Consistent Pricing of Spot Options and Forward Start Options

The idea of this paper is to present how we can use a specific form of local volatility, namely a forward start local vol, that has already been evoked in different sources, and use it in order to fit a specific Forward Start Smile. A local formula, i.e. a formula similar to a Dupire formula, is available for these models and will be used to fit the Forward Smile Market. These models are therefore particularly adapted for pricing Cliquet Products or generally Forward Starting Products with given frequencies. In particular, it is possible to fit different options like Vanilla options and Forward Start Options using them. We will also consider an extension of these models adapted for the pricing of lookback options