This thesis aims to investigate the potential effects on Value at Risk (VaR) measurements when including social media sentiments from Reddit and Twitter. The investigated stock companies are Apple, Alphabet and Tesla. Furthermore, the VaR measurements will be computed through volatility forecasts and assumptions about the return distributions. The volatility will be forecasted by two different models and each model will both include and exclude social media sentiments, so there will be four different volatility forecasts for each stock. Moreover, the volatility models will be the Heterogeneous autoregression (HAR) model and the Heterogeneous autoregression Neural Network (HAR-NN) model. The assumptions of return distributions are a log-logistic distribution and a log-normal distribution. In addition to this, the VaR measurements are computed and evaluated through number of breaches for each of the volatility forecasts and for both assumptions of a return distribution. The result shows that there is an improvement in forecasting volatility for Apple and Alphabet, as well as fewer VaR breaches for both assumptions of log-return distributions. However, the results for Tesla showed that the volatility forecasts were better when excluding social media sentiment. A possible reason for this might be due to Twitter posts made by influential people, like Elon Musk that would have a larger effect on the volatility than the average sentiment score over that day. Another possible explanation to this might be due to multicollinearity. Overall, the results showed that the assumption of a log-logistic distribution was more suitable over a log- normal return distribution for all three stocks. / Den här studien undersöker de potentiella effekterna av att inkludera sentiment från Reddit och Twitter vid beräkning av det finansiella riskmåttet VaR. De undersökta aktierna är Apple, Alphabet och Tesla. VaR måtten beräknas genom att förutspå volatiliteten samt genom att göra antaganden om aktiernas avkast- ningsfördelning. Volatiliteten förutspås genom två olika modeller och bägge modeller kommer både att inkludera sentiment från sociala medier samt exkludera sentimenten. Därav kommer det totalt vara fyra olika volatilitets prognoser för vardera aktie. Volatilitetsmodellerna som används i denna studie är HAR modellen och HAR-NN modellen. De antaganden som görs om logartimen av avkastningsfördelningarna är att de följer en logistik fördelning samt en normalfördelning. Dessutom är VaR måtten beräknade och eval- uerade genom antalet gånger portföljen överskrider VaR måttet för varje volatilitetsprognos och för vardera antagande om avkastningsfördelning. Resultaten av denna studie visar att inkludering av sentiment från sociala medier förbättrar volatilitetsprognosen för Apple och Alphabet, samt att portföljen överskrider dessa VaR mått för båda fördelningsantaganden. Däremot, visar resultaten för Tesla att volatilitetsprog- nosen är sämre då sentiment från sociala medier inkluderas i modellerna. En möjlig anledning till detta skulle kunna vara på grund av inflyelserika personer, så som Elon Musk vars Twitter inlägg har större påverkan på aktievolatiliteten än medelsentimentet. En annan anledning till detta skulle kunna vara på grund av multikollinearitet, ifall sentimenten till Tesla är starkt korrelerade med volatiliteten. Samman- taget visade resultaten att antagandet av att logaritmen av avkastningarna följer en logistikt fördelning var mer passande än antagandet av en normalfördelning för alla tre aktier.

股票市場的報酬率相對於金融市埸是非常重要的，因為其背後的真實機率分配對各種資產定價及選擇權的評價模型都有決定性的影響。本文考慮台灣股票市埸具有漲跌停板的限制來驗證實證中股票報酬機率分配的「厚尾」的現象，希望透過我們的研究能對財務理論在國內金融市埸的應用有更進一步的了解。我們選定了常態分配、對數常態分配及一般化第二種貝它分配 (GB2)來當作是台灣股票報酬率的真實機率分配，以動差法比較再以概似比檢定法(LR test)選出一表現最好的機率分配。由選取的25支國內股票中發現一般化第二種貝它分配 (GB2)可以解釋偏態和峰態對報酬率的影響並且也是概似比檢定法所選出的最適報酬率分配，由此可知一般化第二種貝它分配 (GB2)較為適合作為台灣股票報酬的真實機率分配。

In this paper, we first present the basic principles of set theory and combinatorial analysis which are the most useful tools in computing probabilities. Then, we show some important properties derived from axioms of probability. Conditional probabilities come into play not only when some partial information is available, but also as a tool to compute probabilities more easily, even when partial information is unavailable. Then, the concept of random variable and its some related properties are introduced. For univariate random variables, we introduce the basic properties of some common discrete and continuous distributions. The important properties of jointly distributed random variables are also considered. Some inequalities, the law of large numbers and the central limit theorem are discussed. Finally, we introduce additional topics the Poisson process.

依據之前的文獻研究，市場上主要是在LHP (Large Homogeneous Portfolio) 假設下利用單因子常態關聯結構模式(One factor double Gaussian copula model) 評價擔保債權憑證 (Collateralized debt obligation, CDO)。但這會造成擔保債權憑證的評價與市場報價的差距過大，且會造成base correlation偏斜的情況。Kalemanova et al. (2007) 提出用Normal inverse Gaussian (NIG) 取代常態分配評價擔保債權憑證，此模型不但計算快速而且可以準確估計權益分券 (equity tranche) 的價格，但是它也過於高估了其它的分券的價格。 在本文中使用多變量封閉常態分配(Closed skew normal, 簡稱CSN) 分配取代NIG分配作擔保債權憑證分券的評價，CSN分配具有常態分配的性質，其線性組合仍具有封閉性的特質，且具有較多的參數以控制分配的偏態與峰態。但是與單因子常態關聯結構模式相同，多變量封閉常態分配的單因子關聯結構模式仍然無法估計的很準確，僅有在最高等級分券(senior tranche)的評價上有明顯的改進。 因此在本文中我們使用NIG與CSN複合分配之單因子關聯結構模式評價擔保債權憑證分券，在實例分析時得到極佳的評價結果，並且比單因子常態關聯結構模型具有更多的的參數以使模型更符合實際的需求。 / This article extends the Large Homogeneous Portfolio (LHP) and one factor double Gaussian copula approach for pricing CDOs. In the literature, the one factor double Gaussian copula model under LHP assumption fails to fit the prices of CDO tranches, moreover, it leads to the implied base correlation skew. Some researchers proposed using one factor double NIG copula model to price CDO tranches. It not only economizes on time but also fits the equity tranches exactly, but NIG models do not price other tranches well simultaneously. On the other hand, we substitute the NIG distribution with the Closed Skew normal (CSN) distribution. This family also has properties similar to the normal distribution, which is closure under convolution, and has extra parameters to control the shape. By using this model we get a better fit in the senior tranches, but it seriously overprices subordinate tranches. Thus we consider a mixture distribution of NIG and CSN distributions. The employments of this mixture distribution are comparatively well, and furthermore it brings more flexibility to the dependence structure.

This thesis deals with several statistical and probabilistic aspects of symmetry and asymmetry, both in a univariate and multivariate context, and is divided into three distinct parts. The first part, composed of Chapters 1, 2 and 3 of the thesis, solves two conjectures associated with multivariate skew-symmetric distributions. Since the introduction in 1985 by Adelchi Azzalini of the most famous representative of that class of distributions, namely the skew-normal distribution, it is well-known that, in the vicinity of symmetry, the Fisher information matrix is singular and the profile log-likelihood function for skewness admits a stationary point whatever the sample under consideration. Since that moment, researchers have tried to determine the subclasses of skew-symmetric distributions who suffer from each of those problems, which has led to the aforementioned two conjectures. This thesis completely solves these two problems. The second part of the thesis, namely Chapters 4 and 5, aims at applying and constructing extremely general skewing mechanisms. As such, in Chapter 4, we make use of the univariate mechanism of Ferreira and Steel (2006) to build optimal (in the Le Cam sense) tests for univariate symmetry which are very flexible. Actually, their mechanism allowing to turn a given symmetric distribution into any asymmetric distribution, the alternatives to the null hypothesis of symmetry can take any possible shape. These univariate mechanisms, besides that surjectivity property, enjoy numerous good properties, but cannot be extended to higher dimensions in a satisfactory way. For this reason, we propose in Chapter 5 different general mechanisms, sharing all the nice properties of their competitors in Ferreira and Steel (2006), but which moreover can be extended to any dimension. We formally prove that the surjectivity property holds in dimensions k>1 and we study the principal characteristics of these new multivariate mechanisms. Finally, the third part of this thesis, composed of Chapter 6, proposes a test for multivariate central symmetry by having recourse to the concepts of statistical depth and runs. This test extends the celebrated univariate runs test of McWilliams (1990) to higher dimensions. We analyze its asymptotic behavior (especially in dimension k=2) under the null hypothesis and its invariance and robustness properties. We conclude by an overview of possible modifications of these new tests./ Cette thèse traite de différents aspects statistiques et probabilistes de symétrie et asymétrie univariées et multivariées, et est subdivisée en trois parties distinctes. La première partie, qui comprend les chapitres 1, 2 et 3 de la thèse, est destinée à la résolution de deux conjectures associées aux lois skew-symétriques multivariées. Depuis l'introduction en 1985 par Adelchi Azzalini du plus célèbre représentant de cette classe de lois, à savoir la loi skew-normale, il est bien connu qu'en un voisinage de la situation symétrique la matrice d'information de Fisher est singulière et la fonction de vraisemblance profile pour le paramètre d'asymétrie admet un point stationnaire quel que soit l'échantillon considéré. Dès lors, des chercheurs ont essayé de déterminer les sous-classes de lois skew-symétriques qui souffrent de chacune de ces problématiques, ce qui a mené aux deux conjectures précitées. Cette thèse résoud complètement ces deux problèmes. La deuxième partie, constituée des chapitres 4 et 5, poursuit le but d'appliquer et de proposer des méchanismes d'asymétrisation très généraux. Ainsi, au chapitre 4, nous utilisons le méchanisme univarié de Ferreira and Steel (2006) pour construire des tests de symétrie univariée optimaux (au sens de Le Cam) qui sont très flexibles. En effet, leur méchanisme permettant de transformer une loi symétrique donnée en n'importe quelle loi asymétrique, les contre-hypothèses à la symétrie peuvent prendre toute forme imaginable. Ces méchanismes univariés, outre cette propriété de surjectivité, possèdent de nombreux autres attraits, mais ne permettent pas une extension satisfaisante aux dimensions supérieures. Pour cette raison, nous proposons au chapitre 5 des méchanismes généraux alternatifs, qui partagent toutes les propriétés de leurs compétiteurs de Ferreira and Steel (2006), mais qui en plus sont généralisables à n'importe quelle dimension. Nous démontrons formellement que la surjectivité tient en dimension k > 1 et étudions les caractéristiques principales de ces nouveaux méchanismes multivariés. Finalement, la troisième partie de cette thèse, composée du chapitre 6, propose un test de symétrie centrale multivariée en ayant recours aux concepts de profondeur statistique et de runs. Ce test étend le célèbre test de runs univarié de McWilliams (1990) aux dimensions supérieures. Nous en analysons le comportement asymptotique (surtout en dimension k = 2) sous l'hypothèse nulle et les propriétés d'invariance et de robustesse. Nous concluons par un aperçu sur des modifications possibles de ces nouveaux tests.

Diss. (sammanfattning) Umeå : Umeå universitet, 2010.

Η εκπόνηση της συγκεκριμένης Μεταπτυχιακής Εργασίας, εξετάζει, καταρχήν, την έννοια της πιθανότητας και τις βασικές ιδιότητές της, όπως την τυχαία μεταβλητή και τη συνάρτηση κατανομής. Παράλληλα όμως, παρουσιάζει στοιχεία βασικών διακριτών και συνεχών κατανομών, όπως της κανονικής, της ομοιόμορφης, της Poisson, και άλλων κατανομών της γραμμικής στατιστικής. Στη συνέχεια, αναφέρεται στις βασικές έννοιες της περιγραφικής στατιστικής, όπως οργάνωση και γραφική αναπαράσταση στατιστικών δεδομένων, ομαδοποίηση παρατηρήσεων, ιστόγραμμα συχνοτήτων, καθώς και περιγραφικά μέτρα γραμμικών δεδομένων. Κυρίως, όμως, η παρούσα μελέτη αποτελεί μία γενική επισκόπηση των στατιστικών μεθόδων παρουσίασης και ανάλυσης των περιοδικών δεδομένων. Με τον όρο "περιοδικά δεδομένα", εννοούμε τυχαίες διευθύνσεις και κατευθύνσεις προσανατολισμού. Η παρουσίασης των τυχαίων γωνιών, των γραφικών αναπαραστάσεων των περιοδικών δεδομένων καθώς και των περιγραφικών μέτρων - μέτρα θέσεως, διασποράς, λοξότητας, κυρτώσεως - θα μας οδηγήσουν σε μία καλύτερη προσέγγιση, κατανόηση των περιοδικών κατανομών. Επιπλέον, θα παρουσιαστούν αναλυτικά οι βασικές περιοδικές κατανομές, ομοιόμορφη και Von Mises κατανομή. Όμως, θα εξεταστούν και άλλες κατανομές μονοκόρυφες ή πολυκόρυφες, όπως οι περιελιγμένες κατανομές , η συνημίτονο και η καρδιοειδής κατανομή, οι λοξές κατανομές κ.ά. Τέλος, η εργασία θα αναφερθεί σε μία οικογένεια συμμετρικών περιοδικών κατανομών που προτάθηκε από τον κύριο Παπακωνσταντίνου και αποτελεί επέκταση της καρδιοειδούς κατανομής,σύμφωνα με εργασία των επιστημόνων Toshihiro Abe,Arthur Pewsey,Kunio Shimizu, παρέχοντας σημαντικά πλεονεκτήματα σε σχέση με άλλες οικογένειες κατανομών. / The preparation of this thesis examines, in principle,the concept of probability and its basic properties, such as the random variable and distribution function and presents data of basic discrete and continuous distributions, including normal, uniform, the Poisson, and other distributions of linear statistics. Then it refers to the basic concepts of descriptive statistics, such as the organization and the graphic representation of statistical data, grouping observations Frequency histogram as well as descriptive measures of linear data. Mostly, though, this study represents an overview of statistic methods of presentation and analysis of periodic data. By the term "periodic data" we mean random addresses and directions orientation. The presentation of random angles, graphic representations of periodic data and descriptive measures - measures of location, dispersion, skewness and kurtosis - will lead us to a better approach and understanding of periodic distributions. Furthermore, we present in detail the basic periodic distributions, the uniform and the Von Mises distribution. But other unimodal and multimodal distributions will be examined such as wrapped distributions, the cosine and cardioid distribution, skewed distributions, etc. Finally, this thesis will mention a family of symmetric periodic distributions proposed by Mr. Papakonstantinou and an extension of the cardioid distribution, according to the paper published by the scientists Toshihiro Abe,Arthur Pewsey and Kunio Shimizu, where significant advantages are provided over other families of distributions.

Submitted by Haroudo Xavier Filho (haroudo.xavierfo@ufpe.br) on 2016-04-05T14:28:43Z No. of bitstreams: 2 license_rdf: 1232 bytes, checksum: 66e71c371cc565284e70f40736c94386 (MD5) dissertacao_Raphaela(CD).pdf: 1067677 bytes, checksum: 6d371901336a7515911aeffd9ee38c74 (MD5) / Made available in DSpace on 2016-04-05T14:28:43Z (GMT). No. of bitstreams: 2 license_rdf: 1232 bytes, checksum: 66e71c371cc565284e70f40736c94386 (MD5) dissertacao_Raphaela(CD).pdf: 1067677 bytes, checksum: 6d371901336a7515911aeffd9ee38c74 (MD5) Previous issue date: 2015-07-31 / CAPES / Este trabalho analisa propriedades da família de distribuições de probabilidade Composta N e propõe a sub-família Composta Poisson-Truncada como um meio de compor distribuições de probabilidade. Suas propriedades foram estudadas e uma nova distribuição foi investigada: a distribuição Composta Poisson-Truncada Normal. Esta distribuição possui três parâmetros e tem uma flexibilidade para modelar dados multimodais. Demonstramos que sua densidade é dada por uma mistura infinita de densidades normais em que os pesos são dados pela função de massa de probabilidade da Poisson-Truncada. Dentre as propriedades exploradas desta distribuição estão a função característica e expressões para o cálculo dos momentos. Foram analisados três métodos de estimação para os parâmetros da distribuição Composta Poisson-Truncada Normal, sendo eles, o método dos momentos, o da função característica empírica (FCE) e o método de máxima verossimilhança (MV) via algoritmo EM. Simulações comparando estes três métodos foram realizadas e, por fim, para ilustrar o potencial da distribuição proposta, resultados numéricos com modelagem de dados reais são apresentados. / This work analyzes properties of the Compound N family of probability distributions and proposes the sub-family Compound Poisson-Truncated as a means of composing probability distributions. Its properties were studied and a new distribution was investigated: the Compound Poisson-Truncated Normal distribution. This distribution has three parameters and has the flexibility to model multimodal data. We demonstrated that its density is given by an infinite mixture of normal densities where in the weights are given by the Poisson-Truncated probability mass function. Among the explored properties of this distribution are the characteristic function end expressions for the calculation of moments. Three estimation methods were analyzed for the parameters of the Compound Poisson-Truncated Normal distribution, namely, the method of moments, the empirical characteristic function (ECF) and the method of maximum likelihood (ML) by EM algorithm. Simulations comparing these three methods were performed and, finally, to illustrate the potential of the proposed distribution numerical results with real data modeling are presented.

This thesis deals with several statistical and probabilistic aspects of symmetry and asymmetry, both in a univariate and multivariate context, and is divided into three distinct parts.<p><p>The first part, composed of Chapters 1, 2 and 3 of the thesis, solves two conjectures associated with multivariate skew-symmetric distributions. Since the introduction in 1985 by Adelchi Azzalini of the most famous representative of that class of distributions, namely the skew-normal distribution, it is well-known that, in the vicinity of symmetry, the Fisher information matrix is singular and the profile log-likelihood function for skewness admits a stationary point whatever the sample under consideration. Since that moment, researchers have tried to determine the subclasses of skew-symmetric distributions who suffer from each of those problems, which has led to the aforementioned two conjectures. This thesis completely solves these two problems.<p><p>The second part of the thesis, namely Chapters 4 and 5, aims at applying and constructing extremely general skewing mechanisms. As such, in Chapter 4, we make use of the univariate mechanism of Ferreira and Steel (2006) to build optimal (in the Le Cam sense) tests for univariate symmetry which are very flexible. Actually, their mechanism allowing to turn a given symmetric distribution into any asymmetric distribution, the alternatives to the null hypothesis of symmetry can take any possible shape. These univariate mechanisms, besides that surjectivity property, enjoy numerous good properties, but cannot be extended to higher dimensions in a satisfactory way. For this reason, we propose in Chapter 5 different general mechanisms, sharing all the nice properties of their competitors in Ferreira and Steel (2006), but which moreover can be extended to any dimension. We formally prove that the surjectivity property holds in dimensions k>1 and we study the principal characteristics of these new multivariate mechanisms.<p><p>Finally, the third part of this thesis, composed of Chapter 6, proposes a test for multivariate central symmetry by having recourse to the concepts of statistical depth and runs. This test extends the celebrated univariate runs test of McWilliams (1990) to higher dimensions. We analyze its asymptotic behavior (especially in dimension k=2) under the null hypothesis and its invariance and robustness properties. We conclude by an overview of possible modifications of these new tests./<p><p>Cette thèse traite de différents aspects statistiques et probabilistes de symétrie et asymétrie univariées et multivariées, et est subdivisée en trois parties distinctes.<p><p>La première partie, qui comprend les chapitres 1, 2 et 3 de la thèse, est destinée à la résolution de deux conjectures associées aux lois skew-symétriques multivariées. Depuis l'introduction en 1985 par Adelchi Azzalini du plus célèbre représentant de cette classe de lois, à savoir la loi skew-normale, il est bien connu qu'en un voisinage de la situation symétrique la matrice d'information de Fisher est singulière et la fonction de vraisemblance profile pour le paramètre d'asymétrie admet un point stationnaire quel que soit l'échantillon considéré. Dès lors, des chercheurs ont essayé de déterminer les sous-classes de lois skew-symétriques qui souffrent de chacune de ces problématiques, ce qui a mené aux deux conjectures précitées. Cette thèse résoud complètement ces deux problèmes.<p><p>La deuxième partie, constituée des chapitres 4 et 5, poursuit le but d'appliquer et de proposer des méchanismes d'asymétrisation très généraux. Ainsi, au chapitre 4, nous utilisons le méchanisme univarié de Ferreira and Steel (2006) pour construire des tests de symétrie univariée optimaux (au sens de Le Cam) qui sont très flexibles. En effet, leur méchanisme permettant de transformer une loi symétrique donnée en n'importe quelle loi asymétrique, les contre-hypothèses à la symétrie peuvent prendre toute forme imaginable. Ces méchanismes univariés, outre cette propriété de surjectivité, possèdent de nombreux autres attraits, mais ne permettent pas une extension satisfaisante aux dimensions supérieures. Pour cette raison, nous proposons au chapitre 5 des méchanismes généraux alternatifs, qui partagent toutes les propriétés de leurs compétiteurs de Ferreira and Steel (2006), mais qui en plus sont généralisables à n'importe quelle dimension. Nous démontrons formellement que la surjectivité tient en dimension k > 1 et étudions les caractéristiques principales de ces nouveaux méchanismes multivariés.<p><p>Finalement, la troisième partie de cette thèse, composée du chapitre 6, propose un test de symétrie centrale multivariée en ayant recours aux concepts de profondeur statistique et de runs. Ce test étend le célèbre test de runs univarié de McWilliams (1990) aux dimensions supérieures. Nous en analysons le comportement asymptotique (surtout en dimension k = 2) sous l'hypothèse nulle et les propriétés d'invariance et de robustesse. Nous concluons par un aperçu sur des modifications possibles de ces nouveaux tests. / Doctorat en Sciences / info:eu-repo/semantics/nonPublished

The thesis presents methods and criteria for creation and optimization of design of computer experiments. Using the core of a program Freet the optimized designs were created by combination of these methods and criteria. Then, the suitability of the designs for statistical analysis of the tasks vith input random variables was assessed by comparison of the obtained results of six selected functions and the exact (analytically obtained) solutions. Basic theory, definitions of the evaluated functions, description of the setting of optimization and the discussion of the obtained results, including recommendations related to identified weaknesses of certain designs, are presented. The thesis also contains a description of an application that was created to display the results.