0,t)). $$ So once we have calculated $hat{sigma}(t,X(t))$ we can get $sigma(t,S)$ by going backwards $$ sigma(t,S(t)) = hat{sigma}(t, S(t) / F(0,t)). $$ There is one problem remaining here and that is that we want option prices for the variable $X(t)$ when the market prices are given for the variable $S(t)$. This is easily obtained by a rescaling though. Let us write the expression for the Black (undiscounted) call price as the usual expected value $$ begin{eqnarray*}E[textrm{max}(X(t) - K,0)] &=& E[textrm{max}(frac{S(t)}{F(0,t)} - K, , 0)] \ &=& frac{1}{F(0,t)}E[textrm{max}(S(t) - K cdot F(0,t), , 0)]. end{eqnarray*} $$ The expectations $E[textrm{max}(S(t) - K cdot F(0,t), , 0)]$ are just Black call option prices for the underlying $S(t)$ with strike $K cdot F(0,t)$, which are observable on the market. So using this relation we can go back and forth between option prices in the real market and the zero rate market.

So by using this we can make or setup in the zero rate economy, calculate the local vol surface for $X(t)$ and then go back to $S(t)$ using the relations above.

2. This rough one step proxy of the local volatility surface, was as I expected, not a good candidate for the real local vol surface and it does not satisfy the Dupire equation. One should one use construct and use it as a tool to obtain a denser option price grid which we then can use to construct the real sample points of the local volatility surface (using Dupires equation).

3. The final question of interpolation of the dense grid of points on the local vol surface is still unknown to me. Or course it will depend on the application where we use the constructed local vol surface. If different kids of greeks need to be computed we need a smooth interpolation of some kind. But if we only need rough values, then I guess linear interpolation might be good enough.