SABR Model

Abstract

We have introduced in Chapter 4 the normal and lognormal models. They play an important role as they are intuitive, simple and their parameters can be adjusted quickly to obtain a price in agreement with the market. However, these simple models cannot be calibrated to more than one volatility per expiry. The authors of two papers, Derman and Kani (1994) and Dupire (1994), proposed a model, where the volatility is dependent on the current state of the underlying process. These so-called local volatility models could be calibrated, with some extra effort, to the entire volatility surface. One shortcoming of the local volatility models is that they predict a dynamic behaviour of the smile and the skew (the smile moves in the opposite direction as the underlying) that is different from what is observed in the market (where the smile moves in the same direction as the underlying).