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The local vol model has exactly enough freedom to match the individual densities $X_t.$ There is no additional freedom in the local vol model to match even a joint density for a pair of times $(X_t,X_s).$ When you ask about the joint density across the continuum of times $t \in [0,T]$ it is pretty easy to show that any local vol model differs from any ...

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A general model (with continuous paths) can be written $$\frac{dS_t}{S_t} = r_t dt + \sigma_t dW_t^S$$ where the short rate $r_t$ and spot volatility $\sigma_t$ are random processes. In the Black-Scholes model both $r$ and $\sigma$ are deterministic functions (actually constant). This produces a flat smile for any $$C(t,S;T,K) = ... 3 Yes, there is a unique time homogeneous local vol model. This is proven in http://www.sciencedirect.com/science/article/pii/S0304414912002487. There is a slight generalization required that if the option-implied density is zero somewhere, the corresponding local vol is infinite in that region, giving a "gap diffusion". No, there is no nice formula for the ... 2 One does not estimate the local volatility at a given T and K. Instead, Dupire's formula actually gives \sigma(T,K) for all T and K.$$ \sigma^2(t_0,S_0;T,K)= \frac{\frac{\partial C}{\partial T} + (r - q)K \frac{\partial C}{\partial K} + qC}{\frac{1}{2} K^2 \frac{\partial^2C}{\partial K^2}} where C(t_0,S_0;T,K) are the call prices for ... 2 I'll address your questions in order: 1a) For TSRV constructed using high frequency returns from NYSE market open to market close on a single day, the output should be numbers on the order of magnitude of 1e-4 to 1e-5. In other words, your numbers look about right. I got these number from calculating TSRV for IBM data myself using Kevin Sheppard's MatLab ... 1 Local volatility trats volatility as a function of price (St) and time (t), considering volatility constant. It means that you are building a volatility surface depending on price and time. What you are doing is given a certain value of volatility, what will be the price in Xdays long, or viceversa. If you are dong market making with plain vanilla ... 1 Instead of just considering a parallel shift of the whole volatility surface, you can decompose the surface into maturities/strikes domains, so called buckets and consider Vega buckets which are sensitivities wrt to bumps of each of these domains. The vol smile is often inter/extra-polated using a model calibrated to market prices, e.g. the SABR model or ... 1 I know one article (download) that explaining how to calculate local vol surface from IV surface and also chapter 18 of this book is very good In this context. However you know that Dupire’s (1994) formula for local volatility is \begin{align} \sigma_L(k,T)=\sqrt\frac{\frac{\partial C}{\partial T}}{\frac{1}{2}K^2\frac{\partial^2 C}{\partial K^2}} \end{align} ... 1 Here "dynamics" means the assumed future behaviour of the spot process, namely that it follows the SDE dS/S = r dt + \sigma_{loc}(S,t) dW_t .$$There are various ways to see that these dynamics are unrealistic. One is to look for time homogeneity. In normal cases, you expect the market to follow the same rules in one week and in one year from today. ... 1 Loosely speaking: Local volatility is the instantaneous volatility after time T if the spot is S at that time. Implied volatility is the expected integrated volatility from today up to time T if the spot ends up at S at that time. 1 You should not expect the local vol to be equal to the implied vol except in the trivial case where both are constant (Black-Scholes model). I haven't read the Derman articles but it is quite clear using Dupire's formula (see Gatheral's book for example). Local volatility can be computed in terms of call prices using Dupire's formula$$ \sigma^2(T,K) = ...

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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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