Keliami uždaviniai: GARCH modelių klasės taikymas ilgo periodo finansiniams duomenims: modelių parametrų paieška, jų vertinimas, testavimas ir taikymas. Ilga atmintis sąlyginiame variantiškume yra viena iš empirinių savybių, kurią turi daugelis finansinių laiko eilučių. Viena modelių klasė, kuri atvaizduoja šį elgesį yra vadinama Dalinai Integruotu GARCH (Baillie, Bollerslev ir Mikkelsen 1996). Dalinės integracijos idėją pateikė ir ją pritaikė GARCH struktūrai Granger (1980) ir Hosking (1981). Šiame darbe bus surastos analitinės FIGARCH proceso antros eilės logaritminės tikėtinumo funkcijos išvestinės. Ilgo diapazono priklausomybė bus apskaičiuota parametriniu dalinai integruotu GARCH modeliu. Finansinių laiko eilučių duomenys bus įvertinti GARCH (CGARCH(1), CGARCH(2)) ir FIGARCH(1,d,1)) modeliais maksimalaus tikėtinumo metodu. Taip pat bus sukurtas NASDAQ- NYSE santykinio stiprumo indikatorius bei patikrintos jo panaudojimo sąlygos. Iiustracija yra pateikta 5 akcijų indeksais, 2 valiutų santykiais, aukso bei NNSS duomenims. / The paper deals with the problems of applying GARCH model/framework to a long term financial data, the search of the models, their evaluation, testing/validation and application. Long memory in conditional variance is one of the empirical features exhibited by many financial time series. One class of models that was suggested to capture this behavior is the so-called Fractionally Integrated GARCH (Baillie, Bollerslev and Mikkelsen 1996) in which the ideas of fractional integration originally introduced by Granger (1980) and Hosking (1981) for processes of the mean are applied to GARCH framework. In this paper we derive analytic expressions for the second-order derivatives of the log-likelihood function of FIGARCH processes. Long-range dependence is assessed through the parametric fractionally integrated GARCH model. Financial time series data will be estimated Component GARCH (CGARCH(1), CGARCH(2)) and FIGARCH models maximum likelihood method. Also we built NASDAQ- NYSE relative strength indicator and tested its usage conditions. An illustration is provided on 2 exchange rate, 5 stock index, gold and NNSS data.

This thesis consists of three studies which cover topics in the trading volume-market return volatility linkage, stock market return-aggregate mutual fund flow relationship as well as market return volatility-aggregate mutual fund flow interaction. Chapter 2 investigates the issue of volume-volatility linkage in the US market for the period 1990-2012 (S&P 500) and 1992-2012 (Dow Jones). We construct four sub-samples depending on three different structural points (the Asian Financial Crisis, the Dot-Com Bubble and the 2007 Financial Crisis). By employing univariate and bivariate GARCH processes, we find positive (negative) bidirectional linkages between these two aforementioned variables in various cases of the estimation, while a mixed one is observed in the remainder of these cases. Chapter 3 examines the issue of temporal ordering of the range-based stock market return (S&P 500 index) and aggregate mutual fund flow in the U.S. market for the period 1998-2012. We construct nine sub-samples represented by three fundamental cases of the whole data set. In addition, we take into consideration three essential indicators when splitting the whole data set, which are the 2000 Dot-Com Bubble, the 2007 Financial Crisis as well as the 2009 European Sovereign Debt Crisis. We examine the dynamics of the return-flow interaction by employing bivariate VAR model with various specifications of GARCH approach. Our principal findings display a bidirectional mixed feedback between stock market return and aggregate mutual fund flow for the majority of the sub-samples obtained. Nevertheless, we provide limited evidence of a positive bi-directional causality between return and flow. Chapter 4 investigates the dynamic relation between S&P 500 return volatility and U.S. aggregate mutual fund flow for the period spanning between 1998 and 2012. We assess the dynamics of the volatility-flow linkage by employing a bivariate VAR model with the GARCH approach which allows for long memory in the mean and the variance equations. In addition to the sub-samples obtained in chapter 3, we generate two measurements of volatility. Our baseline results indicate a variety of bidirectional mixed causalities between market return volatility and aggregate mutual fund flow in several sub-samples. In addition, we observe a negative/positive bi-directional relationship between volatility and flow in the rest of the sub-periods. Summarizing, a range of our findings are in line with the empirical underpinnings that most likely predict a significant linkage between the aforementioned variables. Finally, most of the bidirectional effects are found to be quite robust to the dynamics of the various GARCH processes employed in this thesis.

The aim of this master thesis is to introduce models belonging to ARCH(∞) representation where a time series volatility is modelled as a linear function of squared residuals. Specifically, the thesis deals with models IGARCH, FIGARCH and HYGARCH that are used to analyse, model and predict a development of financial time series. Definition and graphical illustration of individual models together with their application on real data, is supplemented by a simulation study of first-order FIGARCH model.

No description available.

在這篇論文中，我們運用馬可夫鏈蒙地卡羅(MCMC)方法來估計沒有價格限制下的白銀期貨價格。接著我們採用FIGARCH模型來計算VaR值，以進而評估估計成果。在本文中我們分別對三種不同分配下的FIGARCH模型計算VaR值，而實證結果顯示出在沒有價格限制下，白銀期貨有較好的估計結果。 / In this paper, we try to implement the MCMC method to simulate the price of the silver futures without price limits. Then we compute the VaR by using the FIGARCH model because of the long memory properties in our data. There are three distributions we use to estimate model and compute VaR. The empirical results show that the silver futures without price limits performs better in computing in-sample VaR.

O objetivo deste trabalho é apresentar e comparar diferentes métodos de modelagem da volatilidade (variância condicional) de séries temporais financeiras. O modelo ARFIMA é empregado para capturar o comportamento de memória longa observado na volatilidade de séries financeiras. Por sua vez, o modelo GARCH é utilizado para modelar a volatilidade variando no tempo destas séries. Finalmente, o modelo FIGARCH é utilizado para modelar a dinâmica dos retornos de séries temporais financeiras juntamente com sua volatilidade. Serão apresentados alguns estimadores para os parâmetros dos modelos estudados. Foram realizadas simulações dos três tipos de modelos com o objetivo de comparar o comportamento dos estimadores para diferentes valores dos parâmetros. Por fim, serão apresentadas aplicações em séries reais. / The goal of this project is to present and compare differents methods of modeling volatility (conditional variance) in financial time series. ARFIMA model is applied to capture long memory behavior of volatility in financial time series. GARCH model is used to model the temporal variation in financial volatility. Finally, FIGARCH model is used to model dynamic of financial time series returns as well as its volatility behavior. We present some estimators for the studied models. Estimators behavior of the three types of models for different parameters is assessed through a simulation study. At last, applications to real data are presented.

O objetivo deste trabalho é apresentar e comparar diferentes métodos de modelagem da volatilidade (variância condicional) de séries temporais financeiras. O modelo ARFIMA é empregado para capturar o comportamento de memória longa observado na volatilidade de séries financeiras. Por sua vez, o modelo GARCH é utilizado para modelar a volatilidade variando no tempo destas séries. Finalmente, o modelo FIGARCH é utilizado para modelar a dinâmica dos retornos de séries temporais financeiras juntamente com sua volatilidade. Serão apresentados alguns estimadores para os parâmetros dos modelos estudados. Foram realizadas simulações dos três tipos de modelos com o objetivo de comparar o comportamento dos estimadores para diferentes valores dos parâmetros. Por fim, serão apresentadas aplicações em séries reais. / The goal of this project is to present and compare differents methods of modeling volatility (conditional variance) in financial time series. ARFIMA model is applied to capture long memory behavior of volatility in financial time series. GARCH model is used to model the temporal variation in financial volatility. Finally, FIGARCH model is used to model dynamic of financial time series returns as well as its volatility behavior. We present some estimators for the studied models. Estimators behavior of the three types of models for different parameters is assessed through a simulation study. At last, applications to real data are presented.

Time series forecasting and volatility forecasting is a particularly active research field within financial mathematics. More recent studies extend well-established forecasting methods with machine learning. This thesis will evaluate and compare the standard Generalized Autoregressive Conditional Heteroskedasticity (GARCH) model and some of its extensions to a proposed Long Short-Term Memory (LSTM) model on historic data from five Swedish stocks. It will also explore hybrid models that combine the two techniques to increase prediction accuracy over longer horizons. The results show that the predictability increases when switching from univariate GARCH and LSTM models to hybrid models combining them both. Combining GARCH, Glosten, Jagannathan, and Runkle GARCH (GJR-GARCH), and Fractionally Integrated GARCH (FIGARCH) yields the most accurate result with regards to mean absolute error and mean square error. The forecasting errors decreased with 10 to 50 percent using the hybrid models. Comparing standard GARCH to the hybrid models, the biggest gains were seen at the longest horizon, while comparing the LSTM to the hybrid models, the biggest gains were seen for the shorter horizons. In conclusion, the prediction ability increases using the hybrid models compared to the regular models. / Tidsserieprognostisering, och volatilitetsprognostiering i synnerhet, är ett växande fält inom finansiell matamatik som kontinereligt står inför implementation av nya tekniker. Det som en gång startade med klassiksa tidsseriemodeller som ARCH har nu utvecklats till att dra fördel av maskininlärning och neurala nätverk. Detta examensarbetet uvärderar och jämför Generalized Autoregressive Conditional Heteroskedasticity (GARCH) modeller och några av dess vidare tillämpningar med Long Short-Term Memory (LSTM) modeller på fem svenska aktier. ARbetet kommer även gå närmare inpå hybridmodeller som kombinerar dessa två tekniker för att öka tillförlitlig prognostisering under längre tidshorisonter. Resultaten visar att förutsägbarheten ökar genom att byta envariata GARCH och LSTM modeller till hybridmodeller som kombinerar båda delarna. De mest korrekta resultaten kom från att kombinera GARCH, Glosten, Jagannathan, och Runkle GARCH (GJR-GARCH) och Fractionally Integrated GARCH (FIGARCH) modeller med ett LSTM nätverk. Prognostiseringsfelen minskade med 10 till 50 procent med hybridmodellerna. Specifikt, vid jämförelse av GARCH modellerna till hybridmodellerna sågs de största förbättringarna för de längre tidshorisonterna, medans jämförelse mellan LSTM och hybridmodellerna sågs den mesta förbättringen hos de kortare tidshorisonterna. Sammanfattningsvis öker prognostiseringsförmågan genom användning av hybridmodeller i jämförelse med standardmodellerna.

在本篇文章中，我們採用長期記憶模型來估計S&P500、Nasdaq100和Dow Jones Industrial Index三個股票指數期貨的日收盤價的風險值。為了更準確地計算風險值，本文採用常態分配、t分配以及偏斜t分配來做模型估計以及風險值之計算。有鑒於大多數探討風險值的文獻只考慮買入部位的風險，本研究除了估計買入部位的風險值，也估計放空部位的風險值，以期更能全面性地估算風險。實證結果顯示，ARFIMA-FIGARCH模型配合偏斜t分配較其他兩種分配更能精確地估算樣本內的風險值。基於ARFIMA-FIGARCH模型配合偏斜t分配在樣本內風險值計算的優異表現，我們利用此模型搭配來實際求算樣本外風險值。結果如同樣本內風險值一般，ARFIMA-FIGARCH模型配合偏斜t分配在樣本外也有相當好的風險預測能力。 / In this thesis, we estimate Value-at-Risk (VaR) for daily closing price of three stock index futures contracts, S&P500, Nasdaq100, and Dow Jones, using the double long memory models. Due to the existence of a long-term persistence characterized in our data, the ARFIMA-FIGARCH models are used to compute the VaR. In order to investigate better, three kinds of density distributions, normal, Student-t, and skewed Student-t distributions, are used for estimating models and computing the VaR. In addition to the VaR for the long trading positions which most researches focus on to date, the VaR for the short trading positions are calculated as well in this study. From the empirical results we show that for the three stock index futures, the ARFIMA-FIGARCH models with skewed Student-t distribution perform better in computing in-sample VaR both in long and short trading positions than symmetric models and has a quite excellent performance in forecasting out-of-sample VaR as well.

近幾年，風險值已經成為金融機構風險控管的重要工具。它的明確及簡單易懂是其讓人接受的原因，加上巴塞爾銀行監理委員會在1996提出的巴塞爾協定修正，規定銀行將市場風險因素納入考量，並允許銀行自行發展內部模型，以風險值模型衡量市場風險後，各種風險值的估算方法相繼被提出。 本篇論文是使用部分整合自回歸條件變異數（Fractional Integrated Generalized Autoregressive Conditional Heteroskedasticity，簡稱FIGARCH）計算長期利率期貨多空部位的每日風險值。選取的三支長期利率期貨是在芝加哥期貨交易所掛牌的三十年期美國政府債券期貨（TB）、十年期美國政府債券期貨（TN） 與十年期市政債券指數期貨（MNI）。 利率期貨的研究在過去文獻中，甚少被提及。但隨著利率型商品日新月異的發展，以利率期貨避險的需求也與日遽增。尤其在台灣，利率期貨更是今年新登場的期貨商品。因此，我選擇利率期貨作為研究標的，藉由以FIGARCH模型來配適波動性，提供避險者一個估算風險值的方法。 FIGARCH模型係由Baillie、Bollerslev與Mikkelsen於1996所提出，與傳統GARCH模型所不同的是，FIGARCH模型特別適用於描述具有波動性長期記憶（Long Memory）性質的資料。所謂長期記憶性，是指衝擊所造成的持續性是以緩慢的雙曲線速率衰退。而許多市場實證分析均指出，FIGARCH較適合用來描述金融市場上的波動性。此外，本研究的風險值計算，除了一般實務界常用的常態分配以外，還考慮了t分配與偏斜t分配，以捕捉財務資料常見的厚尾與偏斜的特性。 而實證結果顯示，長期利率期貨報酬率的波動性確實存在長期記憶性，所以FIGARCH(1,d,1)模型可以適切地估算長期利率期貨的每日風險值，不論在樣本內或樣本外的風險值計算均優於傳統GARCH(1,1)模型的計算結果。至於各種不同分配的比較，在樣本內的風險值計算，當α=0.05時，常態分配FIGARCH(1,d,1)模型表現較佳；當α=0.025到0.0025時，t分配與偏斜t分配FIGARCH(1,d,1)模型表現較佳，而偏斜t分配FIGARCH又稍微優於t分配FIGARCH(1,d,1)模型。 而樣本外的風險值預測，則有不同的結果，當α=0.05，t分配與偏斜t分配FIGARCH(1,d,1)模型表現較佳；而α=0.01時，常態分配FIGARCH(1,d,1)模型表現較佳。而且t分配與偏斜t分配FIGARCH(1,d,1)模型在α=0.01會出現太過保守的情形，出現失敗率（failure rate）為零，高估風險值。 / Value-at-Risk (VaR) has become the standard measure used to quantify market risk recently, and it is defined as the maximum expected loss in the value of an asset or portfolio, for a given probability α at a determined time period. This article uses the FIGARCH(1,d,1) models to calculate daily VaR for long-term interest rate futures returns for long and short trading positions based on the normal, the Student-t, and the skewed Student-t error distributions. The U.S. Treasury bonds futures, Treasury notes futures, and municipal notes index futures of daily frequency are studied. The empirical results show that returns series for three interest rate futures all have long memory in volatility, and should be modeled using fractional integrated models. Besides, the in-sample and out-of-sample VaR values generated using FIGARCH(1,d,1) models are more accurate than those generated using traditional GARCH(1,1) models. For different distributions among FIGARCH(1,d,1) models, the normal FIGARCH(1,d,1) models are preferred for in-sample VaR computing whenα=0.05, and the Student-t and skewed Student-t models perform better for in-sample VaR computing whenα=0.025-0.0025. Nonetheless, for out-of-sample VaR, the Student-t and skewed Student-t FIGARCH(1,d,1) models perform better in the case α=0.05 while the normal FIGARCH(1,d,1) models perform better in the case α=0.01. The VaR values obtained by the Student-t and skewed Student-t FIGARCH(1,d,1) models are too conservative whenα=0.01.