The main objective of our study is to employ copula methodology to develop Bayesian hierarchical models to study the dependencies exhibited by temporal, spatial and spatio-temporal processes. We develop hierarchical models for both discrete and continuous outcomes. In doing so we expect to address the dearth of copula based Bayesian hierarchical models to study hydro-meteorological events and other physical processes yielding discrete responses. First, we present Bayesian methods of analysis for longitudinal binary outcomes using Generalized Linear Mixed models (GLMM). We allow flexible marginal association among the repeated outcomes from different time-points. An unique property of this copula-based GLMM is that if the marginal link function is integrated over the distribution of the random effects, its form remains same as that of the conditional link function. This unique property enables us to retain the physical interpretation of the fixed effects under conditional and marginal model and yield proper posterior distribution. We illustrate the performance of the posited model using real life AIDS data and demonstrate its superiority over the traditional Gaussian random effects model. We develop a semiparametric extension of our GLMM and re-analyze the data from the AIDS study. Next, we propose a general class of models to handle non-Gaussian spatial data. The proposed model can deal with geostatistical data that can accommodate skewness, tail-heaviness, multimodality. We fix the distribution of the marginal processes and induce dependence via copulas. We illustrate the superior predictive performance of our approach in modeling precipitation data as compared to other kriging variants. Thereafter, we employ mixture kernels as the copula function to accommodate non-stationary data. We demonstrate the adequacy of this non-stationary model by analyzing permeability data. In both cases we perform extensive simulation studies to investigate the performances of the posited models under misspecification. Finally, we take up the important problem of modeling multivariate extreme values with copulas. We describe, in detail, how dependences can be induced in the block maxima approach and peak over threshold approach by an extreme value copula. We prove the ability of the posited model to handle both strong and weak extremal dependence and derive the conditions for posterior propriety. We analyze the extreme precipitation events in the continental United States for the past 98 years and come up with a suite of predictive maps.

Credit risk and market risk have already been explored intensively and the reliable models of credit risk and market risk have also been developed progressively. This study try to find a method pricing the CDO (Collateralized Debt Obligation) based on Incomplete information credit model. For the various approaches to CDO valuation, the most widely accepted is the Copula approach. The Copula approach is considered suitable for describing default correlation. Combining with Monte Carlo Simulation, it can price CDO effectively.

Cholesky Decomposition is usually used to deal with the correlation problem among a financial product's underlying assets. However, Cholesky Decomposition inherently suffers from the requirement that all eigenvalues must be positive. Therefore, Cholesky Decomposition can't work very well when the number of the underlying assets is high. The report takes a diffrent approach called spectral Decomposition in attempt to solve the problem. But it turns out that although Spectral Decomposition can meet the requirement of all-positive eigenvalue, the decomposision error will be larger as the number of underlying asset getting larger. Thus, although Spectral Decomposition does offer some help, it works better when the number of underlying assets is not very large.

Copula functions represent a methodology which can describe the dependence structure of multi-dimension random variable, and has recently become the most significant new tool to handle risk factors in finance such as Value-at Risk( VaR) which was probably the most widely used risk measure in financial institutions. In this paper, Copula and the forecast function of Garch model are well combined, and a new method Conditional-Copula-Garch is built for measure the dependence of financial data and compute the VaR of portfolios. Copula-Garch models allow for very flexible joint distribution by splitting the marginal behaviors form the dependence relation unlike the traditional approaches for the estimation VaR, such as variance-covariance, and the Monte Carlo approaches whereas demand the joint distribution to be known. This work presents an application of the Copula-Garch model in the estimation of VaR of a portfolio composed by NASDAQ and TAIEX (Taiwan stock exchanged capitalization weighted index) stock indices.

信用連結債券的價值主要取決於所連結資產池內的資產違約狀況，使得原始信用風險債券在到期時的本金償付受到其他債券的信用風險影響，因此如何準確且客觀的估計資產池內違約機率便一個很重要的課題，而過去文獻常以給定參數的方式，並且假設資產間的違約狀況彼此獨立下進行評價，對於聯合違約機率的捕捉並不明顯，因此本文延伸Factor Copula模型，建立信用連結債券之評價模型，該模型考慮了資產間的違約相關程度，以期達到符合市場的效果，同時配合統計之因素分析法，試圖找出影響商品價格背後的市場因子。 本研究利用延伸的評價模型以及Copula法，對實際商品做一訂價探討，結果發現，不管是使用樣本內或樣本外的資料去評價時，本研究的評價模型表現都優於Copula法，表示說評價時額外加入市場因子的考慮，對於評價是有正向的幫助；而在因子選取方面，我們選取18項因子後，經由因素分析共可萃取出三大類因素，藉由觀察期望價格與市場報價的均方根誤差，發現國家因素以及產業因素均對於商品價格有所影響，而全球因素對於商品不但沒有顯著影響，同時加入後還會使得計算出的商品期望價格更偏離市場報價，代表說並不是盲目的加入許多因子就能使得模型計算出的價格貼近市場報價，則是要視加入的因子對於資產的影響程度而定。 對於後續研究的建議：由於本研究的實證中存在一些假設，使得評價過程中並不完全符合現實市場現況，若能得到市場上的真實數據，或是改以隨機的方式來計算，相信結果會更貼近市場報價；同時，藉由選取不同的因子來評價，希望能找出國家因素、產業因素以外的其他影響因子，可助於我們更了解此項商品背後的影響因素，使得投資人能藉由觀察市場因子數據來判斷商品未來價格走勢。 / Value of the credit-linked notes depend on the pool of assets whether default or not, so the promised payoff of credit-linked notes is affected by other risky underlying assets. Therefore, how to estimate the probability of default asset pool accurately and objectively will be a very important issue. In the past literature, researchers usually use given parameters, and assume assets probability of default are independent from each other under valuation. Furthermore, it is not obvious to capture the joint probability of default. Thus, this article extends the Factor Copula Model to provide a new methodology of pricing credit-linked notes, which consider the default correlation between the extent of assets in order to achieve result in line with market and with Factor Analysis method added, trying to figure out the impact of commodity price factor behind the market. In the empirical analysis, pricing the actual commodity issued by LB Baden-Wuerttemberg using extend model and Copula model, we found that no matter choose in-the-sample or out-the-sample data to valuation, the models in this article are superior to Copula model by compare the root-mean-square deviation(RMSE). It means add the market factors into our valuation is beneficial. In terms of selection factors, we select eighteen factors prepared by Morgan Stanley Capital International, and three categories of factors may be extracted from Factor Analysis method. By observing RMSE, both national factors and industry factors will influence on the commodity, but world factors not only did not significantly impact on the commodity, but also add it to calculate the expected price further from the market price. Representative said not blind join the many factors can make the model to calculate the price close to the market price, it is a factor depending on the degree of influence of the added asset. For the suggestion of future research. The fact that the presence of empirical assumptions in this study, result in the evaluation process is not entirely realistic to market situation. We suggest to get the real data on the market or use random way to calculate, we believe that the outcome will be closer to the market price. Meanwhile, by selecting different factors to evaluate, trying to discover further factors which significantly impact on the commodity; it will help us better to understand the factors behind the commodity, so investors can predict commodity future prices by observing the market data.

Aggregating financial assets together to form a portfolio, commonly referred to as "asset pooling", is a standard practice in the banking and insurance industries. Determining a suitable probability distribution for this portfolio with each underlying asset is a challenging task unless several distributional assumptions are made. On the other hand, imposing assumptions on the distribution inhibits its ability to capture various idiosyncratic behaviors. It limits the model's usefulness in its ability to provide realistic risk metrics of the true portfolio distribution. In order to conquer this limitation, we propose two methods to model a pool of assets with much less assumptions on the correlation structure by way of finding analytical bounds. Our first method uses the Fre??chet-Hoeffding copula bounds to calculate model-free upper and lower bounds for aggregate assets evaluation. For the copulas with specific constraints, we improve the Fre??chet- Hoeffding copula bounds by providing bounds with narrower range. The improvements proposed are very robust for different types of constraints on the copula function. However, the lower copula bound does not exist for dimension three and above. Our second method tackles the open problem of finding lower bounds for higher dimensions by introducing the concept of Complete Mixability property. With such technique, we are able to find the lower bounds with specified constraints. Three theorems are proposed. The first theorem deals with the case where all marginal distributions are identical. The lower bound defined by the first theorem is sharp under some technical assumptions. The second theorem gives the lower bound in a more general setup without any restriction on the marginal distributions. However the bound achieved in this context is not sharp. The third theorem gives the sharp lower bound on Conditional VaR. Numerical results are provided for each method to demonstrate sharpness of the bounds. Finally, we point out some possible future research directions, such as looking for a general sharp lower bound for high dimensional correlation structures.

abstract: Arabic is widely known for the lack of copulas in nominal sentences in the present tense. Arabic employs a copula ‘kana’ in the past and future tenses. However, in some constructions the presence of a third person pronoun is necessary for the purpose of emphasis or ambiguity reduction. The data investigated in this thesis was from Classical Arabic, Standard Arabic, and the Western Saudi ‘Hijazi’ dialect. The thesis briefly discussed the grammaticalization of a transitive verb to a non-present tense copula in Classical Arabic. In addition, the thesis discussed the process of copularization that was a result of grammaticalization of the demonstrative third person pronoun ‘huwa’ to a present tense copula in Standard Arabic. It was shown that the pronoun went through a process of reanalysis from the specifier to the head position of PredP driven by Feature Economy and the Head Preference Principle. The result was the loss of the person feature. The new copula developed and attached to the negative particle ‘ma’ in the Hijazi dialect losing all its phi-features. These phenomena are known as the copula and negative cycles, respectively. The analysis was based on the Generative Grammar framework and the Minimalist program. This study attempted to shed light on Arabic copulas and contribute to more understanding of the use of these copulas in question and negative constructions. It may also help in typological studies, which may lead to a better understanding of the linguistic theory and the language faculty. / Dissertation/Thesis / Masters Thesis English 2015

Magíster en Economía Aplicada / El desafío en la evaluación de proyectos es que los flujos de caja son inciertos. El enfoque convencional para abordar este problema consiste en descontar los flujos de caja con una tasa ajustada por riesgo. Sin embargo, la literatura de evaluación de proyectos ha encontrado varios problemas, conceptuales y prácticos, con la utilización de este método. Un enfoque alternativo es el uso del análisis probabilístico. Este enfoque es más razonable dado que se incorpora el riesgo donde se genera, esto es, en los flujos de caja. La nueva herramienta propuesta en este trabajo se enmarca dentro del contexto del análisis probabilístico. La idea es modelar los flujos de caja a trav\'es de sus medias y matriz de covarianzas. Esta nueva herramienta utiliza también la Copula Gaussiana que a su vez se basa en los dos primeros momentos de la distribución del vector de flujos de caja. Con estos elementos se pueden hacer simulaciones de Monte Carlo que permiten estimar las distribuciones de una gran variedad de indicadores tales como el Valor Presente Neto (VPN), la tasa interna de retorno (TIR), la tasa interna de retorno modificada (MIRR), y el payback period (PBP). Las ventajas de este nuevo enfoque se demuestran aplicándolo a tres proyectos de infraestructura. Estos ejemplos dejan de manifiesto los méritos del método propuesto en comparación con el método tradicional de descontar los flujos con una tasa ajustada por riesgo.

The main objective of this thesis is to examine the dependency structure among different agricultural and energy commodity markets in the United States. For achieving this goal, the paper makes use of the Copula-GARCH(1,1) model to study the financial return volatility and the co-movement between pair of commodities including corn, soybean and gasoline over the pre-COVID 19 pandemic period (from 01-01-2018 to 01-01-2020) and the ongoing COVID 19 pandemic period (from 01-01-2020 to 01-04-2021). First, the study has shown that the time-dependent volatilities of commodity returns display volatility clustering effect in the two periods and the volatility of volatility of commodity markets is higher during the pandemic period. Second, it is observed that the correlations among different commodities have increased significantly in the ongoing pandemic period and we also find that the strongest co-movement is between returns of corn and soybean over the two periods. Finally, the results suggest that the (extreme) co-movements between agricultural commodities (corn and soybean) are governed by symmetry; that is they tend to boom and crash together during extreme shocks or events. On the other hand, the (extreme) co-movements between an agricultural commodity (corn or soybean) and the energy commodity (gas) appear to co-move asymmetrically and they tend to experience the market crash together but not the market boom.

As intelligent investors, we should always consider holding assets of different classes. Investing in assets from various classes allows us to minimize portfolio risks. In this paper, we recommend a better way of devoting money, especially for the investors who are interested in the agricultural sector. Historically fund managers use Markowitz framework to create financial portfolios. However, that framework has some fundamental limitations. A copula is a modern approach that counters the disadvantages of the Markowitz framework, to deal with portfolio construction. Copula also identifies the downside risk (the maximum amount of money you can lose) of a portfolio. We found that farmland is the best asset to have in an agricultural portfolio. However, farmland is scarce. So, we introduce copula, which can be used to find alternative assets. We also found that the portfolio composition does not change during agricultural boom or bust. Currently, the US agricultural sector is going through a slump period. Funds invested in a portfolio during the good seasons (given it was correctly invested) should not be altered during the bad times.