This dissertation research focuses on modeling and evaluating multivariate uncertainties and the dependency between the uncertainties. Managing risk and making strategic decisions under uncertainty is critically important for both individual and corporate success. In this dissertation research, we present two new methodologies, the implied binomial tree approach and the dependent decision tree approach, to modeling multivariate decision making problems with practical applications in real options valuation. First, we present the implied binomial tree approach to consolidate the representation of multiple sources of uncertainty into univariate uncertainty, while capturing the impact of these uncertainties on the project’s cash flows. This approach provides a nonparametric extension of the approaches in the literature by allowing the project value to follow a generalized diffusion process in which the volatility may vary with time and with the asset prices, therefore offering more modeling flexibility. This approach was motivated by the Implied Binomial Tree (IBT) approach that is widely used to value complex financial options. By constructing the implied recombining binomial tree in a way so as to be consistent with the simulated market information, we extended the finance-based IBT method for real options valuation — when the options are contingent on the value of one or more market related uncertainties that are not traded assets. Further, we present a general framework based on copulas for modeling dependent multivariate uncertainties through the use of a decision tree. The proposed dependent decision tree model allows multiple dependent uncertainties with arbitrary marginal distributions to be represented in a decision tree with a sequence of conditional probability distributions. This general framework could be naturally applied in decision analysis and real options valuations, as well as in more general applications of dependent probability trees. While this approach to modeling dependencies can be based on several popular copula families as we illustrate, we focus on the use of the normal copula and present an efficient computational method for multivariate decision and risk analysis that can be standardized for convenient application. / text

Obtaining the distribution of the profit and loss (PL) of a portfolio is a key problem in market risk measurement. However, existing methods, such as those based on the Normal distribution, and historical simulation methods, which use empirical distribution of risk factors, face difficulties in dealing with at least one of the following three problems: describing the distributional properties of risk factors appropriately (description problem); deriving distributions of risk factors with time horizon longer than one day (time aggregation problem); and deriving the distribution of the PL given the distributional properties of the risk factors (risk aggregation problem). Here, we show that expansion methods can provide reasonable solutions to all three problems. Expansion methods approximate a probability density function by a sum of orthogonal polynomials multiplied by an associated weight function. One of the most important advantages of expansion methods is that they only require moments of the target distribution up to some order to obtain an approximation. Therefore they have the potential to be applied in a wide range of situations, including in attempts to solve the three problems listed above. On the other hand, it is also known that expansions lack robustness: they often exhibit unignorable negative density and their approximation quality can be extremely poor. This limits applications of expansion methods in existing studies. In this thesis, we firstly develop techniques to provide robustness, with which expansion methods result in a practical approximation quality in a wider range of examples than investigated to date. Specifically, we investigate three techniques: standardisation, use of Laguerre expansion and optimisation. Standardisation applies expansion methods to a variable which is transformed so that its first and second moments are the same as those of the weight function. Use of Laguerre expansions applies those expansions to a risk factor so that heavy tails can be captured better. Optimisation considers expansions with coefficients of polynomials optimised so that the difference between the approximation and the target distribution is minimised with respect to mean integrated squared error. We show, by numerical examples using data sets of stock index returns and log differences of implied volatility, and GARCH models, that expansions with our techniques are more robust than conventional expansion methods. As such, marginal distributions of risk factors can be approximated by expansion methods. This solves a part of the description problem: the information on the marginal distributions of risk factors can be summarised by their moments. Then we show that the dependence structure among risk factors can be summarised in terms of their cross-moments. This solves the other part of the description problem. We also use the fact that moments of risk factors can be aggregated using their moments and cross-moments, to show that expansion methods can be applied to both the time and risk aggregation problems. Furthermore, we introduce expansion methods for multivariate distributions, which can also be used to approximate conditional expectations and copula densities by rational functions.

The increase in the use of copulas has introduced implementation issues for both practitioners and researchers. One of the issues is to obtain a copula function for a given set of data. The most common approaches for the estimation of the parameters of the copula functions have been the Maximum Likelihood Estimator (MLE) and the Inference Functions for Margins (IFM) methods. Archimedean copulas are one of the most important classes of copulas that are widely used in both finance and insurance for modelling dependent risks. However, simulating multivariate Archimedean copulas has always been a difficult task as the number of dimensions increases. The assessment of capital requirements has always been an important application of stochastic modelling. Capital requirements can vary significantly depending on the model adopted. Several professional bodies have recently discussed the concept of dependencies between insurance risks. They suggest that insurers should use a technique based on copulas to describe the dependence of risks within an insurance company in the context of solvency assessment. The first contribution of this thesis is to provide an insight into the efficiency of parameter estimation methods. This thesis uses numerical experiments to assess the performance of the two common approaches. The second contribution of this thesis is to present a new algorithm to simulate multivariate Exchangeable Archimedean copulas. This algorithm provides a practical solution for simulating one-parameter multivariate Archimedean copulas. Numerical experiments are used to apply this algorithm to determine the "additional" economic capital for an insurance company with multiple lines of business that wants to expand its business by adding another line of business and where the businesses are dependent. The third contribution of this thesis is to quantify the impact of the choice of copulas on the solvency measure of a general insurer within a Dynamic Financial Analysis modelling framework. The results of our experiments provide important guidance for the capital assessment for general insurers.

Thesis (Ph. D.)--University of California, San Diego, 2005. / Vita. Includes bibliographical references (leaves 118-122).

O grande número de publicações na área de finanças atualmente utilizando a modelagem de cópulas pode ser explicada pela capacidade de esta técnica estatística conseguir lidar com a evidência de não normalidade das séries de retornos de ativos financeiros. A não normalidade é evidenciada através do “sorriso de volatilidade” presente em séries de opções de ações perto do vencimento; existência de “caudas pesadas” em carteiras de instituições e consequentemente no gerenciamento de risco das Instituições. Particularmente com relação ao Value at Risk (VaR), que é uma técnica estatística que tem por objetivo calcular a perda máxima de uma carteira em dado horizonte de tempo considerando um nível de significância adotado, a existência de caudas pesadas nas séries gera um problema para a determinação da distribuição de probabilidade conjunta, implicando em grande dificuldade na mensuração do grau de exposição aos fatores de risco. Esse fato acaba por dificultar o correto e efetivo gerenciamento de risco de uma carteira, pois em tese, devido à existência de não normalidade, não é possível separar os efeitos de ativos de diferentes características. Em casos de crises e bolhas, o portfólio pode ser mais arriscado que o desejado ou excessivamente conversador. Neste sentido, a utilização de Cópulas torna-se atrativa, pois com esta técnica é possível separar as distribuições marginais de cada ativo da estrutura de dependência das variáveis. O objetivo do trabalho é propor uma modelagem de risco a partir do uso de Cópulas para o cálculo do Value at Risk (VaR), utilizando os métodos de volatilidade GARCH (1,1), EWMA e HAR. A aplicação empírica do modelo foi efetuada a partir de uma amostra de uma série de retornos de uma carteira teórica composta por ativos de renda variável (ações preferenciais) das empresas Petrobras, Vale, Usiminas e Gerdau. A amostra utilizada corresponde aos preços diários entre o período de 03 de março de 2006 até 30 de abril de 2010, representando 1.026 observações diárias. Os resultados apurados para a amostra demonstraram que as cópulas tendem a gerar um Value at Risk (VaR) significativo para a maioria das famílias de Cópulas, quando testado pelo Teste de Kupiec (1995). / The large number of publications in finance using currently copulas can be explained by the ability of this technique to deal with statistical evidence of non-normality of the return series of financial assets. The non-normality is evidenced by the "volatility smile" in the series of stock options near expiration, the existence of "heavy tails" in portfolios of institutions and consequently the risk management of the institutions. Especially regarding the Value at Risk (VaR), which is a statistical technique that aims to calculate the maximum loss a portfolio at a given time horizon considering a significance level, the existence of heavy tails in the series creates a problem for determining the joint probability distribution, resulting in great difficulty in measuring the degree of exposure to risk factors. This fact makes difficult the correct and effective risk management of a portfolio, because in theory, due to the existence of non-normality, it is not possible to separate the effects of assets with different characteristics. In cases of crises and bubbles, the portfolio may be riskier than desired or overly chatty. In this regard, the use of copulas becomes attractive, because with this technique is possible to separate the marginal distributions of each dependence structure of the variables. The objective is to propose a model of risk using copulas for the calculation of Value at Risk (VaR), using the methods of volatility GARCH (1,1), EWMA and HAR. The empirical application of the model was made from a sample of a series of returns of a theoretical portfolio of assets in equities (shares) of Petrobras, Vale, Usiminas and Gerdau. The sample corresponds to the daily prices in the period between March 3rd, 2006 until April 30th, 2010, representing 1026 daily observations. The results obtained showed that copulas tend to generate a Value at Risk (VaR) for the most significant families of copulas, when tested by the Test of Kupiec (1995).

vi Abstract Proper understanding of the dependence between assets is a crucial ingredient for a number of portfolio and risk management tasks. While the research in this area has been lively for decades, the recent financial crisis of 2007-2008 reminded us that we might not understand the dependence properly. This crisis served as catalyst for boosting the demand for models capturing the dependence structures. Reminded by this urgent call, literature is responding by moving to nonlinear de- pendence models resembling the dependence structures observed in the data. In my dissertation, I contribute to this surge with three papers in financial econo- metrics, focusing on nonlinear dependence in financial time series from different perspectives. I propose a new empirical model which allows capturing and forecasting the conditional time-varying joint distribution of the oil - stocks pair accurately. Em- ploying a recently proposed conditional diversification benefits measure that con- siders higher-order moments and nonlinear dependence from tail events, I docu- ment decreasing benefits from diversification over the past ten years. The diver- sification benefits implied by my empirical model are, moreover, strongly varied over time. These findings have important implications for asset allocation, as the benefits of...

As diversas crises financeiras e econômicas ocorridas nas últimas décadas geraram uma demanda pelo estudo da propagação destes efeitos entre as economias. Neste sentido este trabalho tem como objetivo estudar o efeito contágio (Shift Contagion) tal como definido em Rigobon (2002) do mercado financeiro do Brasil para quatro países para o período de 2011 a 2016, que inclui a recente crise econômica no Brasil. Tais países são Argentina, Colômbia, Chile, e Peru. Para tal, utilizou-se metodologia de cópulas paramétricas estáticas. Com base nos resultados obtidos, não é possível identificar indícios da ocorrência de contágio do mercado financeiro brasileiro para os mercados financeiros dos países analisados no período do estudo. / Different financial and economic crises that occurred in the last decades have generated a demand of studies on propagation of their effects between economies. In this sense, this work has the goal to study the contagion effect (shift contagion), as defined by Rigobon (2002), from the Brazilian financial market to four countries in the period from 2011 to 2016 that includes the recent Brazilian economic crisis. These countries are Argentina, Colombia, Chile, and Peru. For this purpose, the methodology of parametric static copulas is used. Accordingly with the results, it is not possible to identify evidences of the contagion effect from the Brazilian financial market to the analyzed countries.

A modelagem da estrutura de dependência é de grande importância em todos os ramos da economia onde há incerteza. Ela é um elemento crucial na análise de risco e para a tomada de decisão sob incerteza. As cópulas oferecem aos agentes que se deparam com este problema um poderoso e flexível instrumento para modelar a estrutura de dependência entre variáveis aleatórias e que é preferível ao instrumento tradicional baseado na correlação linear. Neste estudo, nós analisamos a dinâmica temporal da estrutura de dependência entre índices de mercados financeiros internacionais e propomos um novo procedimento para capturar a estrutura de dependência ao longo do tempo. Adicionalmente, estudamos alguns fatos estilizados sobre índices de mercados financeiros como a relação entre volume-volatilidade e retorno-volatilidade. / Modelling dependence is of key importance to all economic fields in which uncertainty plays a large role. It is a crucial element of risk analysis and decision making under uncertainty. Copulas offer economic agents facing uncertainty a powerful and flexible tool to model dependence between random variables and often are preferable to the traditional, correlation-based approach. In this work we analyze the time dynamics of the dependence structure between broad stock market indices and propose a novel procedure to capture dependence structure over time. Additionally, we study some stylized facts about stock market indexes such as volume-volatility and return-volatility relations.

This thesis is concerned with the modelling of integer-valued time series. The data naturally occurs in various areas whenever a number of events are observed over time. The model considered in this study consists of a Gaussian copula with autoregressive-moving average (ARMA) dependence and discrete margins that can be specified, unspecified, with or without covariates. It can be interpreted as a 'digitised' ARMA model. An ARMA model is used for the latent process so that well-established methods in time series analysis can be used. Still the computation of the log-likelihood poses many problems because it is the sum of 2^N terms involving the Gaussian cumulative distribution function when N is the length of the time series. We consider an Monte Carlo Expectation-Maximisation (MCEM) algorithm for the maximum likelihood estimation of the model which works well for small to moderate N. Then an Approximate Bayesian Computation (ABC) method is developed to take advantage of the fact that data can be simulated easily from an ARMA model and digitised. A spectral comparison method is used in the rejection-acceptance step. This is shown to work well for large N. Finally we write the model in an R-vine copula representation and use a sequential algorithm for the computation of the log-likelihood. We evaluate the score and Hessian of the log-likelihood and give analytic solutions for the standard errors. The proposed methodologies are illustrated using simulation studies and highlight the advantages of incorporating classic ideas from time series analysis into modern methods of model fitting. For illustration we compare the three methods on US polio incidence data (Zeger, 1988) and we discuss their relative merits.

A modelagem da estrutura de dependência é de grande importância em todos os ramos da economia onde há incerteza. Ela é um elemento crucial na análise de risco e para a tomada de decisão sob incerteza. As cópulas oferecem aos agentes que se deparam com este problema um poderoso e flexível instrumento para modelar a estrutura de dependência entre variáveis aleatórias e que é preferível ao instrumento tradicional baseado na correlação linear. Neste estudo, nós analisamos a dinâmica temporal da estrutura de dependência entre índices de mercados financeiros internacionais e propomos um novo procedimento para capturar a estrutura de dependência ao longo do tempo. Adicionalmente, estudamos alguns fatos estilizados sobre índices de mercados financeiros como a relação entre volume-volatilidade e retorno-volatilidade. / Modelling dependence is of key importance to all economic fields in which uncertainty plays a large role. It is a crucial element of risk analysis and decision making under uncertainty. Copulas offer economic agents facing uncertainty a powerful and flexible tool to model dependence between random variables and often are preferable to the traditional, correlation-based approach. In this work we analyze the time dynamics of the dependence structure between broad stock market indices and propose a novel procedure to capture dependence structure over time. Additionally, we study some stylized facts about stock market indexes such as volume-volatility and return-volatility relations.