In this work we want to examine selected models with nonlinear volatility and their properties. At the beginning we define models with non-constant variance, especially ARCH, GARCH and EGARCH models. Then we study the probability distributions that are mainly used in the EGARCH model. Then we focus on the EGARCH model, describe the conditions for stationarity and invertibility of the model, define diagnostic tests and QMLE estimates of parameters. In the last chapter we perform simulation studies of the selected models and their application to real data. 1

This paper investigates the impact of the European Union membership referendum in the UK on the correlations and volatility between three different broad stock market indices, utilizing an econometric time series model called DCC GARCH. Findings support the claim of higher volatility peaks on the stock market as an immediate response to the event. Evidence indicate higher shortrun correlations between the indices as a response to higher volatility. In addition, the study present evidence that the correlation between the UK stock index and the other two indices declines after the referendum in 2016. / Studien undersöker konsekvenserna av folkomröstningen om Storbritanniens medlemskap i EU. Korrelationen och volatiliteten mellan tre olika aktiemarknadsindex jämförs med hjälp av en ekonometrisk modell för tidsserier kallad DCC GARCH. Resultaten från studien visar på omedelbart högre nivåer av volatilitet på aktiemarknaden dagarna efter omröstningen. Analysen ger stöd för hypotesen om högre nivåer av kortsiktiga korrelationer mellan indexen som en konsekvens av högre nivåer av volatilitet. Resultat visar även på att korrelationen mellan det brittiska aktieindexet och de övriga två minskar efter det undersökta eventet.

Volatility is a measure of the risk of an investment and plays an essential role in several areas of finance, including portfolio management and pricing of options. In this thesis, we have implemented and evaluated several so-called GARCH models for volatility prediction based on historical price series. The evaluation builds on different metrics and uses a comprehensive data set consisting of many assets of various types. We found that more advanced models do not, on average, outperform simpler ones. We also found that the length of the historical training data was critical for GARCH models to perform well and that the length was asset-dependent. Further, we developed and tested a method for taking exogenous variables into account in the model to improve the predictive performance of the model. This approach was successful for some of the large US/European indices such as Russell 2000 and S&P 500. / Volatilitet är ett mått på risken i en investering och spelar en viktig roll inom flera olika områden av finans, såsom portföljteori och prissättning av optioner. I det här projektet har vi implementerat och utvärderat olika, så kallade, GARCH modeller för prediktering av volatiliteten givet historisk prisdata. Utvärderingen av modellerna bygger på olika metriker och använder ett omfattande dataset med prishistorik för tillgångar av olika typer. Vi fann att mer komplexa modeller inte i allmänhet ger bättre resultat än enklare modeller. Vidare fann vi att en kritisk parameter för att erhålla goda resultat är att välja rätt längd på tidshistoriken av data som används för att träna modellen, och att den längden skiljer sig mellan olika tillgångar. Slutligen, vidareutvecklade vi modellen genom att inkorporera exogena variabler på olika sätt. Vi fann att det gick att förbättra GARCH modellerna främst med hjälp av några av de stora amerikanska och europeiska index som Russell 2000 och S&P 500.

In this study within quantitative portfolio optimization, stochastic programming is investigated as an investment decision tool. This research takes the direction of scenario based Mean-Absolute Deviation and is compared with the traditional Mean-Variance model and widely used Risk Parity portfolio. Furthermore, this thesis is done in collaboration with the First Swedish National Pension Fund, AP1, and the implemented multi-asset portfolios are thus tailored to match their investment style. The models are evaluated on two different fund management levels, in order to study if the portfolio performance benefits from a more restricted feasible domain. This research concludes that stochastic programming over the investigated time period is inferior to Risk Parity, but outperforms the Mean-Variance Model. The biggest aw of the model is its poor performance during periods of market stress. However, the model showed superior results during normal market conditions. / I denna studie inom kvantitativ portföljoptimering undersöks stokastisk programmering som ett investeringsbeslutsverktyg. Denna studie tar riktningen för scenariobaserad Mean-Absolute Deviation och jämförs med den traditionella Mean-Variance-modellen samt den utbrett använda Risk Parity-portföljen. Avhandlingen görs i samarbete med Första AP-fonden, och de implementerade portföljerna, med era tillgångsslag, är därför skräddarsydda för att matcha deras investeringsstil. Modellerna utvärderas på två olika fondhanteringsnivåer för att studera om portföljens prestanda drar nytta av en mer restrektiv optimeringsmodell. Den här undersökningen visar att stokastisk programmering under undersökta tidsperioder presterar något sämre än Risk Parity, men överträffar Mean-Variance. Modellens största brist är dess prestanda under perioder av marknadsstress. Modellen visade dock något bättre resultat under normala marknadsförhållanden.

金融資產報酬的波動一直都是財務市場熱衷研究的主題, 由於真正報酬的波動無法確知, 造成無法判斷何者為衡量報酬波動最佳的模型, 進而導致預測未來報酬的風險增加。因此, 本文利用靴帶抽樣法(Bootstrap)反覆抽樣的估計方式, 建立報酬與報酬波動的預測區間來衡量由估計模型參數產生的不確定性, 希望能藉此更瞭解資產報酬的變化以降低投資風險。鑒於目前衡量報酬波動的模型眾多, 文中將採用文獻上普遍最能掌握金融資產報酬波動現象的GARCH模型, 作為衡量報酬波動的方法, 再以靴帶抽樣方法估計其報酬與報酬波動的預測區間, 透過有限樣本的模擬將估計模型參數不確定性的靴帶抽樣方法與其他方法比較, 證明靴帶抽樣法最能適當的捕捉報酬波動真實的情況。最後, 由台灣上市股票市場中選取四支不同類股的各股以日報酬進行實證研究, 結果顯示各股的日報酬都具有波動變異的現象, 進一步估計樣本外不同範圍的波動預測區間, 發現利用估計模型參數不確定性的靴帶抽樣方法可以適當地涵蓋波動的變化。

本研究使用相關係數隨時間變動的雙變量GARCH(1,1)模型(time-varying correlation bivariate GARCH(1,1) model)，討論台灣股票市場中，大公司與小公司之間的報酬、衝擊(shock)、波動(volatility)是否互為影響為主軸。其次，為了了解不同估計方法、相關係數的設定和解釋變數對結果造成的影響，亦設立了3種模型，作為本研究的比較模型。 本研究發現大公司與小公司過去的報酬，存在雙向的報酬外溢效果。換句話說，大公司與小公司過去的報酬分別都對「本身報酬」有影響外，對「對方的報酬」也有影響。進一步發現到：大公司過去受到的衝擊和波動不僅對本身的條件變異數造成影響，也影響到小公司的條件變異數。但相反地，小公司過去受到的衝擊和波動，只對本身的條件變異數有影響，對大公司的條件變異數沒有影響，所以大、小公司間的衝擊外溢效果和波動外溢效果有不對稱的現象。 從不同模型之比較也發現，在討論大公司與小公司的報酬及波動時，應重視兩者彼此相互影響的關係，在估計時使用多變量的方法，以捕捉彼此相依的條件共變異數及條件變異數之動態過程。除此之外，也應考量兩者的相關係數隨時間變動的特性，及過去的波動對描述對方條件變異數的重要性。 關鍵字：多變量、GARCH模型、波動性、外溢效果、不對稱性

L'objectif de cette thèse est d'étudier le problème de la modélisation des changements de régime dans les modèles a corrélations conditionnelles dynamiques en nous intéressant plus particulièrement a l'approche Markov-switching. A la différence de l'approche standard basée sur le modèle à chaîne de Markov caché (HMM) de base, nous utilisons des extensions du modèle HMM provenant des modèles graphiques probabilistes. Cette discipline a en effet proposé de nombreuses dérivations du modèle de base permettant de modéliser des structures complexes. Cette thèse se situe donc a l'interface de deux disciplines: l'économétrie financière et les modèles graphiques probabilistes.Le premier essai présente un modèle construit a partir d'une structure hiérarchique cachée markovienne qui permet de définir différents niveaux de granularité pour les régimes. Il peut être vu comme un cas particulier du modèle RSDC (Regime Switching for Dynamic Correlations). Basé sur le HMM hiérarchique, notre modèle permet de capter des nuances de régimes qui sont ignorées par l'approche Markov-Switching classique.La seconde contribution propose une version Markov-switching du modèle DCC construite a partir du modèle HMM factorise. Alors que l'approche Markov-switching classique suppose que les tous les éléments de la matrice de corrélation suivent la même dynamique, notre modèle permet à tous les éléments de la matrice de corrélation d'avoir leur propre dynamique de saut. Markov-switching. A la différence de l'approche standard basée sur le modèle à chaîne de Markov caché (HMM) de base, nous utilisons des extensions du modèle HMM provenant des modèles graphiques probabilistes. Cette discipline a en effet propose de nombreuses dérivations du modèle de base permettant de modéliser des structures complexes. Cette thèse se situe donc a l'interface de deux disciplines: l'économétrie financière et les modèles graphiques probabilistes.Le premier essai présente un modèle construit a partir d'une structure hiérarchique cachée markovienne qui permet de définir différents niveaux de granularité pour les régimes. Il peut ^etre vu commeun cas particulier du modele RSDC (Regime Switching for Dynamic Correlations). Base sur le HMMhierarchique, notre modele permet de capter des nuances de regimes qui sont ignorees par l'approcheMarkov-Switching classique.La seconde contribution propose une version Markov-switching du modele DCC construite a partir dumodele HMM factorise. Alors que l'approche Markov-switching classique suppose que les tous les elementsde la matrice de correlation suivent la m^eme dynamique, notre modele permet a tous les elements de lamatrice de correlation d'avoir leur propre dynamique de saut.Dans la derniere contribution, nous proposons un modele DCC construit a partir d'un arbre dedecision. L'objectif de cet arbre est de relier le niveau des volatilites individuelles avec le niveau descorrelations. Pour cela, nous utilisons un arbre de decision Markovien cache, qui est une extension de HMM. / The objective of this thesis is to study the modelling of change in regime in the dynamic conditional correlation models. We focus particularly on the Markov-switching approach. Unlike the standard approach based on the Hidden Markov Model (HMM), we use extensions of HMM coming from probabilistic graphical models theory. This discipline has in fact proposed many derivations of the basic model to model complex structures. Thus, this thesis can be view at the interface of twodisciplines: financial econometrics and probabilistic graphical models.The first essay presents a model constructed from a hierarchical hidden Markov which allows to increase the granularity of the regimes. It can be seen as a special case of RSDC model (Regime Switching for Dynamic Correlations). Based on the hierarchical HMM, our model can capture nuances of regimes that are ignored by the classical Markov-Switching approach.The second contribution proposes a Markov-switching version of the DCC model that is built from the factorial HMM. While the classical Markov-switching approach assumes that all elements of the correlation matrix follow the same switching dynamic, our model allows all elements of the correlation matrix to have their own switching dynamic.In the final contribution, we propose a model DCC constructed based on a decision tree. The objective of this tree is to link the level of volatility with the level of individual correlations. For this, we use a hidden Markov decision tree, which is an extension of HMM.

This diploma thesis is devoted to the linkages among stock, bond and foreign exchange markets in the Czech Republic, Austria, Germany and Poland during the period from the beginning of the year 2007 to the end of the year 2014. In order to complexly describe the interconnections among the markets, we utilized two kinds of spillover indices (from the generalized and structural VAR model), dynamic correlation coefficients obtained from the multivariate GARCH model and contemporaneous coefficients from the structural VAR model that was identified through heteroskedasticity in structural shocks. These methods enabled us to describe the linkages among the markets from different angles, to capture their time evolution and to obtain a notion about the transmission mechanism among these markets in Central Europe. The results, inter alia, indicate an intensifying interconnection among the markets during crisis periods, lowering impact of stock markets, increasing influence of bonds and a dominant role of German bonds and Austrian stocks. At the same time, we were able to capture the influence of the European sovereign debt crisis on the spillovers and on the intensity of linkages among the markets. We showed that the intensity of linkages among bond markets relented, probably as a result of higher emphasis on the...

本文以1997年6月至2016年7月臺灣上櫃股票市場做為研究樣本，透過建立一Bivariate Diagonal BEKK GARCH (1,1)-in-mean模型，並以大盤週轉率形成之總合流動性指標與大盤超額報酬率之共變異數做為系統流動性風險之衡量指標，觀察系統流動性風險在臺灣上櫃股票市場是否有被訂價。結論除發現系統流動性風險有確實被訂價外，系統流動性風險溢價還兼具穩定性，且對市場超額報酬率有顯著的影響力。 / By constructing a bivariate diagonal BEKK Garch (1,1)-in-mean model and using the covariance between the excess market return and turnover rate as aggregate systematic liquidity proxy, the study tries to examine whether systematic liquidity risk was priced on Taiwan OTC stock market during the period of June 1997-July 2016. Based on monthly data, the findings suggest that not only the systematic liquidity risk was well priced on Taiwan OTC stock market, but the phenomenon also possessed stability and could have significant impact on stock returns.

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.