DM - Teses de Doutoramento / Ph.D. Thesis
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- Option pricing in generalized rough Bergomi modelsPublication . Guerreiro, Henrique Manuel Emídio Lourenço; Guerra, João; Grossinho, Maria do RosárioThe present thesis concerns generalizations of the rough Bergomi model which are able to t observed VIX implied volatility smiles. First, we propose a new stochastic change of measure based on a fractional Ornstein-Uhlenbeck process with a regime switching long term mean. We solve the relevant fractional stochastic di erencial equation and obtain a semi-closed formula for the forward variance curve. Moreover, we employ two variance reduction methods which substantially reduce the computational cost of simulation and pricing. Then, we consider stochastic Volterra models, where the variance follows a truncated Brownian semi-stationary process with stochastic volatility (of volatility). We device a least squares Monte Carlo method which does not require running regressions on an in nite dimensional predictor variable. In general, this would be the case for non-Markovian models. This least squares Monte Carlo method constitutes a new way for pricing VIX options in a setting where volatility and vol-of-vol are not independent. We provide numerical experiments which attest to the accuracy and e ciency gain in the numerical methods we propose. Moreover, we compare the outputs of both generalizations of the rough Bergomi to market data. The models prove to be able to reproduce key characteristics of both SP500 and VIX option markets. Finally, we discuss a possible framework for a (pseudo) rough vol-of-vol through a multi-factor Markovian approximation of the vol-of-vol process. We identify a key martingale condition which may allow to express the VIX in terms of the solution of a certain Riccati ordinary di erencial equation. We derive this equation and provide su cient conditions for the existence of solutions. We also provide some partial results regarding the martingale condition. In particular, we verify a local martingale condition.
- Theoretical and Practical Advances in Actuarial Science : cyber risk: an analysis of self-protection, the prediction of claims, risk evaluation and managementPublication . Azevedo, Alana Katielli Nogueira; Reis, Alfredo Egídio; Bergel, Agnieszka IzabellaEsta tese apresenta avanços teóricos e práticos na ciência atuarial. É composta por cinco capítulos que consistem em três artigos sobre temas da ciência atuarial, ou seja, tópicos em teoria do risco, dados de seguros e análise de risco. Começamos com uma análise com foco na ocorrência de ataques cibernéticos, à luz das características individuais de empresas brasileiras em relação ao uso de tecnologias e gerenciamento de riscos cibernéticos. Na primeira parte, obtemos informações se a proteção de segurança avançada ajuda a prevenir incidentes cibernéticos, comparando o número de ataques em empresas usando um método de correspondência de pontuação de propensão. A seguir introduzimos uma estrutura de modelo de rede neural para estudar essas diferenças, usando as características da empresa, a fim de prever a ocorrência de um sinistro e gerar informações sobre empresas que podem ter alto risco em relação a ataques cibernéticos. Em sequência, considerando metolodologias de aprendizagem de máquina, trabalhamos com dados reais constituídos por 683 incidentes de riscos cibernéticos identificados num conjunto de 2718 empresas do setor bancário. Desenvolvemos uma análise da frequência (número de ocorrências) do risco cibernético, utilizando toda a informação disponível e introduzimos uma estrutura de árvore de decisão capaz de identificar se uma determinada empresa está sujeita a sinistros cibernéticos. Todas essas informações sobre frequência podem orientar o processo de aceitação do risco pelas seguradoras, bem como conscientizar sobre o risco cibernético a que as empresas estão expostas. Apresentamos então o último artigo, uma modelagem de risco cibernético, estimando as tabelas de prêmios puros de seguros em função de incidentes passados (criação de tarifas a priori) e de variáveis significativas. Primeiramente realizamos um ajuste considerando a Abordagem de Distribuição de Perdas (abreviadamente LDA), uma abordagem estatística para calcular distribuições de perdas agregadas, sem especificar covariáveis. Em seguida propomos uma análise das perdas de risco cibernético, introduzindo uma estrutura de Modelos Aditivos Generalizados de Localização, Escala e Forma (GAMLSS) com objetivo de buscar um modelo atuarial para a cobertura de perdas por riscos cibernéticos utilizando todas as informações disponíveis na estimativa da distribuição agregada de perdas.
- Theoretical and practical advances in actuarial science : risk, ruin theory and life pensionsPublication . Alcoforado, Renata Gomes; Reis, Alfredo Egídio dosEsta tese apresenta avanços teóricos e práticos na ciência atuarial. É composta por seis capítulos que consistem em seis artigos sobre temas da ciência atuarial, ou seja, tópicos em teoria do risco, teoria da ruína, fundos de pensão, dados de seguros e análise de risco. Começamos com text mining em big data em um estudo de caso sobre modelos de risco com dependência. Em seguida, apresentamos um programa de micro pensão pública no Brasil com duas bases de dados contendo 4.644.698 e 1.332.080 pessoas, respectivamente. Neste capítulo, apresentamos a heterogeneidade entre as Unidades Federativas e propomos um ajuste à idade de benefício. Em sequência, avançamos em um tópico teórico, estudamos medidas de ruína e dividendos no modelo de risco dual com renovação. Na primeira parte, calculamos a probabilidade de ruína, os dividendos esperados descontados, o valor do dividendo único, a probabilidade de dividendo, o número de ganhos para atingir uma determinada meta superior e o número de ganhos até a ruína. Finalizamos este capítulo trabalhando alguns exemplos numéricos ilustrativos. A seguir apresentamos uma função de penalização para o modelo de risco dual com renovação citado anteriormente, neste problema trabalhamos com um modelo em que os ganhos aleatórios ocorrem sob um processo de renovação. Apresentamos fórmulas, explicando como alcançá-las, resolvemos a equação integro-diferencial para casos positivos e negativos e, por fim, mostramos exemplos numéricos. Em seguida, trabalhamos em outro avanço prático, consideramos um modelo de risco com frequência e montante de sinistros dependentes para seguros de responsabilidade civil e habitacional, que são dois tipos de seguro, que ao contrário do seguro de automóvel, geralmente não são objeto de estudo. Para tal, trabalhamos com uma base de dados composta por 15.665 sinistros, sendo 966 de responsabilidade civil e 14.699 de seguros habitacionais. Neste capítulo, mostramos novos resultados na área e trabalhamos com modelos GAMLSS capazes de capturar a dependência entre a frequência e o montante dos sinistros. Apresentamos então o último artigo, uma modelagem de risco para commodities no Brasil, este é um trabalho aplicado para os preços à vista e futuros do Boi Gordo. Neste trabalho, usamos cinco métodos diferentes e comparamos 12 modelos distintos como forma de retratar e prever o comportamento da commodity BGI. Para finalizar, fazemos algumas observações finais na conclusão.
- Pricing American Options by the Black-Scholes Equation with a Nonlinear Volatility FunctionPublication . Kord, Yaser Faghannataj; Grossinho, Maria do Rosário; Sevcovic, DanielIn this thesis we are concerned with the study of American-style options in presence of variable transactions costs. This leads to consider some generalized Black-Scholes equations with a nonlinear volatility function depending on the product of the underlying asset price and the second derivative of the option price. Mathematically, this involves the study of a free boundary problem for a nonlinear parabolic equation. The fully nonlinear character of the corresponding differential operator induces increased difficulties. By overcoming adequately those difficulties, we obtain qualitative and quantitative results regarding both types of American-style options, that is put and call options, as described next. Firstly, we investigate the qualitative and quantitative behaviour of a solution to the problem of pricing American style perpetual put options. We assume the option price is a solution to a stationary generalized Black-Scholes equation with a nonlinear volatility function. We prove existence and uniqueness of a solution to the free boundary problem. We derive a single implicit integral equation for the free boundary position and a closed form formula for the option price. It is a generalization of the well-known explicit closed form solution derived by Merton for the case of constant volatility. We also present results of numerical computations for the free boundary position, option price and their dependence on model parameters. Secondly, we analyze a nonlinear generalization of the Black{Scholes equation for pricing American-style call options, with nonlinear volatility. This model generalizes the well-known Leland model with constant transaction costs. Due to the fully nonlinear nature of the differential operator that appears in the model, the direct computation of the nonlinear complementarity problem becomes harder and unstable. Therefore, we propose a new approach to reformulate the nonlinear complementarity problem in terms of the new transformed variable for which the differential operator has the form of a quasilinear parabolic operator. We derive the nonlinear complementarity problem for the transformed variable in order to apply the Gamma transformation for American style options. We then solve the variacional problem by means of the modi ed projected successive over relaxation (PSOR) for constructing an effective numerical scheme for discretization of the Gamma variacional inequality. Finally, we present several computational examples of the nonlinear Black- Scholes equation for pricing American-style call options in the presence of variable transaction costs.
- Essays on Econometrics : Nonlinearities and NonnormalitiesPublication . Zsurkis, Gabriel Florin; Nicolau, João; Rodrigues, PauloThis Dissertation consists of three independent papers on econometrics, having in common the fact that each of them proposes a new methodology to deal with issues caused by the departure from linearity and gaussianity assumptions. We start by introducing a simple and easy to implement procedure to test for multiple structural changes in persistence. An in-depth Monte Carlo analysis shows that the new procedure performs well under various DGPs with persistence changes. The application of the proposed test to OECD countries inflation reveals relevant statistical evidence of breaks in persistence for all countries. Overall, the persistence was high and non-mean-reverting until the early 80’s and subsequently decreased, which coincides with the beginning of the Great Moderation. Then, the second paper introduce a flexible framework able to capture some aspects of the potential nonlinear causal relationships between economic variables. More precisely, the proposed procedure estimates the expected time (ET) an outcome variable takes to cross a fixed threshold given a starting value and conditional on covariates. An application to the economic activity-yield spread relationship for the U.S. suggests that the yield spread may have an important role in stimulating a faster return to desirable growth rates when the economy is in contraction or faces weak growth. Moreover, negative yield spread values in the presence of positive and high industrial production growth rates leads to a quick return to negative growth rates and may trigger a recession. Finally, the third paper proposes a simple framework that allows us to take into account the magnitude of potential losses incurred throughout the investment horizon, denoted intra-horizon risk, in portfolio optimization. To this end, we introduce a novel nonparametric method to estimate the first passage probability function that only make use of the Markovian property of the returns. An empirical application is provided considering equity, bond and commodity Exchange Traded funds (ETFs). Our results suggest that the proposed framework indicates portfolios with lower expected time to reach the target return than those indicated by the Markowitz’ mean-variance approach with similar levels of intra-horizon risk, which may result in higher expected annualized return if the lower threshold that triggers a stop-loss decision is not crossed.
- Uncertainty quantification with a Gaussian Process Prior : an example from macroeconomicsPublication . Tavares, Ivo Alberto Valente; Paulo, RuiThis thesis may be broadly divided into 4 parts. In the first part, we do a literature review of the state of the art in misspecification in Macroeconomics, and what so far has been the contribution of a relatively new area of research called Uncertainty Quantification to the Macroeconomics subject. These reviews are essential to contextualize the contribution of this thesis in the furthering of research dedicated to correcting non-linear misspecifications, and to account for several other sources of uncertainty, when modelling from an economic perspective. In the next three parts, we give an example, using the same simple DSGE model from macroeconomic theory, of how researchers may quantify uncertainty in a State-Space Model using a discrepancy term with a Gaussian Process prior. The second part of the thesis, we used a full Gaussian Process (GP) prior on the discrepancy term. Our experiments showed that despite the heavy computational constraints of our full GP method, we still managed to obtain a very interesting forecasting performance with such a restricted sample size, when compared with similar uncorrected DSGE models, or corrected DSGE models using state of the art methods for time series, such as imposing a VAR on the observation error of the state-space model. In the third part of our work, we improved on the computational performance of our previous method, using what has been referred in the literature as Hilbert Reduced Rank GP. This method has close links to Functional Analysis, and the Spectral Theorem for Normal Operators, and Partial Differential Equations. It indeed improved the computational processing time, albeit just slightly, and was accompanied with a similarly slight decrease in the forecasting performance. The fourth part of our work delved into how our method would account for model uncertainty just prior, and during, the great financial crisis of 2007-2009. Our technique allowed us to capture the crisis, albeit at a reduced applicability possibly due to computational constraints. This latter part also was used to deepen the understanding of our model uncertainty quantification technique with a GP. Identifiability issues were also studied. One of our overall conclusions was that more research is needed until this uncertainty quantification technique may be used in as part of the toolbox of central bankers and researchers for forecasting economic fluctuations, specially regarding the computational performance of either method.
- Real World Economic Scenario GeneratorPublication . Lopes, Sara Bárbara Dutra; Vásquez Cendón, Carlos; Grossinho, Maria do RosárioNeste trabalho apresentamos uma metodologia para simular a evolução das taxas de juros sob medida de probabilidade real. Mais precisamente, usando o modelo de mercado Shifted Lognormal LIBOR multidimensional e uma especificação do vetor do preço de mercado do risco, explicamos como realizar simulações das taxas de juro futuras, usando o método de Euler-Maruyama com preditor-corretor. A metodologia proposta permite acomodar a presença de taxas de juro negativas, tal como é observado atualmente em vários mercados. Após definir a estrutura livre de default, generalizamos os resultados para incorporar a existência de risco de crédito nos mercados financeiros e desenvolvemos um modelo LIBOR para obrigações com risco de crédito classificadas por ratings. Neste trabalho modelamos diretamente os spreads entre as classificações de ratings de acordo com uma dinâmica estocástica que garante a monotonicidade dos preços dos títulos relativamente às classificações por ratings.
- Option pricing in exponential Lévy models with transaction costsPublication . Cantarutti, Nicola; Guerra, João; Guerra, Manuel CastroIn this thesis we present a new model for pricing European call options in presence of proportional transaction costs, when the stock price follows a general exponential Lévy process. The model is a generalization of the celebrated work of Davis, Panas and Zariphopoulou, where the value of the option is defined as the utility indifference price. This approach requires the solution of a stochastic singular control problem in finite time. We introduce the general formulation of the problem, and derive the associated Hamilton-Jacobi-Bellman equation (HJB), which is a nonlinear partial integro-differential equation, with the form of a variational inequality. We prove that the value function of the problem is a solution of the HJB equation in the viscosity sense. The original problem is then simplified for the specific case of the exponential utility function, under the assumption of absence of default for the investor's portfolio. We solve numerically the optimization problems using the Markov chain approximation method. We also apply the multinomial method to the Variance Gamma process, which is an alternative and more efficient approach to discretize the continuous time process. We provide a numerical scheme and prove that it is monotone, stable and consistent and that the solution converges to the viscosity solution of the original HJB equation. Several numerical solutions are presented for both the original problem and the simplified problem. Numerical results are obtained for the cases of diffusion, Merton and Variance Gamma processes. We provide convergence and time complexity analysis and comparisons with option prices computed using the standard martingale pricing theory.
- Option pricing in illiquid markets with jumpsPublication . Cruz, José Manuel Teixeira dos Santos; Sevcovic, Daniel; Grossinho, Maria do Rosário
- Essays in Spatial EconometricsPublication . Santos, Luís Filipe Ávila da Silveira dos; Proença, IsabelThis thesis addresses the specification and estimation of Spatial Lag Models for dichotomous or fractional responses. Three essays are presented. The first essay suggests a new method to approximate the inverse of the spatial lag operator, used in the estimation of Spatial Lag Models for binary outcomes. Related matrix operations are approximated, as well. Closed formulas for the elements of the approximated matrices are deduced. Computational time and complexity is greatly reduced. The second essay focus on the specification of Spatial Lag Models for fractional responses. Observations at the corners, zero and one, are allowed. Two specifications are proposed. The Fractional Response Spatial Lag Model (FRSLM), extends the seminal approach of Papke and Wooldridge (1996) to spatial frameworks. The approximate Fractional Response Spatial Lag Model (aFRSLM), allows to write the FRSLM as an approximate reduced form. Of particular relevance is the interpretation of policy effects. The third essay extends the second essay to the panel data setting. The individual unobserved effects are allowed to be correlated with the explanatory variables. The treatment of the unobserved heterogeneity is addressed as a central issue. Estimation is based on an iterative Generalized Method of Moments (iGMM) procedure, with well-known instruments. Inference is robust to spatial heteroskedasticity and spatial autocorrelation. The performance of the iGMM estimator is evaluated through detailed simulation studies. Results show that the iGMM estimator tends to perform well in terms of computational time, accuracy and precision. The adequacy of the proposed approaches is also assessed through empirical applications on the U.S. Metropolitan Statistical Areas. The first essay analyzes environmental effects over regional competitiveness and the degree of competitiveness spillovers. A new definition for binary competitiveness is introduced. Results show that competitiveness is significantly affected by air quality. Also, being competitive plays an important role in the competitiveness of neighboring areas. The third essay discusses regional knowledge and innovation spillovers, based on the proportion of high-tech patents. Results show that human capital plays a major role in regional innovative processes. However, due to regional aggregation, the degree of knowledge spillovers is significantly low.