VIX options in Bergomi models

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Highly relevant for quantitative trading desks dealing with volatility derivatives. VIX options are among the most liquid volatility instruments on CBOE. The closed-form asymptotic formulas enable fast pricing and calibration of VIX options in Bergomi models, which are widely used in practice for volatility modeling. The explicit ATM level and skew predictions (Propositions 15, 22) are directly applicable to trading strategies involving VIX options. The paper provides practical insights into the sign and magnitude of VIX skew as a function of model parameters (correlation ρ, mean reversion speeds k_i, mixing weights θ_i). However, the results are asymptotic and require careful application outside the valid regimes.

The main results are closed-form analytical expressions, making them relatively straightforward to implement. The key computational steps involve: (1) evaluating integrals F²_{a,i}, F²_{b,ij} over the variance curve, (2) computing the determinant D and auxiliary quantity G, (3) solving the constrained optimization for the rate function (requires numerical root-finding for OTM options), and (4) computing the ATM level and skew from Propositions 15/22. The Monte Carlo validation requires simulating correlated Ornstein-Uhlenbeck processes and applying the trapezoidal rule for VIX computation. The N-factor generalization requires matrix inversion of the covariance matrix Σ_T.

The paper provides fully closed-form analytical formulas for all main results (Theorems 1-12, Propositions 13, 15, 19, 21, 22). All technical proofs are included in Appendix B. Numerical parameters for the two-factor Bergomi model are explicitly stated (ξ₀=0.1, k₁=7.54, k₂=0.24, three weight scenarios). Monte Carlo simulation methodology is described with exact discretization scheme. However, no code repository is provided, and the numerical integration for rate functions requires implementation.