Arbitrage-free models for VIX and equity derivatives

Abstract

Arbitrage-free models for VIX and equity derivatives are the primary focus of this thesis. The first major contribution of the thesis is in Chapter 4. The 3/2 plus jumps model is studied in detail and formulae for the valuation of VIX and equity derivatives are derived under such a specification. The results demonstrate that a pure-diffusion 3/2 model is able to capture the observed upward-sloping implied volatility skew in VIX options. The 3/2 plus jumps model is able to better fit short-term index option implied volatilities, while producing more realistic VIX option implied volatilities, without a loss in tractability. The primary contribution of this thesis is in Chapter 5, in which a new modelling approach that directly prescribes dynamics to the term structure of VIX futures is proposed. 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 result is the derivation of necessary conditions for there to be no arbitrage between the joint market of VIX and equity derivatives. The conditions are stated in Theorem 5.5.3 and Theorem 5.5.4. The theorems also address a fundamental open problem related to models that directly prescribe dynamics to the VIX. The chapter 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. The thesis concluded with Chapter 6, which is concerned option and implied volatility surfaces for VIX and equity derivatives. Several original representations concerning option and implied volatility surfaces are presented. The thesis is concluded with a discussion of the open problems and comments regarding future potential research.