VIX Option Pricing under Hybrid Hawkes Jump-Diffusion with Stochastic Rates

This paper studies VIX option pricing when interest rates and volatility are random. It proposes an affine framework based on a mixed jump-diffusion model. This model uses a Vasicek random interest rate process and a jump component for random volatility. This helps capture interest rate risk, volatility risk, and jump clustering in financial markets. Under a consistent pricing framework, we build a combined system. This system includes Hawkes-type price jumps, volatility jumps, and random interest rates. We then derive the related generalized characteristic function. The pricing problem is solved using the Fourier Cosine Series Expansion (COS) method. Compared to traditional models, this extended model lowers the root mean square error in VIX option pricing. It performs especially well during periods of monetary policy changes and financial market stress. This research offers a new theoretical framework and empirical tools for pricing volatility derivatives in complex market settings.