The Heston Model and its Extensions in Matlab and C#, + Website

The Double Heston Model

Abstract

The original Heston (1993) model is not always able to fit the implied volatility smile very well, especially at short maturities. One remedy is to add additional parameters, which allows the model to be more flexible. This can be accomplished by allowing the parameters to be time dependent, as was illustrated by the methods covered in Chapter 9. Another approach is to enrich the variance process. One simple way to do this is to specify a two–factor structure for the volatility. This is the approach of Christoffersen et al. (2009) in their double Heston model, which we present in this chapter.

In this chapter, we first present the multi–dimensional Feynman–Kac theorem, and we show that the double Heston model is affine in the sense of Duffie et al. (2000). These results are used to obtain the characteristic function of the double Heston model. We then show how the double Heston model is a simple extension of its univariate counterpart, and how its extra parameters allow for a better fit of the implied volatility smile at multiple maturities. We also show that the “Little Trap” formulation of the characteristic function of Albrecher et al. (2007) can be applied to the double Heston model. Finally, we present different simulation schemes that can be applied to the double Heston model. Most of these schemes are extensions of univariate schemes presented ...