with current European option prices is known as the local volatility func- tion. It is unlikely that Dupire, Derman and Kani ever thought of local volatil-. So by construction, the local volatility model matches the market prices of all European options since the market exhibits a strike-dependent implied volatility. Local Volatility means that the value of the vol depends on time (and spot) The Dupire Local Vol is a “non-parametric” model which means that it does not.
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When such volatility has a randomness of its own—often described by a different equation driven by a different W —the model volatjlity is called a stochastic volatility model. This page was last edited on 9 Decemberat Since in local volatility models the volatility is a deterministic function of the random stock price, local volatility models voaltility not very well used to price cliquet options or forward start optionswhose values depend specifically on the random nature of volatility itself.
In mathematical financethe asset S t that underlies a financial derivativeis typically assumed to follow a stochastic differential equation of the form. The concept of a local volatility was developed when Bruno Dupire  and Emanuel Derman and Iraj Kani  noted that there is a unique diffusion process consistent with the risk neutral densities derived from the market prices of European options.
options – pricing using dupire local volatility model – Quantitative Finance Stack Exchange
Local volatility models are nonetheless useful in the formulation of stochastic volatility models. The local volatility model is a useful simplification of the stochastic volatility model. LocalVolatility 5, 3 13 Could you guys clarify? I thought I could get away with it.
You write that since volatiliyy is only one price process, there is one fixed implied standard deviation per maturity. I am reading about Dupire local volatility model and have a rough idea of the derivation. Time-invariant local volatilities are supposedly inconsistent with the dynamics of the equity index lpcal volatility surface,   but see Crepey, S The general non-parametric approach by Dupire is however problematic, as one needs to arbitrarily pre-interpolate the input implied volatility surface before applying the method.
Dupir from ” https: This model is used to calculate exotic option valuations which are consistent with observed prices of vanilla options.
You then argue that consequently, we can’t replicate the prices of all European options since the market exhibits a strike-dependent implied volatility. While your statement is correct, your conclusion is not.
Gordon – thanks I agree. If they have exactly the same diffusion, the probability volatilitty function will be the same and hence the realized volatility will be exactly the same for all options, but market data differentiate volatility between strike and option price.
Numerous calibration methods are developed to deal with the McKean-Vlasov processes including the most used particle and bin approach. Derman and Kani described and implemented a local volatility function to model instantaneous volatility. But I can’t reconcile the local volatility surface to pricing using geometric brownian motion process.
And when such volatility is merely a function of the current asset level S t and of time twe have a local volarility model. How does my model know that I changed my strike? The tree successfully produced option valuations consistent with all market prices across strikes and rupire. From Wikipedia, the free encyclopedia. Email Required, but never shown.