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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 $$C(t,S;T,K) = ... 3 You can view the price of an option as the cost to dynamically replicate it. The more volatility, the more costs you will have trading the underlying to keep your delta equal to 0 (I'm assuming you sold the option, hence a negative gamma position). So, if at any spot, any date your local vol is above 0.194, rebalancing the portfolio will be constantly more ... 2 One does not estimate the local volatility at a given T and K. Instead, Dupire's formula actually gives \sigma(T,K) for all T and K.$$ \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}} $$where C(t_0,S_0;T,K) are the call prices for ... 2 First, please make sure that when you resimulate sample paths, you are keeping your underlying random samples constant, as in this answer. For your delta, vega and rho there is some ambiguity in the definition of the greeks. Consider the simple case of delta in the presence of a skew \sigma(K/S), and say that the underlying price right now is S_0. We ... 2 I'll address your questions in order: 1a) For TSRV constructed using high frequency returns from NYSE market open to market close on a single day, the output should be numbers on the order of magnitude of 1e-4 to 1e-5. In other words, your numbers look about right. I got these number from calculating TSRV for IBM data myself using Kevin Sheppard's MatLab ... 1 Here "dynamics" means the assumed future behaviour of the spot process, namely that it follows the SDE$$ dS/S = r dt + \sigma_{loc}(S,t) dW_t .$$There are various ways to see that these dynamics are unrealistic. One is to look for time homogeneity. In normal cases, you expect the market to follow the same rules in one week and in one year from today. ... 1 Loosely speaking: Local volatility is the instantaneous volatility after time T if the spot is S at that time. Implied volatility is the expected integrated volatility from today up to time T if the spot ends up at S at that time. 1 The local volatility is just a \mathbb{R}_+\times[0,T]\mapsto \mathbb{R}_+ function where T is some time horizon. It is the solution of a simple equation so it expression is written as \sigma(K,t) but here K is essentially a notation to denote a strike value as the Dupire equation relates the function \sigma to vanilla market prices at a given ... 1 Yes, there is a unique time homogeneous local vol model. This is proven in http://www.sciencedirect.com/science/article/pii/S0304414912002487. There is a slight generalization required that if the option-implied density is zero somewhere, the corresponding local vol is infinite in that region, giving a "gap diffusion". No, there is no nice formula for the ... 1 No. In practice the local volatility model has a finite number of slices, so a single slice works as well. Now the problem is : how to compute the time derivative ? Well without adding any information you know that$$ 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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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 ...

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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.

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