Heston stochastic vol-of-vol model for joint calibration of VIX and S&P 500 options

A parsimonious generalization of the Heston model is proposed where the volatility-of-volatility is assumed to be stochastic. We follow the perturbation technique of Fouque et al [Multiscale Stochastic Volatility for Equity, Interest Rate, and Credit Derivatives, 2011, Cambridge University Press] to derive a first-order approximation of the price of options on a stock and its volatility index. This approximation is given by Heston’s quasi-closed formula and some of its Greeks. It can be efficiently calculated since it requires to compute only Fourier integrals and the solution of simple ODE systems. We exemplify the calibration of the model with S&P 500 and VIX data.