The aim of the thesis is to identify an appropriate model in forecasting Value-at-Risk on a morevolatile period than that one from which the model is estimated. We estimate 1-day-ahead and10-days-ahead Value-at-Risk on a number of exchange rates. The Value-at-Risk estimates arebased on three models combined with three distributional assumptions of the innovations, andthe evaluations are made with Kupiec's (1995) test for unconditional coverage. The data rangesfrom January 1st 2006 through June 30th 2011. The results suggest that the GARCH(1,1) andGJR-GARCH(1,1) with normally distributed innovations are models adequately capturing theconditional variance in the series.

This thesis aims to construct an optimal portfolio and model as well as forecast its volatility. The performance of the optimal portfolio is then compared to two benchmarks, namely, an equally weighted portfolio and the market index SP 500. The volatility is estimated by employing two GARCH-type models known as standard GARCH, and GJR-GARCH. The GJR-GARCH outperformed its counterpart in terms of Log-likelihood, AIC, and BIC. The forecast performance is compared based on two statistical errors, root mean squared error, and mean absolute error. The optimal portfolio outperformed its counterparts in both statistical errors. Moreover, standard GARCH gave lower statistics than GJR-GARCH. These empirical results are of important significance to portfolio management and risk management processes.

This thesis examines the volatility forecasting performance of six commonly used forecasting models; the simple moving average, the exponentially weighted moving average, the ARCH model, the GARCH model, the EGARCH model and the GJR-GARCH model. The dataset used in this report are three different Nordic equity indices, OMXS30, OMXC20 and OMXH25. The objective of this paper is to compare the volatility models in terms of the in-sample and out-of-sample fit. The results were very mixed. In terms of the in-sample fit, the result was clear and unequivocally implied that assuming a heavier tailed error distribution than the normal distribution and modeling the conditional mean significantly improves the fit. Moreover a main conclusion is that yes, the more complex models do provide a better in-sample fit than the more parsimonious models. However in terms of the out-of-sample forecasting performance the result was inconclusive. There is not a single volatility model that is preferred based on all the loss functions. An important finding is however not only that the ranking differs when using different loss functions but how dramatically it can differ. This illuminates the importance of choosing an adequate loss function for the intended purpose of the forecast. Moreover it is not necessarily the model with the best in-sample fit that produces the best out-of-sample forecast. Since the out-of-sample forecast performance is so vital to the objective of the analysis one can question whether the in-sample fit should even be used at all to support the choice of a specific volatility model.

No / We examine the short-term price reaction of 424 UK stocks to large one-day price changes. Using the GJR-GARCH(1,1), we find no statistical difference amongst the cumulative abnormal returns (CARs) of the Single Index, the Fama–French and the Carhart–Fama–French models. Shocks ⩾5% are followed by a significant one-day CAR of 1% for all the models. Whilst shocks ⩽−5% are followed by a significant one-day CAR of −0.43% for the Single Index, the CARs are around −0.34% for the other two models. Positive shocks of all sizes and negative shocks ⩽−5% are followed by return continuations, whilst the market is efficient following larger negative shocks. The price reaction to shocks is unaffected when we estimate the CARs using the conditional covariances of the pricing variables.

After a long-time preparation, CSI 300 index futures has made a milestone in the financial market in China in the 16 of April, 2010. In order to know what kind of impact will bring to stock market after the appearance of stock index future, the study discusses volatility and volume separately. On one hand, the study applies Modified Levene and GJR-GARCH as the empirical model, and the result indicates that stock return fluctuation is a short-term phenomenon. However, the result shows that the stock return volatility has no difference in the long-run. Furthermore, it not only reduces the asymmetric return fluctuation from good and bad news cause but improve the information efficiency in the spot market after the introduction of the stock index futures. On the other hand, the study applies multiple regression model and panel model to examine the crowding-out effect and the volume difference after the stock index futures enters the market. First, there is no crowding-out effect in the stock market. Second, both the trading volume of the constituent and non-constituent stocks increase after the introduction of the stock index futures, whereas the level of increasing trading volume of the constituent stocks is larger than non- constituent stocks are.

The efficient market hypothesis has been around since 1962, the theory based on a simple rule that states the price of any asset must fully reflect all available information. Yet there is empirical evidence suggesting that markets are too volatile to be efficient. In essence, this evidence seems to suggest that the reaction of the market participants to the information or events that is the crucial factor, rather than the actual information. This highlights the need to include the behavioural finance theory in the pricing of assets. Essentially, the research aims to analyse the efficiency of six key sovereign debt markets during a period of changing volatility including the recent global financial and sovereign debt crises. We analyse the markets in the pre-crisis period and during the financial and sovereign debt crises to determine the impact of the crises on the efficiency of these financial markets. We use two GARCH-based variance bound tests to test the null hypothesis of the market being too volatile to be efficient. Proposing a GJR-GARCH variant of the variance bound test to account for variation in the asymmetrical effect. This leads to an analysis of the changing behaviour of price volatility to identify what makes the market efficient or inefficient. In general, our EMH tests resulted in mixed results, hinting at the acceptance of the null hypothesis of the market being too volatile to be efficient. However, interestingly a number of 2017 observations under both models seem to be hinting at the rejection of the null hypothesis. Furthermore, our proposed GJR-GARCH variant of the variance bound test seems to be more likely to accept the EMH than the GARCH variant of the test.

In the financial industry, it has been increasingly popular to measure risk. One of the most common quantitative measures for assessing risk is Value-at-Risk (VaR). VaR helps to measure extreme risks that an investor is exposed to. In addition to acquiring information of the expected loss, VaR was introduced in the regulatory frameworks of Basel I and II as a standardized measure of market risk. Due to necessity of measuring VaR accurately, this thesis aims to be a contribution to the research field of applying GARCH-models to financial time series in order to forecast the conditional variance and find accurate VaR-estimations. The findings in this thesis is that GARCH-models which incorporate the asymmetric effect of positive and negative returns perform better than a standard GARCH. Further on, leptokurtic distributions have been found to outperform normal distribution. In addition to various models and distributions, various rolling windows have been used to examine how the forecasts differ given window lengths.

摘要 本文是利用1998年底以前券商發行的15支認購權證為研究標的，試圖說明不同波動度的估計方法，會使得認購權證的理論價與市價產生不同的誤差，藉以提供券商在評價認購權證上作一參考。本文的實證結果發現：（1）在波動度的參數估計上，各模型均有波動度群集效果，但是訊息不對稱的效果各模型卻無一致性的結果；（2）在各模型的預測能力比較上，ARCH-M(1,1)模型都比ARCH(1,1)的預測能力佳。歷史波動度對於標的股的波動度小具有較佳的預測能力，而EGARCH-M(1,1)模型與GJR-GARCH-M(1,1)模型在預測波動度較大的標的股時具有較佳的預測結果；（3）以預測誤差百分比來比較各模型在預測認購權證上何者具有較小的誤差，結果發現：不論有無考慮交易成本及間斷性避險，預測能力最差的是歷史波動度，而預測能力最佳的則是隱含波動度模型，此乃因為台灣認購權證市場只有認購權證而無認售權證所致；（4）以市場溢價來比較那一支認購權證較值得投資者購買，結果發現：若權證處於價外，會使得市場溢價過高，而不利投資者購買；相反，若權證價格處於價內，則使得市場溢價較低，投資者購買較有利；（5）利用Delta法及Delta-Gamma法來計算大華01可發現：不同波動度的估計方法會影響該權證的涉險值，由於隱含波動度明顯高於其他方法所估算的值，故以隱含波動度計算的涉險值也就高於其他模型之涉險值。 目錄 謝辭 摘要 第一章 緒論 第一節 研究背景與動機 ………………………………………….1-1 第二節 研究問題與目的 ………………………………………….1-4 第三節 論文架構與流程 ………………………………………….1-5 第二章 文獻回顧 第一節 隨機波動度模型 ……………………………………….2-1 壹 Hull & White（1987）模型 …………………………..2-1 貳 Wiggins（1987）模型 ………………………………..2-3 參 Johnson & Shanno（1987）模型 …………………….2-4 肆 Scott（1987）模型 …………………………………...2-5 伍 Stein & Stein（1991）模型 …………………………..2-6 陸 Heston（1993）模型 …………………………………2-8 第二節 GARCH體系---波動度估計之方法 ……………………2-10 壹 GARCH模型 …………………………………………2-10 貳 EGARCH模型 ………………………………………..2-10 參 GJR-GARCH模型 ……………………………………2-11 肆 N-GARCH模型 ………………………………………2-12 伍 T-GARCH模型 ………………………………………2-12 第三章 研究方法 第一節 波動度之估計方法 ……………………………………….3-1 壹 歷史波動度 ……………………………………………3-1 貳 GARCH(1,1)模型 ……………………………………..3-2 參 EGARCH(1,1)模型 …………………………………..3-3 肆 GJR-GARCH（1,1）模型 …………………………...3-5 伍 ARCH-M（1,1）模型 ………………………………..3-7 陸 隱含波動度模型（Implied Volatility） ………………3-8 第二節 選擇權評價公式之探討 ………………………………….3-24 壹 Black & Scholes的選擇權評價模型 ………………...3-24 貳 考慮交易成本極間斷性避險下的選擇權評價模型 ...3-25 第四章 實證結果與分析 第一節 波動度的估計與預測能力 ………………………………4-1 第二節 選擇權評價理論的實證結果 …………………………4-18 第三節 認購權證涉險值（VAR）之衡量與應用 ………………4-53 第五章 結論與建議 ……………………………………………….5-1 附錄 …………………………………………………………附-1 參考文獻 …………………………………………………………Ⅰ

本篇論文的目的在驗證台股期貨報酬率與其波動度之間的相對應關係是由槓桿效果或是波動度回饋效果之因果關係所驅動，並且分別以日資料以及高頻率日內資料進行實證。實證結果發現在高頻率日內資料的應用下，能夠比日資料揭露出更詳細的波動度資訊，將報酬率與波動度間的對應關係描繪得更加明瞭。且在大多數資料期間內，同期下，台股期貨報酬率與其波動度之間會呈現負相關性，而負相關的程度會隨著報酬率遞延期數越長而逐漸遞減，因此可以發現報酬率與其波動度間呈現一個經由報酬率進而影響波動度的對應關係，與槓桿效果的因果關係雷同。最後，本文亦採用了常見的波動度預測模型，歷史模擬法、GARCH(1,1)模型、EGARCH(1,1)模型以及GJR-GARCH(1,1)模型，觀察這些波動度模型所預測出之波動度是否含有上述驗證的資訊意涵，並比較各波動度模型的預測能力，結果發現GJR-GARCH模型於樣本外期間所預測之波動度，其與報酬率之間不但具有槓桿效果的因果關係，且預測能力亦於四個波動度模型中表現最佳。

Value at Risk (VaR) and Expected Shortfall (ES) are methods often used to measure market risk. Inaccurate and unreliable Value at Risk and Expected Shortfall models can lead to underestimation of the market risk that a firm or financial institution is exposed to, and therefore may jeopardize the well-being or survival of the firm or financial institution during adverse markets. The objective of this study is therefore to examine various Value at Risk and Expected Shortfall models, including fatter tail models, in order to analyze the accuracy and reliability of these models. Thirteen VaR and ES models under three main approaches (Parametric, Non-Parametric and Semi-Parametric) are examined in this study. The results of this study show that the proposed model (ARMA(1,1)-GJR-GARCH(1,1)-SGED) gives the most balanced Value at Risk results. The semi-parametric model (Extreme Value Theory, EVT) is the most accurate Value at Risk model in this study for S&P 500. / October 2014