The 2008 financial crisis provides a valuable opportunity to study empirical data of market volatility during severe financial crisis. In this thesis, we study the implied volatility of VIX options during the crisis (2008) and a relatively calm period (2011). We present a method of calculating the implied volatility of VIX options and fit the implied volatilities using a 4th degree spline interpolation and propose method of extracting risk neutral density from fitted data. We analyze the slope and the level of the fitted implied volatility of VIX options during those periods. The results show that the level of the implied volatility of VIX options is higher and the slope is flatter during the distressed market compared to the relative calm periods.

The thesis investigates the impact of the fundamental news announcements on the movements of the VIX volatility index and the VIX Futures prices. The theoretical part of the thesis explains the construction of the VIX Index and the VIX Futures, describes the most important fundamental news for the US economy and presents a methodology for the modelling of the relationship between the news announcements and the VIX index movements with a simple linear regression model. In the empirical part of the thesis, we analyze the impact of 105 US fundamental news, from the Reuters Eikon database, on the VIX Index movements on theday of the news announcements as well as on the subsequent day. We find a strong relationship between the surprise component of the news and the VIX Index movements on the day of the news announcement, with the statistically significant news explaining 5-10% of the total return variance (for news with small number of observations up to 30-50%) on the announcement day. In the second part of the empirical study, simple trading system is proposed in order to utilize the possible impact of the economic news on the next-day (after announcement) returns of VIX futures in order to achieve speculative profits. Although the models seem to possess some limited out-sample profitability for some of the news, the results are for most of the cases statistically insignificant and the potential profits from the news trading seem to be relatively low.

Many past literatures have examined the predictive power of implied volatility versus that of historical volatility, but they have showed divergent conclusions. One of the major differences among these studies is the methods that they used to obtain implied volatility. The VIX index, introduced in 1993, provides a model-free and directly observable source of implied volatility data. The VIX futures is an actively traded VIX derivative product, and its prices are believed to contain market’s expectation about future volatility. By analyzing the relationship between the VIX futures prices and the realized volatilities of the 30-day period that these VIX futures contracts cover, this paper finds that the VIX futures contracts with shorter maturities have predictive power on future realized volatility, but they are upwardly biased estimates. The predictive power, however, decreases as the time to maturity increases. The outstanding VIX futures contracts with the nearest expiration dates outperform GARCH estimates based on historical return data at predicting future realized volatility.

This thesis analyses the impact of announcements of macroeconomic indicators in United States on price development of the VIX Futures, S&P500 Futures and EUR/USD FX rate. Theoretical part contains construction and description of individual markets. Empirical part investigates the reaction of market prices after 1, 10 and 30 minutes after announcement of an individual indicator value on a market surprise demonstrated as a difference between reported value and analysts' expectations. We tried to find a systematic reaction of market participants and the pace of absorption of new information into the market price. There have been found minimum of situations, where we explained the market move as a linear combination of market surprise. However, there was a several cases, where the market did not adjust to announced information quickly and was inefficient in a short period. In the second part of empirical research we tested all significant models on an out-sample data. The goal was to determine whether the market inefficiencies persisted and stable profit could be achieved. We analysed the brutto performance, then netto performance including all transaction costs. Finally, we defined a simple trading rules with a purpose of profit stabilization and lowering the riskiness of trades. For VIX Futures and EUR/USD markets we achieved a low loss, respectively negligible profit. For S&P 500 Futures we obtained a profit strategies for all selected indicators, total profit was high with a very low volatility of invested capital.

La première partie de cette thèse est consacrée aux problématiques liées à la modélisation markovienne de la courbe de variance forward. Elle est divisée en 3 chapitres. Dans le premier chapitre, nous présentons le cadre général de la modélisation de type HJM-Markov pour la courbe de variance forward. Nous revisitons le cadre affine-markovien modélisation et nous l'illustrons par l'exemple du modèle de Bühler. Dans le deuxième chapitre, nous proposons un nouveau modèle pour la courbe de variance forward qui combine les caractéristiques des deux versions (continue et discrète) du modèle de Bergomi 2008, sans se réduire ni à l'une ni à l'autre. Un des avantages de ce modèle est que les prix des futures et options sur VIX peuvent être exprimés comme des espérances de fonctions déterministes d'une variable aléatoire gaussienne, ce qui réduit le problème de la calibration à l'inversion de certaines fonctions monotones. Dans le troisième chapitre, on propose une méthode d'approximation pour les prix d'options européennes dans des modèles à volatilité stochastique de type multi-factoriels lognormal (comprenant le modèle présenté dans le deuxième chapitre, les modèles de Bergomi et le modèle de Scot 1987). Nous obtenons un développement d'ordre 3 de la densité du sous-jacent par rapport au paramètre de la volatilité de la volatilité. Nous présentons aussi une méthode de réduction de variance de type "variable de contrôle" pour la simulation par la méthode de Monte-Carlo qui utilise l'approximation explicite que nous obtenons de la fonction de répartition de la loi du sous-jacent. La deuxième partie de cette thèse est consacrée à l'étude des propriétés de monotonie des prix d'options européennes par rapport aux paramètres du CIR dans le modèle de Heston. Elle est divisée en deux chapitres. Dans le premier chapitre (cf. chapitre 4), nous donnons quelques résultats généraux sur le processus CIR. Nous montrons d'abord que les queues de distribution d'une combinaison du CIR et de sa moyenne arithmétique se comportent comme des exponentielles. Nous étudions ensuite les dérivées de la solution de ce processus par rapport aux paramètres de sa dynamique. Ces dérivées sont données comme solutions d'équations différentielles stochastiques, qu'on résout pour obtenir des représentations de ces dérivées en fonction des trajectoires du CIR. Le chapitre 5 est consacré à l'étude de la monotonie du prix d'un Put européen par rapport aux paramètres du CIR et à la corrélation dans le modèle de Heston. Nous montrons que, sous certaines conditions, les prix d’options européennes sont monotones par rapport aux paramètres du drift du CIR. Nous montrons ensuite que le paramètre de la volatilité de la volatilité joue le rôle de la volatilité si on prend la variance réalisée comme sous-jacent. En particulier, les prix d'options convexes sur la variance réalisée sont strictement croissants par rapport à la volatilité de la volatilité. Enfin, nous étudions la monotonie du prix du Put européen par rapport à la corrélation. Nous montrons que le prix du put du Put est croissant par rapport à la corrélation pour les petites valeurs du Spot et décroissant pour les grandes valeurs. Nous étudions ensuite les points de changement de monotonie pour les courtes et longues maturités / The first part of this thesis deals with issues related to the Markov-modeling of the forward variance curve. It is divided into 3 chapters. In the first chapter, we present the general framework of the HJM-type modelling for the forward variance curve. We revisit the Affine-Markov framework, and illustrate by the model proposed by B"uhler 2006. In the second chapter, we propose a new model for the forward variance curve that combines features of the continuous and discrete version of Bergomi's model model Bergomi (2008), without being reduced to either of them. One of the strengths of this model is that the prices of VIX futures and options can be expressed as expectations of deterministic functions of a Gaussian random variable, which reduces the problem of calibration to the inversion of some monotonic functions. In the third chapter, we propose an approximation method for pricing of European options under some lognormal stochastic volatility models (including the model presented in the second chapter, Bergomi's model2008 and Scot model 1987). We obtain an expansion (with respect to the the volatility of volatility parameters of order 3) of the density of the underlying. We also propose a control variate method to effectively reduce variances of Monte Carlo simulations for pricing European optionsThe purpose of the second part of this thesis is to study the monotonicity properties of the prices of European options with respect to the CIR parameters under Heston model. It is divided into two chapters. In the first chapter (see Chapter 4), we give some general results related to the CIR process. We first show that the distribution tails of a combination of the CIR and its arithmetic mean behave as exponential. We then study the derivatives of the solution process with respect to the parameters of its dynamics. These data are derived as solutions of stochastic differential equations, which solves for the representations of these derivatives based on trajectories of the CIR. Chapter 5 is devoted to the study of the monotony of the European price of a put with respect to parameters of CIR and correlation in the Heston model. We show that under certain conditions, prices of European options are monotonic with respect to the parameters of the drift of the CIR. We then show that the parameter of the volatility of volatility plays the role of volatility if we take the realized variance as the underlying. In particular, prices of (convex) options on realized variance are strictly increasing with respect to the volatility of volatility. Finally, we study the monotony of the European Put prices with respect to the correlation. We show that the price of the put is increasing with respect to the correlation for small values ​​of Spot and decreasing for large values. We then study the change points of monotonicity for short and long maturities

This study investigates how term structure dynamics of VIX futures can be exploited forabnormal returns. To be able to access volatility as a tradeable asset, the trading strategiesonly trades ETFs which are designed to replicate the movements of VIX futures index. Itis established that such ETFs are unsuitable for buy-and-hold investments because of thenegative roll yield it usually suffers, caused by the slope of the VIX term structure.Consequently, these conditions create opportunities for strategies that use direct andinverse VIX ETFs to be profitable. The study is a quantitative study that uses historicalprice data to back test three different trading strategies. The strategies are tested over theperiod 11-oct-2011 to 31-mar-2020. The authors have deliberately chosen to delimit thestudy by not testing the performance of the ETFs, not statistically test the risk-adjustedreturns and not perform a regression to calculate optimal hedge ratios for the strategies.The results from this study shows that its possible for strategies that exploit the termstructure dynamics of VIX futures to generate abnormal returns.

本研究探討VIX 期貨價格所隱含的資訊對於S&P 500 指數波動度預測的解釋力。過去許多文獻主要運用線性預測模型探討歷史波動度、隱含波動度和風險中立偏態對於波動度預測的資訊內涵。然而過去研究顯示，波動度具有長期記憶與非線性的特性，因此本文主要研究非線性預測模型對於波動度預測的有效性。本篇論文特別著重在不同市場狀態下(高波動與低波動)的實現波動度及隱含波動度異質自我迴歸模型(HAR-RV-IV model)。因此，本研究以考慮馬可夫狀態轉化下的異質自我迴歸模型(MRS-HAR model)進行實證分析。 本研究主要目的有以下三點: (1) 以VIX期貨價格所隱含的資訊提升S&P 500波動度預測的準確性。(2) 結合風險中立偏態與VIX期貨的資訊內涵，進一步提升S&P 500 波動度預測的準確性。(3) 考慮狀態轉換後的波動度預測模型是否優於過去文獻的線性迴歸模型。 本研究實證結果發現: (1) 相對於過去的實現波動度及隱含波動度，VIX 期貨可以提供對於預測未來波動度的額外資訊。 (2) 與其他模型比較，加入風險中立偏態和VIX 期貨萃取出的隱含波動度之波動度預測模型，只顯著提高預測未來一天波動度的準確性。 (3) 考慮狀態轉換後的波動度預測模型優於線性迴歸模型。 / This paper explores whether the information implied from VIX futures prices has incremental explanatory power for future volatility in the S&P 500 index. Most of prior studies adopt linear forecasting models to investigate the usefulness of historical volatility, implied volatility and risk-neutral skewness for volatility forecasting. However, previous literatures find out the long-memory and nonlinear property in volatility. Therefore, this study focuses on the nonlinear forecasting models to examine the effectiveness for volatility forecasting. In particular, we concentrate on Heterogeneous Autoregressive model of Realized Volatility and Implied Volatility (HAR-RV-IV) under different market conditions (i.e., high and low volatility state). This study has three main goals: First, to investigate whether the information extracted from VIX futures prices could improve the accuracy for future volatility forecasting. Second, combining the information content of risk-neutral skewness and VIX futures to enhance the predictive power for future volatility forecasting. Last, to explore whether the nonlinear models are superior to the linear models. This study finds that VIX futures prices contain additional information for future volatility, relative to past realized volatilities and implied volatility. Out-of-sample analysis confirms that VIX futures improves significantly the accuracy for future volatility forecasting. However, the improvement in the accuracy of volatility forecasts is significant only at daily forecast horizon after incorporating the information of risk-neutral skewness and VIX futures prices into the volatility forecasting model. Last, the volatility forecasting models are superior after taking the regime-switching into account.