This thesis deals with stock modelling using ARCH and GARCH time series. Important aspect of stock modelling is to capture volatility correctly. Volatility in finance is usually defined as a standard deviation of asset returns. Many different models, which are summarized in the first part of this thesis, are used to model volatility. This thesis focus on multivariate volatility models including multivariate GARCH models. An approach to constructing a conditional maximum likelihood estimate to these methods is given. Discussed theory is applied on real financial data. In numeric application there is a construction of a volatility estimates for two specific stocks using models described in the first part of this thesis. Using the same financial data various bivariate models are compared. Based on comparison using maximum likelihood a specific model for these stocks is recommended. Powered by TCPDF (www.tcpdf.org)

Βασικός στόχος αυτής τη εργασίας είναι να παρουσιάσει με λεπτομερή και τεκμηριωμένο τρόπο την διαδικασία που ακολουθεί ένας χρηματοοικονομικός αναλυτής έτσι ώστε να προσδιορίσει την σχέση απόδοσης και κινδύνου κάποιων χρεογράφων με απώτερο σκοπό να καταλήξει σε ορθολογικά συμπεράσματα που μπορούν να τον οδηγήσουν στις βέλτιστες αποφάσεις. Οι αποφάσεις αυτές θα αφορούν την δόμηση ενός βέλτιστου χαρτοφυλακίου χρεογράφων το οποίο για δεδομένο κίνδυνο θα αποφέρει την μέγιστη αναμενόμενη απόδοση η αντίστροφα με δεδομένη την επιθυμητή απόδοση θα ενέχει το ελάχιστο ρίσκο . Η χρήση του απλού γραμμικού υποδείγματος , της μεθόδου ελαχίστων τετραγώνων (OLS) , των διαστημάτων εμπιστοσύνης και της στατιστικής συμπερασματολογίας είναι κάποιες από τις μεθόδους που θα εφαρμόσουμε για να προσδιορίσουμε με ακρίβεια την σχέση απόδοσης και κινδύνου χρεογράφων των οποίων έχουμε επιλέξει για τις εφαρμογές μας . Οι διαταράξεις των υποθέσεων του απλού γραμμικού υποδείγματος , όπως η αυτοσυσχέτιση και η ετεροσκεδαστικότητα είναι επίσης αντικείμενα προς εξέταση ,παράγοντες οι οποίοι αλλοιώνουν τις οικονομετρικές εκτιμήσεις της μεθόδου των ελαχίστων τετραγώνων και πρέπει να άρονται από τον αναλυτή, έτσι ώστε να καταλήγει η ανάλυση και η έρευνα των χρηματοοικονομικών εφαρμογών σε αξιόπιστες εκτιμήσεις . Ιδίως στην αντιμετώπιση της ετεροσκεδαστικότητας, η χρήση των μοντέλων ARCH/GARCH, μπορεί να μας οδηγήσει στο ζητούμενο ,το οποίο είναι η εκτίμηση και η πρόβλεψη του μελλοντικού κινδύνου αγοράς ενός χρεογράφου όπως η μετοχή . / The basic goal of this paper is to present in an analytic and trustworthy way, the process that a financial analyst follows so as to evaluate the relationship between risk and return of stocks. This kind of process will lead the analyst to rational conclusions and effective decisions which concern the structure of optimal portfolios of stocks. The optimal portfolio structure offers a maximum expected return if the risk is known and vice versa (minimum risk when the return is known).The use of the simple linear model, least square method and statistical conclusion function, are some of the methods that will help us to measure the relationship between risk and return of the stocks that we have chose to use for our applications. The disorders of the simple linear model assumptions, autocorrelation and heteroscedasticity, are parameters who are making the estimations of least squares method spurious. The analyst has to detect these problems so as the analysis and research of financial applications to conclude rational and effective estimations.Specialy in coping with heteroscedasticity, the use of ARCH/GARCH models can lead us to the objective, the estimation and forecast of the future risk of the stock being studied.

This essay investigates three different GARCH-models (GARCH, EGARCH and GJR-GARCH) along with two distributions (Normal and Student’s t), which are used to forecast the Value at Risk (VaR) for different return series. Seven major international equity indices are examined. The purpose of the essay is to answer which of the three models that is better at forecasting the VaR and which distribution is more appropriate.  The results show that the EGARCH(1,1)  is preferred for all indices included in the study.

Neste trabalho, fazemos uma breve introdução à Econofísica e às grandezas estatísticas relevantes para o estudo de um ativo financeiro. Estas grandezas são estudadas detalhadamente para o índice NYSE Composto. Determinamos o tempo de autocorrelação e o espectro de potência, cujos resultados indicam a presença de uma correlação de curto alcance. Através do expoente de Hurst, investigamos o tipo de correlação presente e detectamos a presença de multifractalidade. A volatilidade do índice NYSE mostrou-se análoga a um processo de Wiener. Por outro lado, a função densidade de probabilidade do índice NYSE foi ajustada por uma distribuição de Lévy simétrica com alpha = 1,47. Apresentamos os modelos de variância autoregressiva ARCH e GARCH. Em particular, focalizamos o modelo Markoviano GARCH(1,1). Este modelo tem três parâmetros de controle. Mostramos que, para o índice NYSE, o uso do tempo de autocorrelação na determinação deste conjunto de parâmetros de controle não é a melhor escolha. Resultados muito mais satisfatórios são obtidos se utilizarmos o sexto momento padronizado, uma vez que o ganho no ajuste da função de autocorrelação temporal é muito mais expressivo. A proposta de utilização do sexto momento é robusta e se aplica tanto ao modelo GARCH Gaussiano quanto ao modelo GARCH Exponencial. Desenvolvemos uma técnica de expansão em série para obter o sexto momento padronizado em função dos três parâmetros de controle. Obtivemos uma expressão analítica exata para a curtose do modelo GARCH Exponencial. Ambas as versões Gaussiana e Exponencial apresentam um desempenho equivalente na descrição da função densidade de probabilidade e da função de autocorrelação temporal. Porém, no que tange às leis de escala temporal, medidas através da probabilidade de retorno à origem, o modelo Exponencial tem, clara e inequivocamente, um melhor desempenho que o modelo Gaussiano, pois apresenta um expoente da lei de escala temporal em bom acordo com o expoente do índice NYSE. / In this thesis, we briefly give an introduction to Econophysics and discuss some important statistical quantities used in the study of a financial asset. This quantities are meticulously studied for the NYSE Composite Index. For its time series, we determine the time autocorrelation and the power spectrum, which show the presence of a short range correlation. By means of the Hurst exponent, we investigate the kind of autocorrelation which is present and we detected the presence of multifractality. The volatility of the NYSE Index show a behavior analogous to a Wiener process. On the other hand, the probability density function was adjusted by a symmetric Lévy distribuition with alpha = 1.47. We present the variance autoregressive ARCH and GARCH models. More specifically, we focus on the Markovian GARCH(1,1) model. This model has three control parameters. We show that, for the NYSE Index, the use of the time autocorrelation to determinate the set of control parameters is not the best choice. Instead, results much more reasonable are obtained if the standardized sixth moment is used, as can be seen by the adjust of the time autocorrelation function. The proposal of the sixth moment is robust and applies for both the Gaussian and the Exponential GARCH models. We developed a series expansion technique to get the standardized sixth moment as a function of the three control parameters. We found an exact analytic expression for the kurtosis of the Exponential GARCH model. Both the Gaussian and the Exponential versions exhibit an equivalent performance in the description of the probability density function and the time autocorrelation function. However, with respect to the time scaling laws (measured by the probability of return to the origin) the Exponential model shows, in a clear and unequivocal way, a better performance than the Gaussian model, since it gives a time horizon exponent much more close to the real NYSE exponent.

Neste trabalho, fazemos uma breve introdução à Econofísica e às grandezas estatísticas relevantes para o estudo de um ativo financeiro. Estas grandezas são estudadas detalhadamente para o índice NYSE Composto. Determinamos o tempo de autocorrelação e o espectro de potência, cujos resultados indicam a presença de uma correlação de curto alcance. Através do expoente de Hurst, investigamos o tipo de correlação presente e detectamos a presença de multifractalidade. A volatilidade do índice NYSE mostrou-se análoga a um processo de Wiener. Por outro lado, a função densidade de probabilidade do índice NYSE foi ajustada por uma distribuição de Lévy simétrica com alpha = 1,47. Apresentamos os modelos de variância autoregressiva ARCH e GARCH. Em particular, focalizamos o modelo Markoviano GARCH(1,1). Este modelo tem três parâmetros de controle. Mostramos que, para o índice NYSE, o uso do tempo de autocorrelação na determinação deste conjunto de parâmetros de controle não é a melhor escolha. Resultados muito mais satisfatórios são obtidos se utilizarmos o sexto momento padronizado, uma vez que o ganho no ajuste da função de autocorrelação temporal é muito mais expressivo. A proposta de utilização do sexto momento é robusta e se aplica tanto ao modelo GARCH Gaussiano quanto ao modelo GARCH Exponencial. Desenvolvemos uma técnica de expansão em série para obter o sexto momento padronizado em função dos três parâmetros de controle. Obtivemos uma expressão analítica exata para a curtose do modelo GARCH Exponencial. Ambas as versões Gaussiana e Exponencial apresentam um desempenho equivalente na descrição da função densidade de probabilidade e da função de autocorrelação temporal. Porém, no que tange às leis de escala temporal, medidas através da probabilidade de retorno à origem, o modelo Exponencial tem, clara e inequivocamente, um melhor desempenho que o modelo Gaussiano, pois apresenta um expoente da lei de escala temporal em bom acordo com o expoente do índice NYSE. / In this thesis, we briefly give an introduction to Econophysics and discuss some important statistical quantities used in the study of a financial asset. This quantities are meticulously studied for the NYSE Composite Index. For its time series, we determine the time autocorrelation and the power spectrum, which show the presence of a short range correlation. By means of the Hurst exponent, we investigate the kind of autocorrelation which is present and we detected the presence of multifractality. The volatility of the NYSE Index show a behavior analogous to a Wiener process. On the other hand, the probability density function was adjusted by a symmetric Lévy distribuition with alpha = 1.47. We present the variance autoregressive ARCH and GARCH models. More specifically, we focus on the Markovian GARCH(1,1) model. This model has three control parameters. We show that, for the NYSE Index, the use of the time autocorrelation to determinate the set of control parameters is not the best choice. Instead, results much more reasonable are obtained if the standardized sixth moment is used, as can be seen by the adjust of the time autocorrelation function. The proposal of the sixth moment is robust and applies for both the Gaussian and the Exponential GARCH models. We developed a series expansion technique to get the standardized sixth moment as a function of the three control parameters. We found an exact analytic expression for the kurtosis of the Exponential GARCH model. Both the Gaussian and the Exponential versions exhibit an equivalent performance in the description of the probability density function and the time autocorrelation function. However, with respect to the time scaling laws (measured by the probability of return to the origin) the Exponential model shows, in a clear and unequivocal way, a better performance than the Gaussian model, since it gives a time horizon exponent much more close to the real NYSE exponent.

No description available.

Yes / It is believed that financial markets are integrated and sensitive to news – including political conflicts in some regions of the world. Furthermore, financial markets seem to react differently to information flows from one region to another. The purpose of this research is to discern the effects of the recent Middle East and North Africa (MENA) conflicts – commonly referred to as the Arab Spring – on the volatility of risks and returns of global and regional stock markets as well as Gold and Oil markets. To be more specific, we consider the main uprisings in Tunisia, Egypt, Libya and Yemen and their impact on financial markets – as measured by the volatility of their risks and returns. In sum, we cluster 53 stock markets into 6 regions; namely, developed, developing, MENA, Asia, Europe, and Latin America countries, and use T-GARCH to assess the reaction of these regions to each uprising event independently. In addition, we use GARCH-M to assess the reaction of these regions stock markets as well as Gold and Oil markets to the uprisings of MENA as a whole. Our empirical findings suggest that the uprising events of MENA have more impact on the volatility of risks and returns of developed, developing, and Europe regions than MENA itself. In addition, although the results show that the volatility of both risks and returns of both developed and MENA regions are significantly affected by general conflicts in MENA, the volatility of MENA is affected during all intervals and with higher significance level. Furthermore, while MENA uprisings as a whole impact on the volatility of risk of oil (after 5 days) and gold (immediately after entering news) significantly, the returns of these markets are not affected by conflicts.

No description available.

There exist a lot of empirical researches, that examine what factors effect the stock market volatility. The concept of investor sentiment is quite popular and is frequently discussed. However, there does not exist any research which would study the relation between the change in election preferences during the presidential campaigns and stock market volatility. The present thesis explores the effect of political sentiment on United States and French models. Here, we construct the model, which examines the effect of change in election preferences on the volatility. The results suggest, that change in election preferences does not affect the stock market volatility during the presidential campaign. Thus, its inclusion to the model does not increase the prediction power.

House Price in Taiwan is very volatile during the past few decades. As Taiwan go into enormous boom, more and more amount of money invest in the house market. Although house investment is considered as a good investment tool with low risk and inflation hedge properties, its risk can not be underestimated. Therefore, by using the GARCH model, this paper tries to analyze volatilities of investment return in the Taipei housing market from 1973 to 2002. For existing housing, we are not able to use GARCH to model investment volatility because of uncorrelated term risks. On the contrary, pre-sale housing contains correlated term risk. We adopt ARMA(4,4)-GARCH(1,1) to model the investment volatility of pre-sale housing. The investment risk of pre-sale housing is not constant but is time-varying. When an unexpected event happened, the shock will persist but decay from 86 percent in the next term to 40 percent in the sixth term. And we can observe volatility cluster phenomenon from the graph of conditional variance. During 1973 to 1975¡B1979 to 1983 and 1987 to 1990, the risks are higher than other period. Because previous studies commonly suggest some structural changes in the Taiwan housing market, we also control the risk premium affected by the structural changes in our model. We found ARMA(4,4)-GARCH(1,1) can still model the investment volatility process of pre-sale housing, but there is no evidence of risk premium caused by structural changes.