Pricing VIX Futures with stochastic volatility and random jumps
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Since the inception of the volatility index (VIX) by the CBOE, 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. Our newly-found analytical solution is written in a simple integral form, which can be numerically computed very easily. Numerical comparisons show that the results from our exact formula perfectly match up those from Monte Carlo simulations, while the popularly used convexity correction approximation may lead to significant pricing errors. 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 to us, especially with its great computational efficiency, we were also able to conduct some empirical studies, aiming at examining the pricing performance of four stochastic volatility models with or without jumps. We demonstrated how to estimate model parameters, upon using the Markov chain Monte Carlo (MCMC) method to analyze a set of coupled VIX and S&P500 data. Through these empirical studies, we find that adding simultaneous jumps in the asset price and volatility processes can indeed enhance the pricing performance in some cases, whereas only adding jumps in volatility process alone produces little pricing improvement. Our empirical studies also provide quantitative evidence that while the price of VIX futures can be well replicated by any of those four models studied with some degree of di erence in terms of pricing performance, they lead to a slight mis-match of the “right-tail” of the VIX observed directly from the empirical data when the same set of parameters determined from the MCMC method is used.