Arbitrage-free models for VIX and equity derivatives

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 with 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. Acknowledgements I most sincerely thank my supervisors, Professors Ben Goldys and Marek Rutkowski, for their excellent support throughout my doctoral studies. I am grateful for their constant readiness to share knowledge, patience, and guidance through the academic world. I would also like to thank my Honours supervisors, Dr. Xiaoping Lu and Professor Song-Ping Zhu, for their guidance in my undergraduate years and their continued friendship in the years that have followed. I thank the A. F. Pillow Trust for supporting my doctoral studies through the A. F. Pillow Top-Up Scholarship. Particular thanks are reserved for Professor Jane Pillow. I would like to thank my colleagues, Andrew Papanicolaou, Michael Roper and Nicholas Yap, for their insight and encouragement. I owe a special debt of gratitude to Jan Baldeaux. It is only with his friendship and guidance that I was able to complete my studies. To my friends. I thank you for the welcomed distractions, for enduring many one-sided conversations about mathematics, and for keeping me sane. Finally, the most gratitude is reserved for my family. While I am unable to explain to you what I do, know that because of you, I am able to do it. This thesis is dedicated to you.