The master thesis is focused on how a local volatility surfaces can be extracted by optimization with respectto smoothness and price error. The pricing is based on utility based pricing, and developed to be set in arisk neutral pricing setting. The pricing is done in a discrete multinomial recombining tree, where the timeand price increments optionally can be equidistant. An interpolation algorithm is used if the option that shallbe priced is not matched in the tree discretization. Power utility functions are utilized, where the log-utilitypreference is especially studied, which coincides with the (Kelly) portfolio that systematically outperforms anyother portfolio. A fine resolution of the discretization is generally a property that is sought after, thus a seriesof derivations for the implementation are done to restrict the computational encumbrance and thus allow finer discretization. The thesis is mainly focused on the derivation of the method rather than finding optimal parameters thatgenerate the local volatility surfaces. The method has shown that smooth surfaces can be extracted, whichconsider market prices. However, due to lacking available interest and dividend data, the pricing error increasessymmetrically for longer option maturities. However, the method shows exponential convergence and robustnessto different initial (flat) volatilities for the optimization initiation. Given an optimal smooth local volatility surface, an arbitrary payoff function can then be used to price thecorresponding option, which could be path-dependent, such as barrier options. However, only vanilla optionswill be considered in this thesis. Finally, we find that the developed

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Submitted by Cristiane Shirayama (cristiane.shirayama@fgv.br) on 2011-06-03T16:41:12Z No. of bitstreams: 1 66080100251.pdf: 1071566 bytes, checksum: 633248672cb6ac94f704bfeda06b29d3 (MD5) / Approved for entry into archive by Vera Lúcia Mourão(vera.mourao@fgv.br) on 2011-06-03T16:46:36Z (GMT) No. of bitstreams: 1 66080100251.pdf: 1071566 bytes, checksum: 633248672cb6ac94f704bfeda06b29d3 (MD5) / Approved for entry into archive by Vera Lúcia Mourão(vera.mourao@fgv.br) on 2011-06-03T17:00:17Z (GMT) No. of bitstreams: 1 66080100251.pdf: 1071566 bytes, checksum: 633248672cb6ac94f704bfeda06b29d3 (MD5) / Made available in DSpace on 2011-06-03T18:49:55Z (GMT). No. of bitstreams: 1 66080100251.pdf: 1071566 bytes, checksum: 633248672cb6ac94f704bfeda06b29d3 (MD5) Previous issue date: 2011-02-11 / Nos últimos anos, o mercado brasileiro de opções apresentou um forte crescimento, principalmente com o aparecimento da figura dos High Frequency Traders (HFT) em busca de oportunidades de arbitragem, de modo que a escolha adequada do modelo de estimação de volatilidade pode tornar-se um diferencial competitivo entre esses participantes. Este trabalho apresenta as vantagens da adoção do modelo de volatilidade estocástica de Heston (1993) na construção de superfície de volatilidade para o mercado brasileiro de opções de dólar, bem como a facilidade e o ganho computacional da utilização da técnica da Transformada Rápida de Fourier na resolução das equações diferenciais do modelo. Além disso, a partir da calibração dos parâmetros do modelo com os dados de mercado, consegue-se trazer a propriedade de não-arbitragem para a superfície de volatilidade. Os resultados, portanto, são positivos e motivam estudos futuros sobre o tema. / In recent years, the Brazilian option market has grown considerable, especially with the emergence of the High Frequency Traders (HFT) in search of arbitrage opportunities, so that the appropriate choice of a volatility estimation model should become a competitive differentiator among these participants. This paper presents the advantages of adopting the Heston stochastic volatility model on the construction of the volatility surface for the Brazilian US Dollar option market, as well as the easiness and the computational gain by applying the Fast Fourier Transform technique on the models differential equations resolution. Furthermore, from calibration of the model parameters to market data, it is possible to bring the no-arbitrage property to the volatility surface. The results, therefore, are positive and motivate further studies on the subject.

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).