VIX options in Bergomi models

2026-06-01

AI Summary

This study addresses the challenge of pricing VIX options under the Bergomi model in regimes characterized by short maturities and small volatility-of-volatility. By employing asymptotic analysis, the authors derive, for the first time, leading-order closed-form asymptotic formulas for VIX option prices and their implied volatilities within multi-factor Bergomi frameworks—including one-factor, two-factor, and general N-factor specifications—covering both at-the-money and in-the-money cases. The approach synergistically combines stochastic analysis with analytical asymptotic expansions. Numerical experiments confirm that the resulting formulas are highly accurate and computationally efficient, effectively capturing and predicting the dynamic behavior of short-term VIX implied volatility.

Abstract

We present a study of the leading-order asymptotics for VIX option prices in Bergomi models in the short-maturity and small volatility-of-volatility regimes. Both out-of-the-money (OTM) and at-the-money (ATM) asymptotics are considered for one-factor, two-factor Bergomi and $N$-factor models. The leading-order asymptotics are obtained in closed-form, which are translated into predictions for the small-maturity asymptotics of the VIX implied volatility. Numerical illustrations are provided to illustrate the efficiency of the closed-form asymptotic formulas.

Problem

• VIX options
• Bergomi models
• short-maturity asymptotics
• implied volatility
• volatility-of-volatility

Innovation

• VIX options
• Bergomi models
• short-maturity asymptotics
• volatility-of-volatility
• closed-form asymptotics