Implied Volatility Expansions for VIX Options in Forward Variance Models

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Directly relevant to quantitative trading desks dealing with VIX options and volatility derivatives. The closed-form implied volatility expansions enable fast calibration (up to 131x faster than quadrature in standard Bergomi, 52,750x faster than Monte Carlo in rough Bergomi), which is critical for real-time risk management, hedging, and pricing. Mixed-kernel models capture market VIX skew effectively. The formulas serve as reliable initial guesses for root-finding in extreme regimes.

Single-kernel case is straightforward with closed-form expressions for all terms. Mixed-kernel case requires computing Hermite expansion coefficients via numerical integration, selecting optimal order N via nonlinear least squares, and handling auxiliary log-spot/log-strike variables. The calibration procedure involves sequential maturity-by-maturity optimization. Overall moderate complexity with well-defined steps and available code.

GitHub repository provided (https://github.com/fbourgey/vix-expansion-bergomi) reproducing numerical results. Detailed parameter settings, quadrature methods, and Monte Carlo procedures are specified. Market data date (21 November 2025) is stated but raw data not explicitly provided. All formulas are closed-form and fully specified.