Heston vol-of-vol and the VVIX

Abstract: The Heston stochastic volatility model is arguably, the most popular stochastic volatility model used to price and risk manage exotic derivatives. In spite of this, it is not necessarily easy to calibrate to the market and obtain stable exotic option prices with this model. This paper focuses on the vol-of-vol parameter and its relation with the volatility of volatility index (VVIX) level. Four different approaches to estimate the VVIX in the Heston model are presented: two based on the known transition density of the variance, one analytical approximation, and one based on the Heston PDE which computes the value directly out of the underlying SPX500. Finally we explore their use to improve calibration stability.