A Market Model for VIX Futures

Abstract: A new modelling approach that directly prescribes dynamics to the term structure of VIX futures is proposed in this paper. The approach is motivated by the tractability enjoyed by models that directly prescribe dynamics to the VIX, practices observed in interest-rate modelling, and the desire to develop a platform to better understand VIX option implied volatilities. The main contribution of the paper is the derivation of necessary conditions for there to be no arbitrage between the joint market of VIX and equity derivatives. The arbitrage conditions are analogous to the well-known HJM drift restrictions in interest-rate modelling. The restrictions also address a fundamental open problem related to an existing modelling approach, in which the dynamics of the VIX are specified directly. The paper is concluded with an application of the main result, which demonstrates that when modelling VIX futures directly, the drift and diffusion of the corresponding stochastic volatility model must be restricted to preclude arbitrage.