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An Ornstein–Uhlenbeck Model with the Stochastic Volatility Process and Tempered Stable Process for VIX Option Pricing

To effectively fit the dynamics and structure of frequent small jumps and sparse large jumps in the VIX time series, we introduce the tempered stable process (the CTS process and CGMY process) into the Ornstein–Uhlenbeck (OU) stochastic volatility model to build an OU model with the stochastic volatility process and tempered stable process. Based on two different assumptions for the underlying assets, we derive the formula of pricing models via two methods. Empirical studies are conducted to prove that our pricing models have a better performance in matching the VIX options. Furthermore, we find that the pricing model via the infinitesimal value method yields better results than pricing models with a measure of change. Overall, our proposed models enrich the derivative pricing theory and help investors understand and hedge risks.