Zurich Open Repository and Archive University of Zurich Main Library Strickhofstrasse 39 CH-8057 Zurich www.zora.uzh.ch Year: 2013 Affine and quadratic models for volatility and interest rates markets Gourier, Elise Posted at the Zurich Open Repository and Archive, University of Zurich ZORA URL: https://doi.org/10.5167/uzh-94023 Dissertation Originally published at: Gourier, Elise. Affine and quadratic models for volatility and interest rates markets. 2013, University of Zurich, Faculty of Economics. Affine and Quadratic Models for Volatility and Interest Rates Markets Dissertation for the Faculty of Economics, Business Administration and Information Technology of the University of Zurich to achieve the title of Doctor of Philosophy in Banking & Finance presented by Elise Gourier from Pontailler-sur-Saˆone, France approved in July 2013 at the request of Prof. Dr. Markus Leippold Prof. Dr. Josef Teichmann The Faculty of Economics, Business Administration and Information Technology of the University of Zurich hereby authorizes the printing of this Doctoral Thesis, without thereby giving any opinion on the views contained therein. Zurich, July 2013. The Chairman of the Doctoral Committee: Prof. Dr. Dieter Pfaff. Acknowledgements I want to express my gratitude to Prof. Dr. Markus Leippold, my thesis supervisor, for his support and advice, and for the great work atmosphere I could benefit from. Furthermore he gave me the chance to prepare and teach the Financial Engineering lecture together with him and Chris Bardgett, which was a very enriching experience. I would like to further acknowledge the support of my co-supervisor Prof. Dr. Josef Teichmann, whose valuable inputs on my conference and seminar presentations have improved my work. I am also very grateful to Prof. Dr. Loriano Mancini and Prof. Dr. Damir Filipovi´cwho gave me the great opportunity to work with them on a very interesting topic. I learnt a lot from them both on a technical and on a more human level, and admire their rigor and continued enthusiasm for research. I would like to thank them for the time they dedicated to me, the ideas they shared and their patience when difficulties arose. I also thank Chris Bardgett who is my coauthor and with whom I worked long hours on the paper, sometimes late at night when we were under time pressure. He was always available, even during his stay in London, and working together was both fruitful and pleasant. I feel privileged to have worked on publications together with Markus, Loriano, Damir and Chris, and hope that we can pursue our collaboration in the future. I would also like to thank Prof. Dr. Walter Farkas and Prof. Dr. Marc Paolella for always being encouraging and for the career advice they gave me. Furthermore, I had the chance to be in a wonderful chair and enjoyed very much my time as a PhD student. Special thanks goes to Felix Matthys for his patience and to my colleagues and friends Lujing Su, Caroline Oehri, Nikola Vasiljevic and Meriton Ibraimi, for always being ready to help and for all the good times we had together. Finally I wish to express deep gratitude to my family and friends who has always supported and encouraged me throughout the ups and downs. Z¨urich, June 2013 Elise Gourier Contents I Introduction 1 Introduction and Summary of Research Results Elise Gourier 3 II Research Papers 7 Inferring volatility dynamics and risk-premia from the S&P 500 and VIX markets Chris Bardgett, Elise Gourier and Markus Leippold 9 Quadratic Variance Swap Models Damir Filipovi´c,Elise Gourier and Loriano Mancini 71 Libor Market Model: How to account for the Crisis? Elise Gourier 127 III Appendix 163 Curriculum Vitae 165 Part I Introduction Introduction and Summary of Research Results Volatility and interest rates share many features. In the model of Black and Scholes (1973) for stock price returns, they are both considered constant. However, modelling them as realistically as possible and investigating their features have become one of the main goals of mathematical finance, producing a sophisticated growing body of literature. Three approaches are predominant in interest rates and volatility modelling. The first one focuses on the instantaneous processes. Both short rate and volatil- ity are commonly represented by a square root process, which ensures their positivity and reflects their stationarity. Because this approach has difficulties to accurately represent the term structure of interest rates and volatility, an alternative is to model the corresponding forward processes. In interest rates modelling this idea was introduced by Heath, Jarrow, and Morton (1992), and more recently Buehler (2006) developed a class of similar models for the forward variance. Unfortunately, the instantaneous interest rate and forward rates (respectively instantaneous/forward variances) are not directly observable quantities in financial markets, making the choice of a particular model and statistical inference complicated. The third class of models, market models, were developed to over- come this issue by directly modelling observable quantities, such as Libor rates. These models are very popular among practitioners working with interest rates derivatives; indeed, in their simplest form where the forward Libor (alternatively swap rate) is modelled by a Geometric Brownian Motion, closed-form expressions are available for options, caps and floors (alternatively swaptions). This doctoral thesis entitled “Affine and Quadratic Models for Volatility and Interest Rates Markets” comprises three papers which investigate the ability of affine models to represent different features of volatility and interest rates, and price derivatives on these underlyings. Interest rates derivatives are by far the largest derivatives market in the world. With a notional amount outstanding of USD 418 billion in 2008,1 they represent more than 70% of the total amount outstanding in the global OTC derivatives market. On the other side, derivatives on volatility have attracted growing attention in the last decade. According to financial press (e.g., Gangahar (2006)), variance swaps have become the preferred tool used by market participants to bet on and/or hedge against volatility movements. Furthermore, since their introduction in 2006, options on the volatility index VIX have gained increasing popularity, and are with options on the S&P 500 among the most liquid worldwide with a daily average volume of 391,992 traded contracts (783,768 on the S&P 500) in 2011. The recent financial crisis has had a tremendous impact on both interest rates and volatility markets. It uncovered the impact of counterparty and liquidity risks on some interest rates spreads and gave 1According to the report from the Bank of International Settlements published in May 2009. 3 rise to phenomena that had never been observed in the past. Because rates with different tenors were affected by different levels of risk, some essential classical arbitrage relationships were violated and new modelling methodologies were needed. On the other side, because of the leverage effect, volatility reacted very strongly to the crisis as well. The VIX index increased from about 20% up to about 80% in less than a month end of 2008, variance swap rates exhibited a similar peak and smiles of volatility shifted upwards. These sudden movements led to a number of analyses that pointed to the presence of jumps in the interest rates and volatility processes. Under the historical measure, these jumps would be justified by the large upward movements in the trajectories of the processes, while under the risk-neutral measure they could explain the steep smiles of volatility which are typically observed for short-maturity options. This thesis mainly answers three questions. First, what information can we infer on volatility from S&P 500 and VIX underlying levels and option prices using affine processes? Second, do quadratic models improve on affine models in representing the variance term structure and how can this be exploited in a trading strategy? Third, how can affine models be used to build a Libor Market Model which is consistent with the stylized facts of interest rates and reflects the spreads that appeared during the crisis? Each paper is presented in a separate chapter, organized as follows. In the first Chapter Inferring volatility dynamics and risk-premia from the S&P 500 and VIX markets, we use a large dataset of S&P 500 and VIX index and option prices with wide ranges of maturities and moneynesses, and analyze the empirical performance of affine jump-diffusion models for S&P 500 returns to jointly represent underlyings’ and derivatives’ prices. Based on the affine relationship of the VIX squared with respect to the latent factors, we extend the Fourier Cosine Expansion to efficiently price VIX derivatives. We build an Auxiliary Particle Filter which sources the information contained in both indices and derivatives’ prices over time and investigate the behavior of the filtered latent processes. We analyze the out-of-sample performance of sub-models depending on which products and markets are considered in the in-sample estimation procedure. We find that a stochastic central tendency is needed to better represent the tails of the returns’ distribution and the term structure of the smiles of volatility on both S&P 500 and VIX markets. Furthermore, jumps in returns and volatility help reproduce the tail of the variance distribution. Finally, we investigate and compare the information that the underlying levels and options contain on latent factors and risk premia. In the second Chapter Quadratic Variance Swap Models, we introduce a novel class of term structure models for variance swaps. The multivariate state variable follows a diffusion process characterized by a quadratic diffusion function. The variance swap curve is quadratic in the state variable, and available in closed form in terms of a linear ordinary differential equation, greatly facilitating empirical analysis. Various goodness-of- fit tests show that quadratic models fit variance swaps on the S&P 500 remarkably well and outperform nested specifications, including popular affine models.