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

Integration Issues, Parameter Effects, and Variance Modeling

Abstract

In this chapter, we investigate several issues around the Heston model. First, following Bakshi and Madan (2000), we show that the Heston call price can be expressed in terms of a single characteristic function. It is well–known that the integrand for the call price can sometimes show high oscillation, can dampen very slowly along the integration axis, and can show discontinuities. All of these problems can introduce inaccuracies in numerical integration. The “Little Trap” formulation of Albrecher et al. (2007) provides an easy fix to many of these problems. Next, we examine the effects of the Heston parameters on implied volatilities extracted from option prices generated with the Heston model. Borrowing from Gatheral (2006), we examine how the fair strike of a variance swap can be derived under the model and present approximations to local volatility and implied volatility from the model. Finally, we examine moment explosions derived by Andersen and Piterbarg (2007) and bounds on implied volatility of Lee (2004b).

REMARKS ON THE CHARACTERISTIC FUNCTIONS

In Chapter 1, it was shown that the in-the-money probabilities P₁ and P₂ are obtained by the inverse Fourier transform of the characteristic functions f₁ and f₂

This form of inversion is due to Gil-Pelaez (1951) and will be derived in Chapter 3. It makes sense ...