Como a volatilidade é a única variável não observada nas fórmulas padrão de apreçamento de opções, o mercado financeiro utiliza amplamente o conceito de volatilidade implícita, isto é, a volatilidade que ao ser aplicada na fórmula de apreçamento resulte no preço correto (observado) das opções negociadas. Por isso, entender como as volatilidades implícitas das diversas opções de dólar negociadas na BM&F, o objeto de nosso estudo, variam ao longo do tempo e como estas se relacionam é importante para a análise de risco de carteiras de opções de dólar/real bem como para o apreçamento de derivativos cambiais exóticos ou pouco líquidos. A proposta de nosso estudo é, portanto, verificar se as observações da literatura técnica em diversos mercados também são válidas para as opções de dólar negociadas na BM&F: que as volatilidades implícitas não são constantes e que há uma relação entre as variações das volatilidades implícitas e as variações do valor do ativo objeto. Para alcançar este objetivo, aplicaremos a análise de componentes principais em nosso estudo. Com esta metodologia, reduziremos as variáveis aleatórias que representam o processo das volatilidades implícitas em um número menor de variáveis ortogonais, facilitando a análise dos dados obtidos. / Volatility is the only unobserved variable in the standard option pricing formulas and hence implied volatility is a concept widely adopted by the financial market, meaning the volatility which would make the formula yield the options real market price. Therefore, understanding how the implied volatility of the options on dollar traded at BM&F, the subject of our study, vary over time is important for risk analysis over dollar option books and for pricing of exotic or illiquid derivatives Our works proposal is to verify if the observations made by the technical literature over several markets could also be applied to the options on dollar traded at BM&F: implied volatilities do vary over time and there is a relation between this variation and the variation of the underlying asset price. In order to fulfill these goals, we will apply principal component analysis in our study. This methodology will help us analyze the data by reducing the number of variables that represent the implied volatility process into a few orthogonal variables.

Como a volatilidade é a única variável não observada nas fórmulas padrão de apreçamento de opções, o mercado financeiro utiliza amplamente o conceito de volatilidade implícita, isto é, a volatilidade que ao ser aplicada na fórmula de apreçamento resulte no preço correto (observado) das opções negociadas. Por isso, entender como as volatilidades implícitas das diversas opções de dólar negociadas na BM&F, o objeto de nosso estudo, variam ao longo do tempo e como estas se relacionam é importante para a análise de risco de carteiras de opções de dólar/real bem como para o apreçamento de derivativos cambiais exóticos ou pouco líquidos. A proposta de nosso estudo é, portanto, verificar se as observações da literatura técnica em diversos mercados também são válidas para as opções de dólar negociadas na BM&F: que as volatilidades implícitas não são constantes e que há uma relação entre as variações das volatilidades implícitas e as variações do valor do ativo objeto. Para alcançar este objetivo, aplicaremos a análise de componentes principais em nosso estudo. Com esta metodologia, reduziremos as variáveis aleatórias que representam o processo das volatilidades implícitas em um número menor de variáveis ortogonais, facilitando a análise dos dados obtidos. / Volatility is the only unobserved variable in the standard option pricing formulas and hence implied volatility is a concept widely adopted by the financial market, meaning the volatility which would make the formula yield the options real market price. Therefore, understanding how the implied volatility of the options on dollar traded at BM&F, the subject of our study, vary over time is important for risk analysis over dollar option books and for pricing of exotic or illiquid derivatives Our works proposal is to verify if the observations made by the technical literature over several markets could also be applied to the options on dollar traded at BM&F: implied volatilities do vary over time and there is a relation between this variation and the variation of the underlying asset price. In order to fulfill these goals, we will apply principal component analysis in our study. This methodology will help us analyze the data by reducing the number of variables that represent the implied volatility process into a few orthogonal variables.

In this thesis, we propose a new and simple approach of extending the single-factor Heston stochastic volatility model to a more flexible one in solving option pricing problems.  In this approach, the volatility process for the underlying asset dynamics depends on the time to maturity of the option. As this idea is inspired by the Heath-Jarrow-Morton framework which models the evolution of the full dynamics of forward rate curves for various maturities, we name this approach as the HJM-type stochastic volatility (HJM-SV)  model. We conduct an empirical analysis by calibrating this model to real-market option data for underlying assets including an equity  (ABB stock) and a market index (EURO STOXX 50), for two separated time spans from Jan 2017 to Dec 2017 (before the COVID-19 pandemic) and from Nov 2019 to Nov 2020 (after the start of COVID-19 pandemic). We investigate the optimal way of dividing the set of option maturities into three classes, namely, the short-maturity, middle-maturity, and long-maturity classes. We calibrate our HJM-SV model to the data in the following way, for each class a single-factor Heston stochastic volatility model is calibrated to the corresponding market data. We address the question that how well the new HJM-SV model captures the feature of implied volatility surface given by the market data.

[pt] O presente trabalho busca explorar a previsibilidade na dinâmica temporal em modelos lineares de superfícies de volatilidade implícita estimados para opções de compra de ações brasileiras. Resultados de estudos anteriores, sob a abordagem usualmente empregada de estimação de modelos lineares em função do preço de exercício e do tempo até o vencimento a partir de dados de corte transversal sobre cada contrato disponível em dado instante, como Dumas, Fleming e Whaley (1998), revelam grande instabilidade nos coeficientes estimados ao longo do tempo. Por conseguinte, a incapacidade desta perspectiva em descrever a dinâmica intertemporal da estrutura, contrariando a observação empírica de volatilidade variável no tempo. A partir destas evidências e das conclusões de Heston e Nandi (2000), que reportaram significativa dependência da trajetória para a volatilidade dos retornos do índice S&P 500, Gonçalves e Guidolim (2006), propuseram um modelo em dois estágios, que aplica vetores autoregressivos para capturar a presença de variação temporal dos coeficientes de um modelo linear. A contribuição deste trabalho está em aplicar o Modelo proposto à realidade do mercado brasileiro de opções de ações, incipiente em liquidez e horizonte de negociação se comparado ao mercado norte americano, adaptando critérios a fim de validar sua aplicabilidade neste contexto em termos estatísticos e econômicos. Os resultados comprovam a superioridade desta abordagem em relação a outras comparáveis na literatura, mas não a capacidade de gerar retornos acima da média na presença de custos de transação contra a referência natural da taxa livre de risco. O que sugere a adequação à hipótese de eficiência de mercado. / [en] O The present study aims to explore predictability in temporal dynamics regarding linear models of the implied volatility surfaces estimated for Brazilian stocks options. Previous results, by usual approach of fitting linear models linking implied volatility to time to maturity and moneyness, available for each cross-section of option contracts at a point in time, as in Dumas, Fleming and Whaley (1998), suggest that estimated parameters of such models are highly unstable over time. Therefore, this approach isn t capable of replicating various IVS s shapes, contrary to the empirical evidence of implied volatility varying with options strike price and date of expiration. Based on these evidences and in Heston and Nandi (2000), that exploit the information on path-dependency in volatility contained in the spot S&P 500 index, Gonçalves e Guidolim (2006) proposed a two-stage approach to modeling and forecasting the S&P 500 index options IVS. In the second-stage they model the dynamics of the cross-sectional first-stage coefficients by means of vector autoregression models. The contribution of this work is to apply the proposed model to the reality of the Brazilian stock options, incipient in terms of liquidity and trading horizon dimensions when compared to the U.S. market, adapting criterians to validate its applicability in this context in statistical and economical sense. The results demonstrate the superiority of this approach over comparable literature, but not the ability to generate abnormal profits in the presence of transaction costs in excess of the benchmark of the risk-free rate. This indicates adaptation to the market efficiency hypothesis.

Seit der Entdeckung der arbitragefreien Bewertung hat sich das Gebiet finance grundlegend geändert - sowohl in der Theorie als auch in der Anwendung. Märkte für Derivate haben sich entwickelt und Optionen dienen heutzutage als Basis- und als Absicherungsinstrumente. In dieser Dissertation betrachten wir einige Märkte für Aktienderivate. Wir beginnen mit statistischen Analysen des Marktes für europäische Optionen und des Marktes für Varianzswaps, weil diese Produkte die hauptsächlichen Absicherungsinstrumente für komplexe Optionen sind. Dann betrachten wir verschiedene Optionspreismodelle und ihre Kalibrierung an beobachtete Preisoberflächen. Schließlich untersuchen wir die Verbindung zwischen Optionspreisen und dem grundlegenden ökonomischen Konzept der Risikoaversion anhand des empirischen Preiskernes. / Since the ideas of arbitrage free pricing were born, finance has changed radically - both in theory and practice. Derivatives markets have evolved and options serve nowadays as underlyings and as hedging instruments. In this thesis, we consider some markets for equity derivatives. We start by statistical analysis of the markets for European options and variance swaps because these products are important for hedging more complex claims. Then we consider different option pricing models and their calibration to observed price surfaces. Finally, we investigate the connection between option prices and the fundamental economic concept of risk aversion by the empirical pricing kernel.

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Synthetic short positions constructed by equity options and stock loan short sells are linked by arbitrage. This thesis analyses the link by considering the implied volatility surface (IVS) at 80%, 100%, and 120% moneyness, and stock loan variables such as benchmark rate (rt), utilization, short interest, and transaction trends to inspect time-dependent structures between the two assets. By applying multiple multivariate time-series analyses in terms of vector autoregression (VAR) and the recurrent neural networks long short-term memory (LSTM) and gated recurrent units (GRU) with a sliding window methodology. This thesis discovers linear and complex relationships between the IVS and stock loan data. The three-day-ahead out-of-sample LSTM forecast of IV at 80% moneyness improved by including lagged values of rt and yielded 19.6% MAPE and forecasted correct direction 81.1% of samples. The corresponding 100% moneyness GRU forecast was also improved by including stock loan data, at 10.8% MAPE and correct directions for 60.0% of samples. The 120% moneyness VAR forecast did not improve with stock loan data at 26.5% MAPE and correct directions for 66.2% samples. The one-month-ahead rt VAR forecast improved by including a lagged IVS, at 25.5% MAPE and 63.6% correct directions. The presented data was optimal for each target variable, showing that the application of LSTM and GRU was justified. These results indicate that considering stock loan data when forecasting IVS for 80% and 100% moneyness is advised to gain exploitable insights for short-term positions. They are further validated since the different models yielded parallel inferences. Similar analysis with other equity is advised to gain insights into the relationship and improve such forecasts. / Syntetiska kortpositioner konstruerade av aktieoptioner och blankning med aktielån är kopplade med arbitrage. Denna tes analyserar kopplingen genom att överväga den implicerade volatilitetsytan vid 80%, 100% och 120% moneyness och aktielånvariabler såsom referensränta rt, låneutnyttjande, låneintresse, och transaktionstrender för att granska tidsberoende strukturer mellan de två tillgångarna. Genom att tillämpa multipel multidimensionell tidsserieanalys såsom vektorautoregression (VAR) och de rekursiva neurala nätverken long short-term memory (LSTM) och gated recurrent units (GRU). Tesen upptäcker linjära och komplexa samband mellan implicerade volatilitetsytor och aktielånedata. Tre dagars LSTM-prognos av implicerade volatiliteten vid 80% moneyness förbättrades genom att inkludera fördröjda värden av rt och gav 19,6% MAPE och prognostiserade korrekt riktning för 81,1% av prover. Motsvarande 100% moneyness GRU-prognos förbättrades också genom att inkludera aktielånedata, resulterande i 10,8% MAPE och korrekt riktning för 60,0% av prover. VAR-prognosen för 120% moneyness förbättrades inte med alternativa data på 26,5% MAPE och korrekt riktning för 66,2% av prover. En månads VAR-prognos för rt förbättrades genom att inkludera en fördröjd implicerad volatilitetsyta, resulterande i 25,5% MAPE och 63,6% korrekta riktningar. Presenterad statistik var optimala för dessa variabler, vilket visar att tillämpningen av LSTM och GRU var motiverad. Därav rekommenderas det att inkludera aktielånedata för prognostisering av implicerade volatilitetsytor för 80% och 100% moneyness, speciellt för kortsiktiga positioner. Resultaten valideras ytterligare eftersom de olika modellerna gav dylika slutsatser. Liknande analys med andra aktier är rekommenderat för att få insikter i förhållandet och förbättra sådana prognoser.

The implied volatility surface plays an important role for Front office and Risk Management functions at Nasdaq and other financial institutions which require mark-to-market of derivative books intraday in order to properly value their instruments and measure risk in trading activities. Based on the aforementioned business needs, being able to calibrate an end of day implied volatility surface based on new market information is a sought after trait. In this thesis a statistical learning approach is used to calibrate the implied volatility surface intraday. This is done by using OMXS30-2019 implied volatility surface data in combination with market information from close to at the money options and feeding it into 3 Machine Learning models. The models, including Feed Forward Neural Network, Recurrent Neural Network and Gaussian Process, were compared based on optimal input and data preprocessing steps. When comparing the best Machine Learning model to the benchmark the performance was similar, indicating that the calibration approach did not offer much improvement. However the calibrated models had a slightly lower spread and average error compared to the benchmark indicating that there is potential of using Machine Learning to calibrate the implied volatility surface. / Implicita volatilitetsytor är ett viktigt vektyg för front office- och riskhanteringsfunktioner hos Nasdaq och andra finansiella institut som behöver omvärdera deras portföljer bestående av derivat under dagen men också för att mäta risk i handeln. Baserat på ovannämnda affärsbehov är det eftertraktat att kunna kalibrera de implicita volatilitets ytorna som skapas i slutet av dagen nästkommande dag baserat på ny marknadsinformation. I denna uppsats används statistisk inlärning för att kalibrera dessa ytor. Detta görs genom att uttnytja historiska ytor från optioner i OMXS30 under 2019 i kombination med optioner nära at the money för att träna 3 Maskininlärnings modeller. Modellerna inkluderar Feed Forward Neural Network, Recurrent Neural Network och Gaussian Process som vidare jämfördes baserat på data som var bearbetat på olika sätt. Den bästa Maskinlärnings modellen jämfördes med ett basvärde som bestod av att använda föregående dags yta där resultatet inte innebar någon större förbättring. Samtidigt hade modellen en lägre spridning samt genomsnittligt fel i jämförelse med basvärdet som indikerar att det finns potential att använda Maskininlärning för att kalibrera dessa ytor.

In vielen Anwendungen ist es notwendig, die stochastische Schwankungen der maximalen Abweichungen der nichtparametrischen Schätzer von Quantil zu wissen, zB um die verschiedene parametrische Modelle zu überprüfen. Einheitliche Konfidenzbänder sind daher für nichtparametrische Quantil Schätzungen der Regressionsfunktionen gebaut. Die erste Methode basiert auf der starken Approximation der empirischen Verfahren und Extremwert-Theorie. Die starke gleichmäßige Konsistenz liegt auch unter allgemeinen Bedingungen etabliert. Die zweite Methode beruht auf der Bootstrap Resampling-Verfahren. Es ist bewiesen, dass die Bootstrap-Approximation eine wesentliche Verbesserung ergibt. Der Fall von mehrdimensionalen und diskrete Regressorvariablen wird mit Hilfe einer partiellen linearen Modell behandelt. Das Verfahren wird mithilfe der Arbeitsmarktanalysebeispiel erklärt. Hoch-dimensionale Zeitreihen, die nichtstationäre und eventuell periodische Verhalten zeigen, sind häufig in vielen Bereichen der Wissenschaft, zB Makroökonomie, Meteorologie, Medizin und Financial Engineering, getroffen. Der typische Modelierungsansatz ist die Modellierung von hochdimensionalen Zeitreihen in Zeit Ausbreitung der niedrig dimensionalen Zeitreihen und hoch-dimensionale zeitinvarianten Funktionen über dynamische Faktorenanalyse zu teilen. Wir schlagen ein zweistufiges Schätzverfahren. Im ersten Schritt entfernen wir den Langzeittrend der Zeitreihen durch Einbeziehung Zeitbasis von der Gruppe Lasso-Technik und wählen den Raumbasis mithilfe der funktionalen Hauptkomponentenanalyse aus. Wir zeigen die Eigenschaften dieser Schätzer unter den abhängigen Szenario. Im zweiten Schritt erhalten wir den trendbereinigten niedrig-dimensionalen stochastischen Prozess (stationär). / In many applications it is necessary to know the stochastic fluctuation of the maximal deviations of the nonparametric quantile estimates, e.g. for various parametric models check. Uniform confidence bands are therefore constructed for nonparametric quantile estimates of regression functions. The first method is based on the strong approximations of the empirical process and extreme value theory. The strong uniform consistency rate is also established under general conditions. The second method is based on the bootstrap resampling method. It is proved that the bootstrap approximation provides a substantial improvement. The case of multidimensional and discrete regressor variables is dealt with using a partial linear model. A labor market analysis is provided to illustrate the method. High dimensional time series which reveal nonstationary and possibly periodic behavior occur frequently in many fields of science, e.g. macroeconomics, meteorology, medicine and financial engineering. One of the common approach is to separate the modeling of high dimensional time series to time propagation of low dimensional time series and high dimensional time invariant functions via dynamic factor analysis. We propose a two-step estimation procedure. At the first step, we detrend the time series by incorporating time basis selected by the group Lasso-type technique and choose the space basis based on smoothed functional principal component analysis. We show properties of this estimator under the dependent scenario. At the second step, we obtain the detrended low dimensional stochastic process (stationary).