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On Dupire local volatility

The Dupire equation is surely one of the biggest technological discovery in Equity derivatives; it has really changed how the market analyse and risk manage structure products. Even though there are many deficiencies on this model, it is being used in an industrial way. By industrial, I really mean like a factory. Even though it may not take into account many risk associated with the products, The Local Volatility Model is the benchmark for pricing any structured products. The local volatility is also the most widely used model for risk management. But why? Because it is so easy to implement a local vol and the simulations are so precise. Nevertheless the model has many deficiencies and is not supposed to price accurately non European products. But when it comes to implementation, the standard Dupire formula is not the best choice. Jim Gatheral implementation of the local vol is surely the most practical one. For the simple reason that we tend to see prices as function of on implied volatilities rather than actual vanilla prices. In Gatheral formula the local vol is a function of the implied volatility rather than call/put prices. This is much practical.

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