自從Engle（1982）觀察金融資產報酬序列具有波動叢聚的現象，進而提出自迴歸條件變異數異質（Autoregressive Conditional Heteroskedasticity）模型後，解決了傳統的時間序列模型如自我迴歸移動平均（Autoregressive Moving Average；簡稱ARMA）模型在財務金融的實證研究上對變異數異質（Heteroskedasticity）的現象不能做有效解釋的問題後。陸續的延申模型如Bollerslev（1986）一般化自我迴歸條件異質變異（General Autoregressive Conditional Heteroskedasticity；簡稱GARCH）模型、Chou（1988）的GARCH - M（General Autoregressive Conditional Heteroskedasticity in Mean；簡稱GARCH - M）模型，已廣泛的應用於分析股票市場股價持續波動的問題上。 同時，由於國際股市間通常存在著訊息傳遞的現象，國際股市波動命題已在學術界廣泛的被討論，因此本文選取與台灣經濟景氣息息相關的美國、日本來分析訊息的傳遞型式，並且討論風險貼水項在解釋股價指數報酬上是否具有解釋能力。此外，若美、日、台三地的股價指數無法擊敗隨機漫步模型Ⅰ，則三地股票市場是效率市場，股價指數將是完全隨機不可預測的，因此本文經由BDS檢定方法來檢定隨機漫步模型Ⅰ是否足以表徵三地的價格過程，做為在是否進行多變數GARCH模型估計的參考。同時，若市場間的非同步現象嚴重的話將對估計結果產生影響，因此，本文也對三地股市非同步現象加以探討，視其是否有非同步調整的必要再加以調整，以避免估計時產生誤差。 接著，本文亦就股價指數歷史資料橫斷面探討國際間股市的互動性，經由Tank and Kwok（1997）模型加以延申，進行台灣、美國、日本國際投資組合多角化效益之分析，探討的命題主要是週末效果、元月效果，和Rogalski效果是否顯著存在於台灣、美國、日本股票市場，即分析上述效果是否對國際投資組合均報酬、報酬波動、風險分散利益產生影響。 總言之，經由上述的實證過程，將使我們對美、日、台股市具有更進一步的了解。同時，三地股票市場訊息的連結與傳遞型式分析，也可做為投資人在進行股票市場投資與國際投資組合建構時的決策參考。

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.

An recent finding shows that heart rate data possess self-similar property, which is characterized by a parameter H, as well as a long range dependent parameter d. We estimate H by the EBP(Embedded Branching Process) method to derive the fractional parameter d in the first part. The heart rate and R-R interval data are found to have high differencing parameter(d=0.8 ~0.9) and against the normality assumption. Thus the heart rate and R-R interval data are first fractionally differenced of order 0.5 to achieve stationarity. In the second part, we analyze the RR-interval data on the physionet and obtain the long range parameters. After fractionally differencing 0.5 order, the EBP method is adapted to estimate the long range parameter d. The EWMA and EWRMS control charts of the I(d) processes are constructed to monitor the heart rate mean level and variability, respectively for the 18 RR-interval data sets from the physionet. For the EWMA control chart the out of control percentages are chosen to the nominal probability. However, the out of control percentages are affected by the skewness and kurtosis of the process distribution for the EWRMS control carts. Generally speaking, the I(d)-EWMA and I(d)-EWRMS control charts provide a proper monitor system for heart rate mean level and variability.

In the literature, many methods are proposed to value American options. However, due to computational difficulty, there are only approximate solution or numerical method to evaluate American options. It is not easy for general investors either to understand nor to apply. In this thesis, we build up an option pricing and virtual asset model system, which provides a friendly environment for general public to calculate early exercise boundary of an American option. This system modularize the well-handled pricing models to provide the investors an easy way to value American options without learning difficult financial theories. The system consists two parts: the first one is an option pricing system, the other one is an asset model simulation system. The option pricing system provides various option pricing methods to the users; the virtual asset model system generates virtual asset prices for different underlying models.

Value-at-risk (VaR) model as a tool to estimate market risk is considered in the thesis. It is a statistical model defined as the maximum future loss due to likely changes in the value of financial assets portfolio during a certain period with a certain probability. A new definition of the aggregated VaR is given and the empirical study about different currencies position VaR estimates’ dependence on data aggregation functions (pointwise, maximum value, minimum value and average value) is provided. Functional ρ−GARCH(1,1) model is introduced and theorems of the stationary solution existence and maximum likelihood estimators of model parameters consistency are proved. Additionally, some examples of the model taking known density function of aggregated observations are given. Next, the general Hilbert space valued time series is presented and GARCH(1,1) model with univariate volatility is investigated. Theorems of the stationary solution existence, maximum likelihood estimators of model parameters consistency and asymptotic normality are proved; the analysis of residuals is provided. In the last chapter of the thesis the empirical study about Hurst index intraday value dependence on data aggregation taking different foreign currencies’ absolute returns is presented. / Disertacijoje nagrinėjamas vertės pokyčio rizikos modelis. Tai toks statistinis modelis, kurį taikant su tam tikra tikimybe įvertinamas didžiausias galimas nustatyto laikotarpio nuostolis, kredito įstaigos patiriamas dėl neigiamų taikomos finansinės priemonės vertės pokyčių. Apibrėžiamas agreguotų duomenų vertės pokyčio rizikos modelis ir pateikiamas praktinis tyrimas apie valiutų pozicijos vertės pokyčio rizikos modelio įvertinių priklausomybę nuo duomenų agregavimo taisyklės (pataškio, didžiausios vertės, mažiausios vertės ir vidutinės vertės). Kitame disertacijos skyriuje pristatomas naujas funkcinis ρ−GARCH(1,1) modelis, įrodomos stacionaraus sprendinio egzistavimo ir didžiausio tikėtinumo metodu įvertintų parametrų suderinamumo teoremos. Taip pat pateikiama keletas apibrėžtojo modelio pavyzdžių, kai žinoma agreguotų grąžų tankio funkcija. Disertacijoje apibrėžiamas Hilberto erdvės GARCH(1,1) modelis su vienmačiu kintamumu. Nagrinėjamos modelio savybės ir įrodomos stacionaraus sprendinio egzistavimo, didžiausio tikėtinumo metodu vertinamų parametrų suderinamumo ir asimptotinio normalumo teoremos, atliekama liekanų analizė. Paskutiniame disertacijos skyriuje aprašomas atliktas empirinis tyrimas apie Hursto indekso, kaip ilgos atminties parametro, priklausomybę nuo agregavimo taisyklės dienos metu, pasitelkiant absoliučias valiutų kursų grąžas.

Disertacijoje nagrinėjamas vertės pokyčio rizikos modelis. Tai toks statistinis modelis, kurį taikant su tam tikra tikimybe įvertinamas didžiausias galimas nustatyto laikotarpio nuostolis, kredito įstaigos patiriamas dėl neigiamų taikomos finansinės priemonės vertės pokyčių. Apibrėžiamas agreguotų duomenų vertės pokyčio rizikos modelis ir pateikiamas praktinis tyrimas apie valiutų pozicijos vertės pokyčio rizikos modelio įvertinių priklausomybę nuo duomenų agregavimo taisyklės (pataškio, didžiausios vertės, mažiausios vertės ir vidutinės vertės). Kitame disertacijos skyriuje pristatomas naujas funkcinis ρ−GARCH(1,1) modelis, įrodomos stacionaraus sprendinio egzistavimo ir didžiausio tikėtinumo metodu įvertintų parametrų suderinamumo teoremos. Taip pat pateikiama keletas apibrėžtojo modelio pavyzdžių, kai žinoma agreguotų grąžų tankio funkcija. Disertacijoje apibrėžiamas Hilberto erdvės GARCH(1,1) modelis su vienmačiu kintamumu. Nagrinėjamos modelio savybės ir įrodomos stacionaraus sprendinio egzistavimo, didžiausio tikėtinumo metodu vertinamų parametrų suderinamumo ir asimptotinio normalumo teoremos, atliekama liekanų analizė. Paskutiniame disertacijos skyriuje aprašomas atliktas empirinis tyrimas apie Hursto indekso, kaip ilgos atminties parametro, priklausomybę nuo agregavimo taisyklės dienos metu, pasitelkiant absoliučias valiutų kursų grąžas. / Value-at-risk (VaR) model as a tool to estimate market risk is considered in the thesis. It is a statistical model defined as the maximum future loss due to likely changes in the value of financial assets portfolio during a certain period with a certain probability. A new definition of the aggregated VaR is given and the empirical study about different currencies position VaR estimates’ dependence on data aggregation functions (pointwise, maximum value, minimum value and average value) is provided. Functional ρ−GARCH(1,1) model is introduced and theorems of the stationary solution existence and maximum likelihood estimators of model parameters consistency are proved. Additionally, some examples of the model taking known density function of aggregated observations are given. Next, the general Hilbert space valued time series is presented and GARCH(1,1) model with univariate volatility is investigated. Theorems of the stationary solution existence, maximum likelihood estimators of model parameters consistency and asymptotic normality are proved; the analysis of residuals is provided. In the last chapter of the thesis the empirical study about Hurst index intraday value dependence on data aggregation taking different foreign currencies’ absolute returns is presented.

[pt] Esta Dissertação apresenta um modelo para extrair fatores capazes de prever a volatilidade do índice de ações IBOVESPA, representativo do mercado de ações brasileiro. Esta metodologia é diferenciada por utilizar fatores que não incluem ativos da classe de ações. São utilizados fatores extraídos de classes de ativos de crédito, taxas de juros, moedas e commodities para precificar a volatilidade de um índice de ações. Além disso, os fatores são extraídos de painéis de volatilidades filtradas por modelos do tipo GARCH. / [en] It will be presented a model that is able to extract factors capable of predicting the volatility of IBOVESPA market index, which is representative of Brazilian equity market. This methodology is different from others because it won t use any inputs from equity asset classes. It will be used factors extracted from credit risk, interest rates, exchange rates and commodities data for pricing the volatility of an equity index. Besides that, those factors will be extracted from panels of volatility filtered by GARCH models.

The diploma thesis deals with modelling of time series (stock and commodities) by using the models of volatility. The theoretical part focuses on the term of volatility and other terms connected to it. There is a theoretical description of the models as well. The practical part of the thesis focuses on the analysis of the time series and modelling of volatility using the program R.

This master's thesis focuses on the problem area of option pricing under stochastic volatility. The theoretical part includes terms that are essential for understanding the problem area of option pricing and explains particular models for both option pricing under stochastic volatility and those under constant volatility. The application of described models is performed in the practical part of the thesis. After that particular models are compared to the real data.

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.