Pricing VIX options based on mean-reverting models driven by information

Abstract

Financial time series are dynamic and influenced by different types of information from the market. In this study, we propose new models for SPX and VIX options using the Hawkes process, jump process with stochastic intensity, and tempered stable process to capture these changes in financial time series based on three distinct characteristics of market information. We calculate the VIX option pricing formula using these models and find that the simplified VIX model based on VIX characteristics has significantly less pricing error than the consistent VIX model derived from the SPX model. Additionally, our findings suggest that the tempered stable process effectively models the volatility of VIX, sparse large jumps, and infinitesimal jumps. It also shows potential as an alternative to Brownian motion for representing volatility. Conversely, jump processes with stochastic jump intensities adeptly describe asymmetric jumps, and their integration with Brownian motion provides a more accurate depiction of the VIX’s volatility and jump dynamics. Finally, the introduction of jump processes into mean-reverting models for the VIX indicates a relatively low correlation between volatility magnitudes and the current VIX levels. This research contributes to the theory of SPX and VIX options and offers guidance for the development of other information-driven economic models.