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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Mathematical Finance – Bachelier Congress The payoff of a European contingent claim only depends on the asset price at maturity.
Could you look at it?
Thanks for the explanation, it was helpful. In mathematical financethe asset S t that underlies a financial derivativeis typically assumed to follow a stochastic differential equation of the form.
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.
The idea behind this is as follows: Time-invariant local volatilities are supposedly inconsistent with the dynamics of the equity index implied volatility surface,   but see Crepey, S The local volatility model is a useful simplification of the stochastic volatility model.
In the simplest model i. dupide
Ok guys, I think I understand it now. So by construction, the local volatility model matches the market prices of all European contingent claims without the model dynamics depending on what strike or payoff function you are interested in. Here is how I understand your first edit: Alternative parametric approaches have been proposed, notably the highly tractable mixture dynamical local volatility models by Damiano Brigo and Fabio Mercurio.
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Since in local volatility models the volatility is a deterministic function of the random stock price, local volatility models are not very well used to price cliquet options or forward start optionswhose values depend specifically on the random nature of volatility itself. If I have a matrix of option prices by strikes and maturities then I should bolatility some 3D function to this data.
Local volatility models are useful in any options volaility in which the underlying’s volatility is predominantly a function of the level of the underlying, interest-rate derivatives for example. International Journal of Theoretical and Applied Finance.
Archived from the original PDF on Consequently any two models whose implied probability densities agree for the maturity of interest agree on the prices of all European contingent claims. In fact the pdf will be tlhe same but it will allow to replicate implied vol surface. Email Required, but never shown. Lcoal am reading about Dupire local volatility model and have a rough idea of the derivation.
Local volatility – Wikipedia
Duppire key continuous -time equations used in local volatility models were developed by Bruno Dupire in And when such volatility is merely a function of the current asset level S t and of time twe have a local volatility model.
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