Heston vol-of-vol and the VVIX

ArXiv ID: 2512.19611 “View on arXiv”

Authors: Jherek Healy

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.

Keywords: Heston Model, Volatility of Volatility (Vol-of-Vol), Calibration, VVIX, SPX 500, Equity Derivatives

Complexity vs Empirical Score

• Math Complexity: 8.0/10
• Empirical Rigor: 3.0/10
• Quadrant: Lab Rats
• Why: The paper is heavily mathematical, featuring stochastic calculus, PDEs, characteristic functions, and numerical integration for option pricing. However, it lacks backtesting, real market data analysis, or implementation code, focusing instead on theoretical approximations and model calibration stability.

  flowchart TD
    A["Research Goal<br>Estimate VVIX using Heston<br>Improve Calibration Stability"] --> B["Methodology<br>Four Estimation Approaches"]
    B --> C["Data Inputs<br>SPX500 Options<br>Market VVIX Index"]
    
    C --> D{"Computational Process"}
    
    D --> E["Method 1 & 2<br>Transition Density<br>of Variance"]
    D --> F["Method 3<br>Analytical<br>Approximation"]
    D --> G["Method 4<br>Heston PDE<br>Direct Computation"]
    
    E --> H["Key Findings<br>Calibration Stability<br>Improved via Vol-of-Vol<br>VVIX Correlation Analysis"]
    F --> H
    G --> H