Since the inception of the volatility index (VIX) by the CBOE in 1993, in particular, the introduction of the VIX futures by CBOE in 2004, various pricing models with stochastic volatilities have been proposed to value VIX futures. However, rarely could an analytic closed-form solution be found, especially for models that include jumps in both VIX and its volatility. This study fills a gap in the field of pricing VIX futures by deriving a closed-form exact solution for the fair value of VIX futures under stochastic volatility model with simultaneous jumps in the asset price and volatility processes. The derivation of this formula for VIX futures with a very general dynamics of VIX represents a substantial progress in identifying and developing more realistic VIX futures models and pricing formulae.

With the newly-found pricing formula available, especially with its great computational efficiency, we were also able to conduct empirical studies, aiming at examining the performance of four different stochastic volatility models with or without jumps. More importantly, using the Markov chain Monte Carlo (MCMC) method to analyze a set of coupled VIX and S&P500 data, we demonstrate how to estimate model parameters, which is a crucial step for any fancy mathematical model to be of practical use. Through these empirical studies, we are able to compare the pricing performance of four models, of which analytical pricing formulae have been found and presented in this paper.

Here is a copy of the presentation slides.

Guang-Hua Lian is completing his PhD at the School of Mathematics and Applied Statistics, University of Wollongong. This is joint work with his supervisor, Professor Song-Ping Zhu.