Yes / This paper studies the predictive ability of corridor implied volatility (CIV) measure. It is motivated by the fact that CIV is measured with better precision and reliability than the model-free implied volatility due to the lack of liquid options in the tails of the risk-neutral distribution. By adding CIV measures to the modified GARCH specifications, the out-of-sample predictive ability of CIV is measured by the forecast accuracy of conditional volatility. It finds that the narrowest CIV measure, covering about 10% of the RND, dominate the 1-day ahead conditional volatility forecasts regardless of the choice of GARCH models in high volatile period; as market moves to non volatile periods, the optimal width broadens. For multi-day ahead forecasts narrow and mid-range CIV measures are favoured in the full sample and high volatile period for all forecast horizons, depending on which loss functions are used; whereas in non turbulent markets, certain mid-range CIV measures are favoured, for rare instances, wide CIV measures dominate the performance. Regarding the comparisons between best performed CIV measures and two benchmark measures (market volatility index and at-the-money Black–Scholes implied volatility), it shows that under the EGARCH framework, none of the benchmark measures are found to outperform best performed CIV measures, whereas under the GARCH and NAGARCH models, best performed CIV measures are outperformed by benchmark measures for certain instances.

本研究利用GARCH動態過程的優點捕捉匯率報酬率之異質變異與波動度叢聚性質，並以GARCH動態過程為基礎，考慮跳躍風險服從Lévy過程，再利用特徵函數與快速傅立葉轉換方法推導出GARCH-Lévy動態過程下的歐式匯率選擇權解析解。以日圓兌換美元(JPY/USD)之歐式匯率選擇權為實證資料，比較基準GARCH選擇權評價模型與GARCH-Lévy選擇權評價模型對市場真實價格的配適效果與預測能力。實證結果顯示，考慮跳躍風險為無限活躍之Lévy過程，即GARCH-VG與GARCH-NIG匯率選擇權評價模型，不論是樣本內的評價誤差或是在樣本外的避險誤差皆勝於考慮跳躍風險為有限活躍Lévy過程的GARCH-MJ匯率選擇權評價模型。整體而言，本研究發現進行匯率選擇權之評價時，GARCH-NIG匯率選擇權評價模型有較小的樣本內及樣本外評價誤差。 / In this thesis, we make use of GARCH dynamic to capture volatility clustering and heteroskedasticity in exchange rate. We consider a jump risk which follows Lévy process based on GARCH model. Furthermore, we use characteristic function and fast fourier transform to derive the currency option pricing formula under GARCH-Lévy process. We collect the JPY/USD exchange rate data for our empirical analysis and then compare the goodness of fit and prediction performance between GARCH benchmark and GARCH-Lévy currency option pricing model. The empirical results show that either in-sample pricing error or out-of-sample hedging performance, the infinite-activity Lévy process, GARCH-VG and GARCH-NIG option pricing model is better than finite-activity Lévy process, GARCH-MJ option pricing model. Overall, we find using GARCH-NIG currency option pricing model can achieve the lower in-sample and out-of sample pricing error.

本文以 GJR-GARCH 為分析模型，針對所選八家台灣上市公司股價所計算之每日對數報酬率(daily log returns)，對於各種不斷到達的新增訊息所引起的波動反應。所納入條件變異數方程式的訊息到達(解釋變數)分別為：(1)同日成交數量(2)成交量變動率(3)星期一與星期五之日曆效應(4)不同權值規模(size-based)投資組合間的波動外溢效果。研究結果發現(1)同日成交量對於台股權值較低的小公司，有能力捕捉其波動性，但是對於權值偏高的大公司，其解釋能力顯有不足(2)成交量變化普遍會導致公司報酬率的波動(3)臺灣股市波動性並不具有星期五效應，至於星期一效應也只出現在部分的小公司(4)不同規模的投資組合間雖然互有波動外溢現象，但其不對稱性非常明顯， 亦即訊息到達後，先造成大公司股價的波動，此波動再進而影響到小公司，引起小公司股價的波動。 / Applying the GJR-GARCH model to the daily returns of eight selected firms from Taiwan stock market, this paper examines response of variance volatility to various information arrivals which separately include (1) concurrent trading volume (2) change in trading volume (3) calendar effects, especially Modnay and Friday effects, and (4) asymmetric volatility spillover between two sized-based portfolios. The results find that concurrent trading volume as a proxy of information arrival dramatically reduces volatility persistence of the small firm's conditional variance, but has little influence on large firm's, and change in trading volume cause significant change in conditional variance. Although there is a conjecture that the volatility in stock markets may be higher on Monday and Friday, it can't be found in this study. The results also strongly support that the volatility spillover effect from larger to small portfolio is more significant than that from smaller to large portfolio.

本文目的在於探討匯率不確定性對台灣出口波動之影響。本文應用Barkoulas et al.(2002)理論架構，利用台灣1989年至2007年的月資料。實證結果發現：美元、日圓兌新台幣的匯率波動對於台灣出口美、日兩國的數量並無明顯的影響。美元兌新台幣的匯率波動對於以美國為進口國的台灣出口波動則有正向的影響；日圓兌新台幣的匯率波動對於以日本為進口國的台灣出口波動卻沒有顯著影響。本文認為：造成美元匯率波動主要支配力量，來自於貨幣政策制定者掌握之資訊優勢差異；造成日圓匯率波動的來源則無主要支配力量的存在。造成此種結果的原因在於貨幣政策制定者長久以來所建立的政策可信度所致，削減了造成美元匯率波動的另外二股力量。因此，新台幣兌換美元匯率波動取決於貨幣政策制定者掌握經濟真實狀況的能力與其貨幣政策方向。 / This paper investigates into the effect of exchange rate uncertainty on Taiwan export volatility. Under the theoretical framework of Barkoulas et al.(2002) and the empirical monthly data of Taiwan exports from 1989 to 2007, it is summarized that the exchange rate volatility of NTD/USD and NTD/JPY had no effect on the Taiwan exporting volume toward U.S. or Japan. However, the exchange rate volatility of NTD/USD did have positive effect on the export volatility of Taiwan to U.S. while that of NTD/JPY had no significant effect on the export volatility of Taiwan to Japan.It is argued that the dominant source of NTD/USD exchange rate volatility resulted from the variance of monetary authorities’ information advantage. On the other hand, it exists no such a dominant source in NTD/JPY exchange rate volatility.

本文提出一個動態隱含波動度函數模型，以改善一般隱含波動度函數難以隨時間的經過而調整波動度曲線且無法描述資料的時間序列特性等缺點。本文模型為兩階段隱含波動度函數模型，分別配適隱含波動度函數的時間穩定(time-invariant)部分與時間不穩定(time-variant)部分。 本文模型在波動度的時間不穩定部分配適非對稱GARCH(1,1)過程，以描述隱含波動度的時間序列特性。本文使用的非對稱GARCH(1,1)過程將標的資產的正報酬與負報酬對價平隱含波動度的影響分別估計，並將蘊含於歷史價平隱含波動度中的訊息及標的資產報酬率與波動度之間的關連性藉由價平隱含波動度過程納入隱含波動度函數中，使隱含波動度函數能納入波動度的時間序列特性及資產報酬與波動度的相關性，藉此納入最近期的市場資訊，以增加隱含波動度模型的解釋及預測能力。時間穩定部分則根據Pena et al.(1999)的研究結果，取不對稱二次函數形式以配適實證上發現的笑狀波幅現象。時間穩定部分並導入相對價內外程度做為變數，以之描述價內外程度、距到期時間、及價平隱含波動度三者的交互關係；並以相對隱含波動度作為被解釋變數，使隱含波動度函數模型除理論上包含了比先前文獻提出的模型更多的訊息及彈性外，還能描繪「隱含波動度函數隨波動度的高低水準而變動」、「越接近到期日，隱含波動度對價內外程度的曲線越彎曲」、「隱含波動度函數為非對稱的曲線」、「波動度和資產價格有很高的相關性」等實證上常發現的現象。 本文以統計測度及交易策略之獲利能力檢定模型的解釋能力及預測能力是否具有統計與經濟上的顯著性。本文歸納之前文獻提出的不同隱含波動度函數模型，並以之與本文提出的模型做比較。本文以台指選擇權五分鐘交易頻率的成交價作為實證標的，以2003年1月1日~2006年12月31日作為樣本期間，並將模型解釋力及AIC作為模型樣本內配適能力之比較標準，我們發現本文提出的模型具有最佳的資料解釋能力。本文以2006年7月1日~2006年12月31日作為隱含波動度模型預測期間，以統計誤差及delta投資策略檢定模型的預測能力是否具有統計及經濟上的顯著性。實證結果指出，本文提出的模型對於預測下一期的隱含波動度及下一期的選擇權價格，皆有相當良好的表現。關於統計顯著性方面，我們發現本文提出的動態隱含波動度函數模型對於未來的隱含波動度及選擇權價格的預測偏誤約為其他隱含波動度函數模型的五分之一，而預測方向正確頻率亦高於預測錯誤的頻率且超過50%。關於經濟顯著性方面，本文使用delta投資組合進行經濟顯著性檢定，結果發現在不考慮交易成本下，本文提出的模型具有顯著的獲利能力。顯示去除標的資產價格變動對選擇權造成的影響後，選擇權波動度的預測準確性確實能經由delta投資組合捕捉；在考慮交易成本後，各模型皆無法獲得超額報酬。最後，本文提出的動態隱含波動度函數模型在考量非同步交易問題、30分鐘及60分鐘等不同的資料頻率、不同的投資組合交易策略後，整體的結論依然不變。 / This paper proposes a new implied volatility function to facilitate implied volatility forecasting and option pricing. This function specifically takes the time variation in the option implied volatility into account. Our model considers the time-variant part and fits it with an asymmetric GARCH(1,1) model, so that our model contains the information in the returns of spot asset and contains the relationship of the returns and the volatility of spot asset. This function also takes the time invariant in the option implied volatility into account. Our model fits the time invariant part with an asymmetric quadratic functional form to model the smile on the volatility. Our model describes the phenomena often found in the literature, such as the implied volatility level increases as time to maturity decreases, the curvature of the dependence of implied volatility on moneyness increases as options near maturity, the implied volatility curve changes as the volatility level changes, and the implied volatility function is an asymmetric curve. For the empirical results, we used a sample of 5 minutes transaction prices for Taiwan stock index options. For the in-sample period January 1, 2003–June 30, 2006, our model has the highest adjusted- and lowest AIC. For the out-of-sample period July 1, 2006–December 31, 2006, the statistical significance shows that our model substantially improves the forecasting ability and reduces the out-of-sample valuation errors in comparison with previous implied volatility functions. We conjecture that such good performance may be due to the ability of the GARCH model to simultaneously capture the correlation of volatility with spot returns and the path dependence in volatility. To test the economic significance of our model, we examine the profitability of the delta-hedged trading strategy based on various volatility models. We find that although these strategies are able to generate profits without transaction costs, their profits disappear quickly when the transaction costs are taken into consideration. Our conclusions were unchanged when we considered the non-synchronization problem or when we test various data frequency and different strategies.

Sammanfattning Examensarbete i finansiering, Civilekonomprogrammet Ekonomihögskolan vid Linnéuniversitetet, VT-2017 Författare: Emma Magnusson & Jonathan Hillgren Handledare: Håkan Locking Examinator: Andreas Stephan Titel: For better or for worse – A study on the impact of exchange rate volatility on trade   Bakgrund: Växelkurssvängningar har studerats av flertalet forskare då detta anses vara en osäkerhet vars effekt inte är säkerställd. Då internationell handel är en viktig faktor för tillväxt och välstånd i en nation är dess samband med volatiliteten betydelsefullt att fastställa för att identifiera huruvida inverkan på landet är positiv eller negativ.   Problemformulering: Har volatiliteten i eurons växelkurs mot rörliga valutor någon påverkan på den bilaterala handeln mellan eurozonen och andra europeiska länder?   Syfte: Att kunna urskilja effekten av växelkursvolatilitetens påverkan på export och import vilket kan gynna företag i dess handelsbeslut, strategier och framtidprognostisering.    Metod: Undersökningens tillvägagångssätt grundar sig i en tidsserieanalys där beräkningar för volatiliteten ligger som grund till förklaringsvariabeln i modellen för att studera dess effekt på handeln, vilket skattas genom en noga utvald ARDL-metod. Regressionerna ger både ett lång- och kortsiktigt samband för att visa skillnader i influenser från volatiliteten på export och import för Sverige och Norge som studeras i rapporten.   Slutsatser: De erhållna resultaten för både Sveriges och Norges export visar att ingen påverkan alls kan urskiljas från växelkurssvängningar vilket innebär att exporten fortskrider oavsett grad av volatilitet vilket kan förklaras genom dess förmodade likheter i handelsmönster, varukategorier och exponering mot euroländerna. En möjlig slutsats är även att det inte är volatiliteten i sig som påverkar handeln, utan underliggande faktorer som inte kontrollerats för, vilket åskådliggjordes när oljeprisindex inkluderandes och eliminerade volatilitetens effekt på Norges export. Importen visade en långsiktig negativ effekt av volatiliteten för Norge och en kortsiktig positiv påverkan för Sverige. Skillnader i importen antas bero på olikheter i valutasäkring, trögheter i ekonomin och relationen till EU. / Abstract   Master Thesis in finance, Business and Administration School of Business and Economics at Linnaeus University, VT-2017 Authors: Emma Magnusson & Jonathan Hillgren Advisor: Håkan Locking Examiner: Andreas Stephan Title: For better or for worse – A study on the impact of exchange rate volatility on trade   Background: Exchange rate fluctuations have been studied by numerous researchers, since it is thought of as an uncertainty whose effect is not guaranteed. Because international trade is an important factor to growth and wealth for a country, its connection to volatility is important to establish in order to identify whether the influence on the nation is positive or negative.   Problem: Does the volatility in the exchange rate between the euro and floating currencies affect bilateral trade between the euro area and other European countries?   Purpose: The purpose of the study is to distinguish the effect of the exchange rate volatility on export and import, which can favor companies in their trade decisions and strategies.   Method: The approach of the study is built on a time series analysis where estimates of volatility are underlying the explanatory variable to find its effect on trade, which is calculated by a carefully selected ARDL method. The regressions obtain both long-term and short-term relationships to show differences in the effect from the volatility on export and import for Sweden and Norway, the studied countries in this report.   Conclusions: The results for the export of both Sweden and Norway do not show any impact from the exchange rate fluctuations, which means the export continues regardless of the level of volatility. This can be explained by their similarities in the pattern of trade, products and exposure to the euro countries. Another possible conclusion is that the volatility itself is not affecting trade but that the underlying factors not being controlled for are, which was shown when the oil price index was included and eliminated the effect of the volatility on Norwegian exports. The import exposed a long-term negative effect of the volatility for Norway and a short-term positive effect for Sweden. The disparities are assumed to be due to differences in the use of hedging, inertia in the economy and the relationship with the European Union.

This thesis investigates the modelling and forecasting of multivariate volatility and dependence in financial time series. The first paper proposes a new model for forecasting changes in the term structure (TS) of interest rates. Using the level, slope and curvature factors of the dynamic Nelson-Siegel model, we build a time-varying copula model for the factor dynamics allowing for departure from the normality assumption typically adopted in TS models. To induce relative immunity to structural breaks, we model and forecast the factor changes and not the factor levels. Using US Treasury yields for the period 1986:3-2010:12, our in-sample analysis indicates model stability and we show statistically significant gains due to allowing for a time-varying dependence structure which permits joint extreme factor movements. Our out-of-sample analysis indicates the model's superior ability to forecast the conditional mean in terms of root mean square error reductions and directional forecast accuracy. The forecast gains are stronger during the recent financial crisis. We also conduct out-of-sample model evaluation based on conditional density forecasts. The second paper introduces a new class of multivariate volatility models that utilizes high-frequency data. We discuss the models' dynamics and highlight their differences from multivariate GARCH models. We also discuss their covariance targeting specification and provide closed-form formulas for multi-step forecasts. Estimation and inference strategies are outlined. Empirical results suggest that the HEAVY model outperforms the multivariate GARCH model out-of-sample, with the gains being particularly significant at short forecast horizons. Forecast gains are obtained for both forecast variances and correlations. The third paper introduces a new class of multivariate volatility models which is easy to estimate using covariance targeting. The key idea is to rotate the returns and then fit them using a BEKK model for the conditional covariance with the identity matrix as the covariance target. The extension to DCC type models is given, enriching this class. We focus primarily on diagonal BEKK and DCC models, and a related parameterisation which imposes common persistence on all elements of the conditional covariance matrix. Inference for these models is computationally attractive, and the asymptotics is standard. The techniques are illustrated using recent data on the S&P 500 ETF and some DJIA stocks, including comparisons to the related orthogonal GARCH models.

This thesis analyzes the effect of the Czech National Bank's (CNB) communica- tion on the interest rate volatility (PRJBOR reference rate). Starting with the literature survey about the central bank communication in the world, I focus on the literature that concerns the CNB. To model the CNB's communication, I use the GARCH(l,1), EGARCH(l,1) and TARCH(l,1) models. I have created a unique data set containing the dummy variables for the CNB communication. The results are as follows: (a) the CNB's communication tends to decrease the volatility, (b) timing of the communication has a key role as the comments closer to the meeting have bigger calming effect, and that (c) there is no clear effect concerning the comments of the Bank Board members in the media. JEL Classification Keywords E43, E44, E52, E58 Czech National Bank, monetary policy signaling, central bank communication, the term structure of interest rates, GARCH analysis Author's email karasp@email.cz Supervisors's email roman. horvath@gmail.com

The thesis describes commonly used measures of risk, such as volatility, Value at Risk (VaR) and Expected Shortfall (ES), and is tasked with creating models for measuring market risk. It is concerned with the risk over daily and over monthly horizons and shows the shortcomings of a square-root-of-time approach for converting VaR and ES between horizons. Parametric models, geometric Brownian motion (GBM) and GARCH process, and non-parametric models, historical simulation (HS) and some its possible improvements, are presented. The application of these mentioned models is demonstrated using real data. The accuracy of VaR models is proved through backtesting and the results are discussed. Part of this thesis is also a simulation study, which reveals the precision of VaR and ES estimates.

Uma grande dificuldade da gestão financeira é conseguir associar métodos quantitativos às formas tradicionais de gestão, em um único arranjo. O estilo tradicional de gestão tende a não crer, na devida medida, que métodos quantitativos sejam capazes de captar toda sua visão e experiência, ao passo que analistas quantitativos tendem a subestimar a importância do enfoque tradicional, gerando flagrante desarmonia e ineficiência na análise de risco. Um modelo que se propõe a diminuir a distância entre essas visões é o modelo Black-Litterman. Mais especificamente, propõe-se a diminuir os problemas enfrentados na aplicação da teoria moderna de carteiras e, em particular, os decorrentes da aplicação do modelo de Markowitz. O modelo de Markowitz constitui a base da teoria de carteiras há mais de meio século, desde a publicação do artigo Portfolio Selection [Mar52], entretanto, apesar do papel de destaque da abordagem média-variância para o meio acadêmico, várias dificuldades aparecem quando se tenta utilizá-lo na prática, e talvez, por esta razão, seu impacto no mundo dos investimentos tem sido bastante limitado. Apesar das desvantagens na utilização do modelo de média-variância de Markowitz, a idéia de maximizar o retorno, para um dado nível de risco é tão atraente para investidores, que a busca por modelos com melhor comportamento continuou e é neste contexto que o modelo Black-Litterman surgiu. Em 1992, Fischer Black e Robert Litterman publicam o artigo Portfolio Optimization [Bla92], fazendo considerações sobre o papel de pouco destaque da alocação quantitativa de ativos, e lançam o modelo conhecido por Black-Litterman. Uma grande diferença entre o modelo Black-Litterman e um modelo média-variância tradicional é que, enquanto o segundo gera pesos em uma carteira a partir de um processo de otimização, o modelo Black-Litterman parte de uma carteira de mercado em equilíbrio de longo prazo (CAPM). Outro ponto de destaque do modelo é ser capaz de fornecer uma maneira clara para que investidores possam expressar suas visões de curto prazo e, mais importante, fornece uma estrutura para combinar de forma consistente a informação do equilíbrio de longo prazo (priori) com a visão do investidor (curto prazo), gerando um conjunto de retornos esperados, a partir do qual os pesos em cada ativo são fornecidos. Para a escolha do método de estimação dos parâmetros, levou-se em consideração o fato de que matrizes de grande dimensão têm um papel importante na avaliação de investimentos, uma vez que o risco de uma carteira é fundamentalmente determinado pela matriz de covariância de seus ativos. Levou-se também em consideração que seria desejável utilizar um modelo flexível ao aumento do número de ativos. Um modelo capaz de cumprir este papel é o GARCH ortogonal, pois este pode gerar matrizes de covariâncias do modelo original a partir de algumas poucas volatilidades univariadas, sendo, portanto, um método computacionalmente bastante simples. De fato, as variâncias e correlações são transformações de duas ou três variâncias de fatores ortogonais obtidas pela estimação GARCH. Os fatores ortogonais são obtidos por componentes principais. A decomposição da variância do sistema em fatores de risco permite quantificar a variabilidade que cada fator de risco traz, o que é de grande relevância, pois o gestor de risco poderá direcionar mais facilmente sua atenção para os fatores mais relevantes. Ressalta-se também que a ideia central da ortogonalização é utilizar um espaço reduzido de componentes. Neste modelo de dimensão reduzida, suficientes fatores de risco serão considerados, assim, os demais movimentos, ou seja, aqueles não capturados por estes fatores, serão considerados ruídos insignificantes para este sistema. Não obstante, a precisão, ao desconsiderar algumas componentes, irá depender de o número de componentes principais ser suficiente para explicar grande parte da variação do sistema. Logo, o método funcionará melhor quando a análise de componentes principais funcionar melhor, ou seja, em estruturas a termo e outros sistemas altamente correlacionados. Cabe mencionar que o GARCH ortogonal continua igualmente útil e viável quando pretende-se gerar matriz de covariâncias de fatores de risco distintos, isto é, tanto dos altamente correlacionados, quanto daqueles pouco correlacionados. Neste caso, basta realizar a análise de componentes principais em grupos correlacionados. Feito isto, obtêm-se as matrizes de covariâncias utilizando a estimação GARCH. Em seguida faz-se a combinação de todas as matrizes de covariâncias, gerando a matriz de covariâncias do sistema original. A estimação GARCH foi escolhida pois esta é capaz de captar os principais fatos estilizados que caracterizam séries temporais financeiras. Entende-se por fatos estilizados padrões estatísticos observados empiricamente, que, acredita-se serem comuns a um grande número de séries temporais. Séries financeiras com suficiente alta frequência (observações intraday e diárias) costumam apresentar tais características. Este modelo foi utilizado para a estimação dos retornos e, com isso, obtivemos todas as estimativas para que, com o modelo B-L, pudéssemos gerar uma carteira ótima em um instante de tempo inicial. Em seguida, faremos previsões, obtendo carteiras para as semanas seguintes. Por fim, mostraremos que a associação do modelo B-L e da estimação GARCH ortogonal pode gerar resultados bastante satisfatórios e, ao mesmo tempo, manter o modelo simples e gerar resultados coerentes com a intuição. Este estudo se dará sobre retornos de títulos de renda fixa, mais especificamente, títulos emitidos pelo Tesouro Nacional no mercado brasileiro. Tanto a escolha do modelo B-L, quanto a escolha por utilizar uma carteira de títulos emitidos pelo Tesouro Nacional tiveram como motivação o objetivo de aproximar ferramentas estatísticas de aplicações em finanças, em particular, títulos públicos federais emitidos em mercado, que têm se tornado cada vez mais familiares aos investidores pessoas físicas, sobretudo através do programa Tesouro Direto. Ao fazê-lo, espera-se que este estudo traga informações úteis tanto para investidores, quanto para gestores de dívida, uma vez que o modelo média-variância presta-se tanto àqueles que adquirem títulos, buscando, portanto, maximizar retorno para um dado nível de risco, quanto para aqueles que emitem títulos, e que, portanto, buscam reduzir seus custos de emissão a níveis prudenciais de risco. / One major challenge to financial management resides in associating traditional management with quantitative methods. Traditional managers tend to be skeptical about the quantitative methods contributions, whereas quantitative analysts tend to disregard the importance of the traditional view, creating clear disharmony and inefficiency in the risk management process. A model that seeks to diminish the distance between these two views is the Black-Litterman model (BLM). More specifically, it comes as a solution to difficulties faced when using modern portfolio in practice, particularly those derived from the usage of the Markowitz model. Although the Markowitz model has constituted the basis of portfolio theory for over half century, since the publication of the article Portfolio Selection [Mar52], its impact on the investment world has been quite limited. The Markowitz model addresses the most central objectives of an investment: maximizing the expected return, for a given level of risk. Even though it has had a standout role in the mean-average approach to academics, several difficulties arise when one attempts to make use of it in practice. Despite the disadvantages of its practical usage, the idea of maximizing the return for a given level of risk is so appealing to investors, that the search for models with better behavior continued, and is in this context that the Black-Litterman model came out. In 1992, Fischer Black and Robert Litterman wrote an article on the Black-Litterman model. One intrinsic difference between the BLM and a traditional mean-average one is that, while the second provides the weights of the assets in a portfolio out of a optimization routine, the BLM has its starting point at the long-run equilibrium market portfolio(CAPM). Another highlighting point of the BLM is the ability to provide one clear structucture that is able to combine the long term equilibrium information with the investors views, providing a set of expected returns, which, together, will be the input to generate the weights on the assets. As far as the estimation process is concerned, and for the purpose of choosing the most appropriate model, it was taken into consideration the fact that the risk of a portfolio is determined by the covariation matrix of its assets and, being so, matrices with large dimensions play an important role in the analysis of investments. Whereas, provided the application under study, it is desirable to have a model that is able to carry out the analysis for a considerable number of assets. For these reasons, the Orthogonal GARCH was selected, once it can generate the matrix of covariation of the original system from just a few univariate volatilities, and for this reason, it is a computationally simple method. The orthogonal factors are obtained with principal components analysis. Decomposing the variance of the system into risk factors is highly important, once it allows the risk manager to focus separately on each relevant source of risk. The main idea behind the orthogonalization consists in working with a reduced dimension of components. In this kind of model, sufficient risk factors are considered, thus, the variability not perceived by the model will be considered insigficant noise to the system. Nevertheless, the precision, when not using all the components, will depend on the number of components be sufficient to explain the major part of the variability. Moreover, the model will provide reasonable results depending on principal component analysis performing properly as well, what will be more likely to happen, in highly correlated systems. It is worthy of note that the Orthogonal GARCH is equally useful and feasible when one intends to analyse a portfolio consisting of assets across various types of risk, it means, a system which is not highly correlated. It is common to have such a portfolio, with, for instance, currency rates, stocks, fixed income and commodities. In order to make it to perform properly, it is necessary to separate groups with the same kind of risk and then carry out the principal component analysis by group and then merge the covariance matrices, producing the covariance matrix of the original system. To work together with the orthogonalization method, the GARCH model was chosen because it is able to draw the main stylized facts which characterize financial time series. Stylized facts are statistical patterns empirically observed, which are believed to be present in a number of time series. Financial time series which sufficient high frequency (intraday, daily and even weekly) usually present such behavior. For estimating returns purposes, it was used a ARMA model, and together with the covariance matrix estimation, we have all the parameters needed to perform the BLM study, coming out, in the end, with the optimal portfolio in a given initial time. In addition, we will make forecasts with the GARCH model, obtaining optimal portfolio for the following weeks. We will show that the association of the BLM with the Orthogonal GARCH model can generate satisfactory and coherent with intuition results and, at the same time, keeping the model simple. Our application is on fixed income returns, more specifically, returns of bonds issued in the domestic market by the Brazilian National Treasury. The motivation of this work was to put together statistical tolls and finance uses and applications, more specifically those related to the bonds issued by the National Treasuy, which have become more and more popular due to the \"Tesouro Direto\" program. In conclusion, this work aims to bring useful information either for investors or to debt managers, once the mean-variance model can be useful for those who want to maximize return at a given level or risk as for those who issue bonds, and, thus, seek to reduce their issuance costs at prudential levels of risk.