The submitted work deals with option pricing. Mathematical approach is immediately followed by an economic interpretation. The main problem is to model the underlying uncertainities driving the stock price. Using two well-known valuation models, binomial model and Black-Scholes model, we explain basic principles, especially risk neutral pricing. Due to the empirical biases new models have been developped, based on pure jump process. Variance gamma process and its special symmetric case are presented.

Hipóteses precisas são características naturais das teorias econômicas de determinação do valor ou preço de ativos financeiros. Nessas teorias, a precisão das hipóteses assume a forma do conceito de equilíbrio ou da não arbitragem. Esse último possui um papel fundamental nas teorias de finanças. Sob certas condições, o Teorema Fundamental do Apreçamento de Ativos estabelece um sistema único e coerente para valorização dos ativos em mercados não arbitrados, valendo-se para tal das formulações para processos de martingal. A análise da distribuição estatística desses ativos financeiros ajuda no entendimento de como os participantes se comportam nos mercados, gerando assim as condições para se arbitrar. Nesse sentido, a tese defendida é a de que o estudo da hipótese de não arbitragem possui contrapartida científica, tanto do lado teórico quanto do empírico. Utilizando-se do modelo estocástico Variância Gama para os preços dos ativos, o teste Bayesiano FBST é implementado com o intuito de se verificar a existência da arbitragem nos mercados, potencialmente expressa nos parâmetros destas densidades. Especificamente, a distribuição do Índice Bovespa é investigada, com os parâmetros risco-neutros sendo estimados baseandose nas opções negociadas no Segmento de Ações e no Segmento de Derivativos da BM&FBovespa. Os resultados aparentam indicar diferenças estatísticas significantes em alguns períodos de tempo. Até que ponto esta evidência é a expressão de uma arbitragem perene nesses mercados ainda é uma questão em aberto. / Precise hypotheses are natural characteristics of the economic theories for determining the value or prices of financial assets. Within these theories the precision is expressed in terms of equilibrium and non-arbitrage hypotheses. The former concept plays an essential role in the theories of finance. Under certain conditions, the Fundamental Theorem of Asset Pricing establishes a coherent and unique asset pricing framework in non-arbitraged markets, grounded on martingales processes. Accordingly, the analysis of the statistical distributions of financial assets can assist in understanding how participants behave in the markets, and may or may not engender conditions to arbitrage. On this regard, the dissertation proposes that the study of non-arbitrage hypothesis has a scientific counterparty, theoretically and empirically. Using a variance gamma stochastic model for prices, the Bayesian test FBST is conducted to verify the presence of arbitrage potentially incorporated on these densities parameters. Specifically, the Bovespa Index distribution is investigated, with risk neutral parameters estimated based on options traded in the Equities Segment and the Derivatives Segment at the BM&FBovespa Exchange. Results seem to indicate significant statistical differences at some periods of time. To what extent this evidence is actually the expression of a perennial arbitrage between the markets still is an open question.

Hipóteses precisas são características naturais das teorias econômicas de determinação do valor ou preço de ativos financeiros. Nessas teorias, a precisão das hipóteses assume a forma do conceito de equilíbrio ou da não arbitragem. Esse último possui um papel fundamental nas teorias de finanças. Sob certas condições, o Teorema Fundamental do Apreçamento de Ativos estabelece um sistema único e coerente para valorização dos ativos em mercados não arbitrados, valendo-se para tal das formulações para processos de martingal. A análise da distribuição estatística desses ativos financeiros ajuda no entendimento de como os participantes se comportam nos mercados, gerando assim as condições para se arbitrar. Nesse sentido, a tese defendida é a de que o estudo da hipótese de não arbitragem possui contrapartida científica, tanto do lado teórico quanto do empírico. Utilizando-se do modelo estocástico Variância Gama para os preços dos ativos, o teste Bayesiano FBST é implementado com o intuito de se verificar a existência da arbitragem nos mercados, potencialmente expressa nos parâmetros destas densidades. Especificamente, a distribuição do Índice Bovespa é investigada, com os parâmetros risco-neutros sendo estimados baseandose nas opções negociadas no Segmento de Ações e no Segmento de Derivativos da BM&FBovespa. Os resultados aparentam indicar diferenças estatísticas significantes em alguns períodos de tempo. Até que ponto esta evidência é a expressão de uma arbitragem perene nesses mercados ainda é uma questão em aberto. / Precise hypotheses are natural characteristics of the economic theories for determining the value or prices of financial assets. Within these theories the precision is expressed in terms of equilibrium and non-arbitrage hypotheses. The former concept plays an essential role in the theories of finance. Under certain conditions, the Fundamental Theorem of Asset Pricing establishes a coherent and unique asset pricing framework in non-arbitraged markets, grounded on martingales processes. Accordingly, the analysis of the statistical distributions of financial assets can assist in understanding how participants behave in the markets, and may or may not engender conditions to arbitrage. On this regard, the dissertation proposes that the study of non-arbitrage hypothesis has a scientific counterparty, theoretically and empirically. Using a variance gamma stochastic model for prices, the Bayesian test FBST is conducted to verify the presence of arbitrage potentially incorporated on these densities parameters. Specifically, the Bovespa Index distribution is investigated, with risk neutral parameters estimated based on options traded in the Equities Segment and the Derivatives Segment at the BM&FBovespa Exchange. Results seem to indicate significant statistical differences at some periods of time. To what extent this evidence is actually the expression of a perennial arbitrage between the markets still is an open question.

Doctor of Philosophy (PhD) / This thesis mainly builds on the Variance Gamma (VG) model for financial assets over time of Madan & Seneta (1990) and Madan, Carr & Chang (1998), although the model based on the t distribution championed in Heyde & Leonenko (2005) is also given attention. The primary contribution of the thesis is the development of VG models, and the extension of t models, which accommodate a dependence structure in asset price returns. In particular it has become increasingly clear that while returns (log price increments) of historical financial asset time series appear as a reasonable approximation of independent and identically distributed data, squared and absolute returns do not. In fact squared and absolute returns show evidence of being long range dependent through time, with autocorrelation functions that are still significant after 50 to 100 lags. Given this evidence against the assumption of independent returns, it is important that models for financial assets be able to accommodate a dependence structure.

Doctor of Philosophy (PhD) / This thesis mainly builds on the Variance Gamma (VG) model for financial assets over time of Madan & Seneta (1990) and Madan, Carr & Chang (1998), although the model based on the t distribution championed in Heyde & Leonenko (2005) is also given attention. The primary contribution of the thesis is the development of VG models, and the extension of t models, which accommodate a dependence structure in asset price returns. In particular it has become increasingly clear that while returns (log price increments) of historical financial asset time series appear as a reasonable approximation of independent and identically distributed data, squared and absolute returns do not. In fact squared and absolute returns show evidence of being long range dependent through time, with autocorrelation functions that are still significant after 50 to 100 lags. Given this evidence against the assumption of independent returns, it is important that models for financial assets be able to accommodate a dependence structure.

Stein's method is a powerful technique that can be used to obtain bounds for approximation errors in a weak convergence setting. The method has been used to obtain approximation results for a number of distributions, such as the normal, Poisson and Gamma distributions. A major strength of the method is that it is often relatively straightforward to apply it to problems involving dependent random variables. In this thesis, we consider the adaptation of Stein's method to the class of Variance-Gamma distributions. We obtain a Stein equation for the Variance-Gamma distributions. Uniform bounds for the solution of the Symmetric Variance-Gamma Stein equation and its first four derivatives are given in terms of the supremum norms of derivatives of the test function. New formulas and inequalities for modified Bessel functions are obtained, which allow us to obtain these bounds. We then use local approach couplings to obtain bounds on the error in approximating two asymptotically Variance-Gamma distributed statistics by their limiting distribution. In both cases, we obtain a convergence rate of order n^-1 for suitably smooth test functions. The product of two normal random variables has a Variance-Gamma distribution and this leads us to consider the development of Stein's method to the product of r independent mean-zero normal random variables. An elegant Stein equation is obtained, which motivates a generalisation of the zero bias transformation. This new transformation has a number of interesting properties, which we exploit to prove some limit theorems for statistics that are asymptotically distributed as the product of two central normal distributions. The Variance-Gamma and Product Normal distributions arise as functions of the multivariate normal distribution. We end this thesis by demonstrating how the multivariate normal Stein equation can be used to prove limit theorems for statistics that are asymptotically distributed as a function of the multivariate normal distribution. We establish some sufficient conditions for convergence rates to be of order n^-1 for smooth test functions, and thus faster than the O(n^-1/2) rate that would arise from the Berry-Esseen Theorem. We apply the multivariate normal Stein equation approach to prove Variance-Gamma and Product Normal limit theorems, and we also consider an application to Friedman's X² statistic.

This thesis focuses on the parameter stability of additive normal tempered stable processes when calibrating a volatility surface. The studied processes arise as a generalization of Lévy normal tempered stable processes, and their main characteristic are their time-dependent parameters. The theoretical background of the subject is presented, where its construction is discussed taking as a starting point the definition of Lévy processes. The implementation of an option valuation model using Fourier techniques and the calibration process of the model are described. The thesis analyzes the parameter stability of the model when it calibrates the volatility surface of a market index (EURO STOXX 50) during three time spans. The time spans consist of the periods from Dec 2016 to Dec 2017 (after the Brexit and the US presidential elections), from Nov 2019 to Nov 2020 (during the pandemic caused by COVID-19) and a more recent time period, April 2023. The findings contribute to the understanding of the model itself and the behavior of the parameters under particular economic conditions.

In this thesis, the structural model in credit risk and the credit derivatives is studied under both Black-Scholes setting and Variance Gamma (VG) setting. Using a Variance Gamma process, the distribution of the firm value process becomes asymmetric and leptokurtic. Also, the jump structure of VG processes allows random default times of the reference entities. Among structural models, the most emphasis is made on the Black-Cox model by building a relation between the survival probabilities of the Black-Cox model and the value of a binary down and out barrier option. The survival probabilities under VG setting are calculated via a Partial Integro Differential Equation (PIDE). Some applications of binary down and out barrier options, default probabilities and Credit Default Swap par spreads are also illustrated in this study.

因為股票市場常具有厚尾、偏態和峰態的特性且在國際的股票市場之間，股票報酬長存在有尾端相依的情況，所以我們的資產模型不能選用Gaussian分配。 近幾年來，常用GH 分配建構單維度的股票報酬。這篇文章將利用多元仿射JD、多元仿射VG 和多元仿射NIG分配去建構風險性資產的報酬並請應用到資產配置。 建構風險性資產的報酬後，我們提供兩種不同形式的投資組合並且可以導出投資組合的期望值、變異數、偏態和峰態。我們嘗試以投資組合的期望值、變異數、偏態和峰態當成我們的目標函數，然後得出未來最佳的投資組合的權重。為了讓我們的資產配置更加動態和有效率，我們重新估計模型的參數、選擇最佳的投資組合權重，然後重新評估最佳的資產配置在每個決策日期。實證結果發現當股票市場的表現好的時候，我們建議資產配置應使用偏態當成我們的目標函數，但是當股票市場的表現太好的時候，我們建議資產配置應使用變異數當成我們的目標函數。 / Since the stock markets always have the characteristics of heavy-tailness, skewness and kurtosis and there exists tail dependence among the international stock markets, we can’t use the Gaussian distribution as our model. Recently, the generalized hyperbolic (GH) distribution has been suggested to fit the single stock returns. This article will use the multivariate affine JD (MAJD), multivariate affine variance gamma (MAVG) and multivariate affine normal inverse Gaussian (MANIG) distributions to construct the risky asset returns, and apply them to asset allocation. After constructing the risky asset returns, we provide two different forms of portfolio and obtain the mean, variance, skewness, kurtosis of portfolio. We can try to select the optimal weights of portfolio by using the mean, variance, skewness, kurtosis of portfolios as our objective functions. To make our asset allocation more dynamic and efficient, we re-estimate all parameters for our models, select the optimal weights of portfolio, and re-assess the optimal asset allocation at each decision date. Empirically, when the performances of stock markets are good, we suggest that our asset allocation uses the skewness as the objective function. When the performances of stock markets are not good, we suggest that our asset allocation uses the variance as the objective function.