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

## TL;DR

This paper introduces a stochastic volatility-of-volatility extension to the Heston model, enabling efficient joint calibration of S&P 500 options and VIX, with practical approximation formulas derived via perturbation techniques.

## Contribution

It proposes a new stochastic volatility-of-volatility model based on the Heston framework, with a first-order approximation for option pricing that simplifies calibration.

## Key findings

- Model accurately fits S&P 500 and VIX data
- Approximation formulas are computationally efficient
- Calibration process is streamlined and effective

## Abstract

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 (2011, CUP) 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 very 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.