A local volatility model such as introduced in the previous chapter has the advantage of being able to fit the market for European vanillas, with little extra computational cost compared with similar numerical techniques for a term-structure Black–Scholes model (both being one-dimensional models). On the other hand, a purely local volatility model will underestimate the forward smile and skew in the market, and as such is a poor candidate for a model designed to match market prices for exotic options. Further, a historical time-series analysis of any FX currency pair immediately shows that the realised historical volatility is far from constant. For these reasons, various stochastic volatility models have been proposed in the literature, many of which have gained some acceptance in the practitioner community.
In this chapter, I will attempt to describe some of these models and, perhaps most crucially from the industry perspective, introduce some of the numerical techniques that can be used for their calibration and for pricing (some of the discussion will necessarily have to wrap over into the next chapter). A model that is difficult to calibrate is of limited use indeed in practical applications.
To begin, we consider a much simpler candidate model than a full continuous time stochastic volatility model, but which shares surprisingly many features – an uncertain volatility model.
6.2 UNCERTAIN VOLATILITY
We already have prices ...