Linking vanillas and VIX options: A constrained martingale optimal transport problem

VIX options traded on the CBOE have become popular volatility derivatives. As S&P 500 vanilla options and VIX both depend on S&P 500 volatility dynamics, it is important to understand the link between these products. In this paper, we bound VIX options from vanilla options and VIX futures. This leads us to introduce a new martingale optimal transportation problem that we solve numerically. Analytical lower and upper bounds are also provided which already highlight some (potential) arbitrage opportunities. We fully characterize the class of marginal distributions for which these explicit bounds are optimal, and illustrate numerically that they seem to be optimal for the market-implied marginal distributions.