Heston Stochastic Vol-of-Vol Model for Joint Calibration of VIX and S&P 500 Options

摘要

A parsimonious generalization of the Heston model is proposed where thevolatility-of-volatility is assumed to be stochastic. We follow theperturbation technique of Fouque et al (2011, CUP) to derive a first orderapproximation of the price of options on a stock and its volatility index. Thisapproximation is given by Heston's quasi-closed formula and some of its Greeks.It can be very efficiently calculated since it requires to compute only Fourierintegrals and the solution of simple ODE systems. We exemplify the calibrationof the model with S&P 500 and VIX data.