A Market Model for VIX Futures

TL;DR

This paper introduces a new model for VIX futures that prescribes their term structure dynamics, deriving no-arbitrage conditions similar to interest-rate models, enhancing understanding of VIX options and volatility modeling.

Contribution

It develops necessary no-arbitrage conditions for VIX and equity derivatives markets, addressing a key open problem in direct VIX dynamics modeling.

Findings

Derived arbitrage-free drift restrictions for VIX futures

Established conditions linking VIX futures and stochastic volatility models

Provided a framework for consistent VIX derivatives modeling

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.