Abstracts

We consider a tractable affine stochastic volatility model that generalizes the seminal Heston (1993) by augmenting it with jumps in the instantaneous variance. This model class can be seen as a particular instance of the double-jump model of Duffie, Pan and Singleton (2000) allowing for jumps only at the variance level. Embedded in this modeling framework is also the case of variance processes described by the pure-jump Ornstein-Uhlenbeck type processes introduced by Barndorff-Nielsen and Shepard (2001).
In this framework, we consider options written on the realized variance and we examine how the particular choice of the distribution of jumps impacts the associated implied volatility smile. In particular, we show that by selecting alternative jump distributions, one obtains fundamentally different shapes of the implied volatility of variance smile - some clearly at odds with the upward-sloping volatility skew observed in variance-derivatives markets. For example, we find that the Gamma distribution leads to a downward-sloping volatility skew, the Inverse Gamma distribution predicts an upward-sloping skew, while the Inverse Gaussian distribution leads to a frown of the implied volatility surface.
Our analysis is based on the classical Tauberian theorems and the more recent results of Lee (2004) and Gulisashvili (2012) on implied volatility asymptotics. Specifically, we derive easy-to-check sufficient conditions for the asymptotic behavior of volatility of variance for small as well as large strikes, given the particular distribution of variance jumps.