In the classical Black–Scholes model volatility is constant. However, in reality options with different strikes need different volatilities to match their market prices. This leads to the notion of implied volatility, which is the unique volatility that must be inserted into the Black–Scholes formula to produce the correct market price. For interest rate derivatives the range of implied volatilities is typically referred to as the volatility smile or skew. The reason for the terms ‘smile’ or ‘skew’ is that the graph of the implied volatility as a function of the strike price is typically smile shaped or downward sloping.

In order to accurately value and manage the risk of a portfolio of options, a model should be capable of reproducing the volatility smile observed in the market. In reality this is difficult to achieve. In this chapter we discuss an extension of the Black–Scholes approach referred to as local volatility models (LVM), where volatility is a function of time and the underlying price or rate. A special type of LVM known as the constant elasticity of variance (CEV) model is an extension of the LMM capable of capturing the skew. Examples of tractable local volatility models such as the normal model, CEV and displaced-diffusion models will be analysed.

Local volatility models are an improvement on Black's approach, but are unable to reproduce smile-shaped curves. Stochastic volatility models were introduced to overcome this limitation. In these models the volatility of the underlying price or rate is itself a stochastic process driven by its own Brownian motion. The stochastic volatility model touched upon in this chapter is known as SABR. It is a model of a single forward interest rate (either the forward LIBOR rate or the forward swap rate), which has become one of the standard approaches when it comes to modelling the volatility smile.