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.

自從2008年金融危機爆發後，黑天鵝事件相繼出現，VIX成為投資人衡量股市波動度的重要指標。但是若投資人想使用VIX避險，僅能透過限專業投資人參與的VIX期貨。而在近年ETF產品盛行的背景下，投資標的更加多元的交易所交易債券（ETN）也應運而生，使一般投資人得以進入以往難以觸及或交易成本高昂的市場。本研究採用兩檔交易量較大之VIX ETN，分別追蹤VIX短期與中期期貨指數之VXX與VXZ，希望透過建構GARCH模型用以預測其隔日價格，並以此預測的價格近一步建構避險策略，目標係在預期空頭即將發生時，提供投資人除了VIX期貨和波動相對平穩的債券以外的避險替代工具。 建構GARCH模型的部分，本研究主要參考Kambouroudis和McMillan（2013）的文獻，在變異數方程式中加入輔助變數，可以增加模型的預測能力，故本研究在VIX ETN之GARCH模型的變異數方程式中加入VIX、短期VIX指數及中期VIX指數。實證結果顯示，在VIX ETN的GARCH模型中同時加入VIX相關指數，確實能提高配適程度並增進預測能力，尤其當加入的輔助變數與VIX ETN追蹤標的的到期期限相符時，此改善模型的效果最為顯著。 本研究接者參考Alexander和Korovilas（2012）的VIX ETN避險研究，文獻顯示，在S&P 500 ETF投資組合中加入VXX與VXZ避險可提高夏普比率。本研究在此基礎上，額外考量了不同的持有期間、進場條件、股債混合的投資組合，並分別比較兩種ETN的避險效果。本研究發現只在VIX大於20時才進場建構避險部位的策略，提前買入VIX ETN確實可以做為良好的避險工具。此外，在此策略下，VIX ETN亦則可達到比持有債券更佳的避險效果。而本研究所測試的兩種VIX ETN中，又以VXX 避險效果更佳，因VXX乃是追蹤VIX短期期貨指數，更能反映市場短期的變化，搭配滾動的避險比率，能更加精準的反應空頭時期劇烈的波動。 / Since the 2008 financial crisis, along with the black swan events, the volatility of global stock market has intensified, and VIX index becomes an important indicator for investors to measure the volatility of the stock market. However, if investors would like to use VIX index for hedge, they could only use VIX futures, which is only for professional investors to participate. In recent years, the prevalence and popularity of the various ETPs lead to the booming of VIX ETNs, which has become an alternative for regular investors to invest in VIX index. Therefore, this study hopes to build GARCH model for VIX ETN and predict their prices of the next day, and use the prediction to build hedging strategies. In this paper, this study mainly refers to the paper of Kambouroudis and McMillan (2013) to construct the VXX and VXZ prediction models. Because the two VIX ETNs track the S&P 500 VIX short-term and medium-term futures index respectively, the study add the VIX index, short-term VIX index and medium-term VIX index in the GARCH models. The empirical results show that the addition of VIX and other relevant VIX indices in the VIX ETN GARCH models can improve the forecasting ability. In particular, when the maturity of the VIX index is consistent with the maturity of the VIX ETN’s tracking target, it would improve the prediction power the most. Based on the predicted VIX ETN prices, this study then constructs the hedging strategies, considering the different holding period, the entry condition and the stock and debt mixed portfolio, and also compares the hedging effect of VXX and VXZ respectively. This study found that under the strategy that only enter the VIX ETN market when VIX was greater than 20, VIX ETN can indeed be a good hedge tool and reduce the standard deviation of the portfolio. In addition, under this strategy, if investors use VIX ETN to hedge, investors can achieve a higher return and lower standard deviation than holding a bond to hedge. Finally, among the two VIX ETNs tested in this study, VXX is a better hedge tool against VXX. It is because VXX tracks the VIX short-term futures index which reflects the short-term changes in the market and hence could reflect the short-term volatility better.

This paper examines the role that gold and other precious metals play in portfolio construction as hedges and safe havens in comparison with the Volatility Index (VIX), a derivative of market volatility. Gold has long been considered to be among the best assets for reducing portfolio volatility due to its lack of correlation with the overall market. However, the major finding of this paper is that while gold and the other precious metals do serve in this role to varying extents, the VIX performs this function far better. Both econometric and portfolio analyses reveal that the VIX improves overall portfolio performance to a greater extent, and most importantly serves as an effective safe haven relative to the market.

In this thesis, I study the variance risk premium in Gold VIX market. Using synthetically created variance swaps, I quantify the variance risk premium to be average -0.068 in absolute terms and -0.358 in log return terms, meaning that purchasing volatility in Gold VIX is generally unprofitable. Although the average negative risk premium is not statistically significant, the mean log return of risk premium is robust with Newey-West test. Furthermore, I attempt to test whether risk premium vary with time or the level of the swap rate, but obtain unclear results.

This thesis uses synthetically created variance swaps on VIX futures to quantify the variance risk premium in VIX options. The results of this methodology suggest that the average premium is -3.26%, meaning that the realized variance on VIX futures is on average less than the variance implied by the swap rate. This premium does not vary with time or the level of the swap rate as much as premiums in other asset classes. A negative risk premium should mean that VIX option strategies that are net credit should be profitable. This thesis tests two simple net credit strategies with puts and calls, and finds that the call strategy is profitable while the put strategy is not.

This dissertation consists of two chapters that examine the construction of financial market volatility indexes and their forecasting efficiency across predictive regression models. Each of the chapter is devoted to diferent volatility measures which are related and evaluated in thframework of forecasting regressions. The first chapter studies the sampling and liquidity issues in constructing volatility indexes, VIX and VXO, in emerging options market like Taiwan. VXO and VIX have been widely used to measure the 22-day forward volatility of the market. However, for an emerging market, VXO and VIX are difficult to measure with accuracy when tradings of the second and next to second nearby options are illiquid. The chapter proposes four methods to sample the option prices across liquidity proxies ¡V five different days of rollover rules ¡V for option trades to construct volatility index series. The paper finds that, based on the sampling method of the average of all midpoints of bid and ask quote option prices, the volatility indexes constructed by minute-tick data have less missing data and more efficient in volatility forecast than the method suggested by CBOE. Additionally, illiquidity in emerging options market does not, based on different rollover rules, lead to substantial biases in the forecasting effectiveness of the volatility indexes.Finally, the forecasting ability of VIX, in terms of naive forecasts and forecasting regressions, is superior to VXO in Taiwan. The second chapter uses high-frequency intraday volatility as a benchmark to measure the efficacy of model-free and model-based econometric models. The realized volatility computed from intraday data has been widely regarded as a more accurate proxy for market volatility than squared daily returns. The chapter adopts several time series models to assess the fore-casting efficiency of future realized volatility in Taiwan stock market. The paper finds that, for 1-day directional accuracy forecast performance, semiparametric fractional autoregressive model (SEMIFAR, Beran and Ocker, 2001) ranks highest with 78.52% hit accuracy, followed by multiplicative error model (MEM, Engle, 2002), and augmented GJR-GARCH model. For 1-day forecasting errors evaluated by root mean squared errors (RMSE), GJR-GARCH model augmented with high-low range volatility ranks highest, followed by SEMIFAR and MEM model, both of which, however, outperform augmented GJR-GARCH by the measure of mean absolute value (MAE) and p-statistics (Blair et al., 2001).

Over the past couple of decades, the number of volatility indices has increased rapidly. These indices seek to represent the market’s expectation of realized volatility over the coming month, based on the prices of options traded on each underlying equity index. Although the dynamics of realized volatility spillover have been studied extensively, very few studies exists that examine the spillover between these volatility indices. By using DAG-based structural vector autoregression, this paper provides evidence that implied volatility spillover differs from realized volatility spillover. Through solving the well-known VAR identification problem for these indices, this paper finds that Asia, more specifically Hong Kong, plays a central role in implied volatility spillover during and after the 2008 financial crisis.

I denna studie undersöks guld som safe haven-tillgång och om den erbjuder tillflykt mot volatilitet, vilket är studiens huvudsakliga syfte. För att åstadkomma detta används data från VIX- och GVZ-indexet samt priset på guld under perioden 1994–2018. Guldets egenskaper testas mot safe haven-teorin via ett kvantitativt angreppsätt där korrelation, avkastning och standardavvikelse är ämne för undersökning i utvalda perioder. Guld visar sig bevara sitt värde under oroligheter – vilket syns genom att den genomsnittliga avkastningen för guld är högre när VIX befinner sig på höga nivåer. Dock misslyckas tillgången att förbli lågvolatil då guldets volatilitet stiger i takt med den förväntade volatiliteten på aktiemarknaden. Den förväntade volatiliteten i guldpriset, uttryckt genom GVZ, korrelerar dessutom med den förväntade volatiliteten på aktiemarknaden mätt av VIX-indexet. Däremot upptäcks knappt någon korrelation mellan guldets pris och förändringar i VIX-indexets värde. / The purpose of this study was to analyse whether gold fulfils the criteria for being a safe haven asset in certain conditions. Through the use of data pooled from CBOE’s VIX, GVZ and the spot price of gold, we employ a quantitative approach to analyse correlation, rate of return and standard deviation during times of market volatility. The chosen period for this analysis is the time between 1994 and 2018. Gold proved its ability to retain value during such conditions, which is evidenced by higher average returns when VIX has been at high levels. This strengthens its role as a safe haven asset. However, gold failed to keep a low level of volatility in periods of rising implied volatility on the stock market, as expressed by the VIX index. Moreover, the implied volatility of gold, expressed through the GVZ index, has shown a strong correlation with the VIX, indicating that gold is not a safe haven. Finally, the gold spot price was shown to have little to no correlation with changes in VIX.

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

[pt] Este trabalho mostra que o VIX e sua versão de mais curto prazo tem poder preditivo sobre a variância realizada do S e P 500 em dias de FOMC (reuniões do Banco Central do Estados Unidos – FED). Apesar disto, o resultado não persiste quando utilizamos o Índice de Volatilidade do S e P 500 de Longo Prazo (3 meses). / [en] This work shows that the VIX and its shorter-term version have predictive power over the variance of the S and P 500 on FOMC days (meetings of the United States Central Bank - FED). Despite this, the result does not persist when we use the Long Term S and P 500 Volatility Index (3 months).