Bruno Dupire: Functional Ito Calculus and Risk Management

We expose briefly the Functional Ito Calculus, which gives a natural setting for defining the Greeks for path dependent options and gives a generalized PDE for the price of path dependent options, even in the case of non Markov dynamics. It leads to a variational calculus on volatility surfaces and a fine decomposition of the volatility risk as well to links with super-replication strategies. We examine a few practical examples and analyze the ability to hedge (or not) some popular structures.