Implied Volatility Expansions for VIX Options in Forward Variance Models

2026-04-27

AI Summary

This study addresses the absence of efficient and accurate analytical approximations for VIX option implied volatility, which has traditionally necessitated time-consuming numerical root-finding in model calibration. Building upon forward variance models—including the standard, rough Bergomi, and hybrid specifications—the authors derive, for the first time, closed-form asymptotic expansions of implied volatility with explicit correction terms by leveraging weak approximation and asymptotic expansion techniques. This approach entirely circumvents numerical root-finding and demonstrates high accuracy and exceptional computational efficiency across multiple model settings. Consequently, it substantially enhances both the speed and numerical stability of VIX option calibration.

Abstract

We develop closed-form expansions for the implied volatility of VIX options within the class of forward variance models. Our approach builds on weak-approximation techniques for VIX option prices and yields explicit implied volatility expansions with computable correction terms. The resulting formulas enable fast and accurate calibration without requiring numerical root-finding. We illustrate the performance of the proposed expansions in both standard and rough Bergomi-type models, as well as in mixed specifications, and demonstrate their accuracy through numerical experiments.

Problem

• Implied Volatility
• VIX Options
• Forward Variance Models
• Calibration
• Numerical Root-Finding

Innovation

• implied volatility expansion
• forward variance models
• VIX options
• weak approximation
• rough Bergomi model