1 A Jump Diffusion Model for VIX Volatility Options and Futures

Implied volatility indices are becoming increasingly popular as a measure of market uncertainty and as a vehicle for developing derivative instruments to hedge against unexpected changes in volatility. Although jumps are widely considered as a salient feature of volatility, their implications for pricing implied volatility options and futures are not yet fully understood. This paper provides evidence indicating that the time series behavior of the VIX equity implied volatility index is well approximated by a mean reverting logarithmic diffusion with jumps. This process is capable of capturing stylized facts of VIX dynamics such as fast mean-reversion at high levels, level effects of volatility and large upward movements during times of market stress. Based on this process, we develop closed form valuation models for volatility futures and options and show that incorrectly omitting jumps may cause considerable problems to pricing and hedging