Neste trabalho propomos um modelo de mercado através de uma discretização aleatória do movimento browniano proposta por Leão & Ohashi (2010). Com este modelo, dada uma função payoff, vamos desenvolver uma estratégia de hedging e uma metodologia para precificação de opções / In this work we propose a market model using a discretization scheme of the random Brownian motion proposed by Leão & Ohashi (2010). With this model, for any given payoff function, we develop a hedging strategy and a methodology to option pricing

Made available in DSpace on 2016-04-25T18:40:44Z (GMT). No. of bitstreams: 1 Carl Douglas De Gennaro Oliveira.pdf: 1342470 bytes, checksum: 9868002de42872f20913eb856aa2b173 (MD5) Previous issue date: 2010-05-24 / The process of Brazil s compliance with the International Financial Reporting Standard (IFRS) took a big step forward, definitively getting on the agenda of regulatory agencies, companies and auditing firms, when Federal Law 11.638 was signed in December 2007, altering the accounting chapter of Brazilian Corporate Law, 6.404/76. This study contributes to Brazil s process of compliance with the IFRS, specifically regarding the applicability of IFRS 2 Share-based Payment, or its Brazilian corollary CPC 10 Pagamento Baseado em Ações, and the impact on accounting and on the disclosure of a long-term compensation program for executives, characterized as phantom stock options. IFRS 2 was published in February 2002 and was required internationally from January 2005, as an outcome of the growing use of commercial transaction payments based on shares, and also the IOSCO´s report that pointed out the lack of an accounting standard dealing with this kind of transaction. The study found that IFRS 2 or CPC 10 can be appropriately applied to guide the accounting treatment given to a phantom stock option program, and was a more informative accounting practice than that which had been used in Brazil, before 2008. The study also found a wide-spread need of financial knowledge regarding the valuation of stock options, such as the Black-Scholes-Merton model, as well as statistical methods for appropriately account and disclose the fair value of share-based incentive plans. Furthermore, in order to understand more fully the economic event which is being accounted, it is highly important to understand its essence. In the case of long-term share-based incentives for executives, the essence of their existence can be found in agency theory / O processo de convergência do Brasil às Normas Internacionais de Contabilidade (IFRSs) deu um grande salto e entrou definitivamente na agenda dos órgãos reguladores, empresas e auditorias, com a sanção da lei federal 11.638 em dezembro de 2007, que alterou o capítulo contábil da Lei das Sociedades Anônimas, 6.404/76. Este estudo contribui para o processo de convergência brasileiro às IFRSs, especificamente quanto à aplicabilidade do IFRS 2 Share Based Payment, ou sua correlação brasileira CPC 10 Pagamento Baseado em Ações, e dos impactos contábeis e de divulgação decorrentes de um programa de compensação de longo prazo a executivos, com as características de phantom stock options, ou opções fantasmas. O IFRS 2 foi publicado em fevereiro de 2002 e requerido internacionalmente a partir de janeiro de 2005, como uma decorrência do crescente uso de pagamento das transações comerciais com base em ações e também do relatório da IOSCO, que identificou como falha a lacuna de norma contábil que tratasse deste tipo de transação. O estudo identificou que o IFRS 2 ou CPC 10 aplica-se adequadamente para orientar o tratamento contábil de um programa de phantom stock option e representou uma prática contábil mais informativa que aquela até então adotada no Brasil, antes do ano de 2008. O estudo também identificou a grande necessidade de conhecimento de finanças relacionado à avaliação de opções, tal como o modelo Black-Scholes-Merton, bem como de métodos estatísticos, para uma apropriada contabilização e divulgação do valor justo dos planos de incentivo baseados em ações. Além disso, para que se entenda com profundidade o evento econômico que se contabiliza, é de suma importância a compreensão de sua essência. No caso de incentivos de longo prazo para executivos, baseados em ações, a essência de sua existência pode ser encontrada na Teoria de Agência

Η διατριβή χωρίζεται σε δύο μέρη. Στο Μέρος Α' χρησιμοποιούνται μαθηματικές μέθοδοι της Θεωρίας Παιγνίων και των Δυναμικών Συστημάτων για να μελετηθεί η κανονική και χαοτική δυναμική διαφόρων μοντέλων της Μικροοικονομίας. Βασικά αποτελέσματα είναι η μετάβαση σε συνθήκες πλήρους ανταγωνισμού και η διαφοροποίηση του παραγόμενου προιόντος σε ένα δυοπώλιο-τριοπώλιο. Στο Μέρος Β', κύριος στόχος της έρευνας ήταν να συνδεθούν ορισμένες από τις πλέον γνωστές μερικές διαφορικές εξισώσεις (ΜΔΕ) που χρησιμοποιούνται στα Οικονομικά Μαθηματικά και Χρηματοοικονομικά, με την εξίσωση της θερμότητας της Μαθηματικής Φυσικής, εφαρμόζοντας την κατά Lie συμμετρίες ανάλυση. Επίσης η ανάλυση αυτή αποδείχθηκε ιδιαίτερα ισχυρή για την εύρεση αλγεβρικών δομών εξισώσεων που περιγράφουν την τιμολόγηση αγαθών. Έτσι, οδηγούμαστε με συστηματικό τρόπο όχι μόνο στην εύρεση νέων λύσεων αλλά και στην ανακάλυψη κομψών γενικεύσεων των εξισώσεων αυτών. / The thesis is divided into two parts. In Part One we use the mathematical methods of Game Theory and Dynamical Systems to study the stable and chaotic dynamics of various models in Microeconomics. Some of our main results are the route to perfect competition and the differentiation of goods in a duopoly and in a triopoly. In Part Two, our main concern was to link some of the most well-known partial differential equations that are encountered in Economics and Financial Mathematics, with the heat equation of Mathematical Physics, using Lie symmetry analysis. More to that, this analysis proved extremely powerful to the finding of interesting algebraic properties for equations that describe the pricing of commodities. In such way, we succeed in presenting, in a systematic fashion, not only new solutions, but also elegant generalisations of the equations under investigation.

Neste trabalho propomos um modelo de mercado através de uma discretização aleatória do movimento browniano proposta por Leão & Ohashi (2010). Com este modelo, dada uma função payoff, vamos desenvolver uma estratégia de hedging e uma metodologia para precificação de opções / In this work we propose a market model using a discretization scheme of the random Brownian motion proposed by Leão & Ohashi (2010). With this model, for any given payoff function, we develop a hedging strategy and a methodology to option pricing

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.

The thesis presents a new method of Symmetry Analysis of the Black-Scholes Merton Finance Model through modi ed Local one-parameter transformations. We determine the symmetries of both the one-dimensional and two-dimensional Black-Scholes equations through a method that involves the limit of in nitesimal ! as it approaches zero. The method is dealt with extensively in [23]. We further determine an invariant solution using one of the symmetries in each case. We determine the transformation of the Black-Scholes equation to heat equation through Lie equivalence transformations. Further applications where the method is successfully applied include working out symmetries of both a Gaussian type partial di erential equation and that of a di erential equation model of epidemiology of HIV and AIDS. We use the new method to determine the symmetries and calculate invariant solutions for operators providing them. / Mathematical Sciences / Applied Mathematics / D. Phil. (Applied Mathematics)

Thesis (MSc)--Stellenbosch University, 2015 / AFRIKAANSE OPSOMMING : Die groeiende belangstelling in kalibrering en modelrisiko is ’n redelik resente ontwikkeling in finansiële wiskunde. Hierdie proefskrif fokusseer op hierdie sake, veral in verband met die prysbepaling van vanielje-en eksotiese opsies, en vergelyk die prestasie van verskeie Lévy modelle. ’n Nuwe metode om modelrisiko te meet word ook voorgestel (hoofstuk 6). Ons kalibreer eers verskeie Lévy modelle aan die log-opbrengs van die S&P500 indeks. Statistiese toetse en grafieke voorstellings toon albei aan dat suiwer sprongmodelle (VG, NIG en CGMY) die verdeling van die opbrengs beter beskryf as die Black-Scholes model. Daarna kalibreer ons hierdie vier modelle aan S&P500 indeks opsie data en ook aan "CGMY-wˆ ereld" data (’n gesimuleerde wÃłreld wat beskryf word deur die CGMY-model) met behulp van die wortel van gemiddelde kwadraat fout. Die CGMY model vaar beter as die VG, NIG en Black-Scholes modelle. Ons waarneem ook ’n effense verskil tussen die nuwe parameters van CGMY model en sy wisselende parameters, ten spyte van die feit dat CGMY model gekalibreer is aan die "CGMYwêreld" data. Versperrings-en terugblik opsies word daarna geprys, deur gebruik te maak van die gekalibreerde parameters vir ons modelle. Hierdie pryse word dan vergelyk met die "ware" pryse (bereken met die ware parameters van die "CGMY-wêreld), en ’n beduidende verskil tussen die modelpryse en die "ware" pryse word waargeneem. Ons eindig met ’n poging om hierdie modelrisiko te kwantiseer / ENGLISH ABSTRACT : The growing interest in calibration and model risk is a fairly recent development in financial mathematics. This thesis focussing on these issues, particularly in relation to the pricing of vanilla and exotic options, and compare the performance of various Lévy models. A new method to measure model risk is also proposed (Chapter 6). We calibrate only several Lévy models to the log-return of S&P500 index data. Statistical tests and graphs representations both show that pure jump models (VG, NIG and CGMY) the distribution of the proceeds better described as the Black-Scholes model. Then we calibrate these four models to the S&P500 index option data and also to "CGMY-world" data (a simulated world described by the CGMY model) using the root mean square error. Which CGMY model outperform VG, NIG and Black-Scholes models. We observe also a slight difference between the new parameters of CGMY model and its varying parameters, despite the fact that CGMY model is calibrated to the "CGMY-world" data. Barriers and lookback options are then priced, making use of the calibrated parameters for our models. These prices are then compared with the "real" prices (calculated with the true parameters of the "CGMY world), and a significant difference between the model prices and the "real" rates are observed. We end with an attempt to quantization this model risk.

The thesis presents a new method of Symmetry Analysis of the Black-Scholes Merton Finance Model through modi ed Local one-parameter transformations. We determine the symmetries of both the one-dimensional and two-dimensional Black-Scholes equations through a method that involves the limit of in nitesimal ! as it approaches zero. The method is dealt with extensively in [23]. We further determine an invariant solution using one of the symmetries in each case. We determine the transformation of the Black-Scholes equation to heat equation through Lie equivalence transformations. Further applications where the method is successfully applied include working out symmetries of both a Gaussian type partial di erential equation and that of a di erential equation model of epidemiology of HIV and AIDS. We use the new method to determine the symmetries and calculate invariant solutions for operators providing them. / Mathematical Sciences / Applied Mathematics / D. Phil. (Applied Mathematics)

Esta dissertação revisa a literatura acadêmica existente sobre a teoria de opções utilizando os modelos de precificação com saltos. Os conceitos foram equalizados, a nomenclatura foi padronizada, sendo gerado um material de referência sobre o assunto. O pressuposto de lognormalidade com volatilidade constante não é aceito pelo mercado financeiro. É freqüente, no meio acadêmico, a busca de modelos que reproduzam os fenômenos observados de leptocurtose ou assimetria dos log-retornos financeiros e que possuam a mesma robustez e facilidade para manipulação analítica do consagrado modelo de Black-Scholes. Os modelos com saltos são uma alternativa para esse problema. Avaliou-se o modelo de Kou no mercado acionário brasileiro composto por um componente de difusão que segue um movimento browniano geométrico e um componente de saltos que segue um processo de Poisson com intensidade do salto descrito por uma distribuição duplamente exponencial. A simulação histórica do modelo aponta, em geral, uma superioridade preditiva do modelo, porém as dificuldades de calibração dos parâmetros e de hedge em mercados incompletos são as principais deficiências para o uso dos modelos com saltos. / This master dissertation reviews the academic literature about option pricing and hedging with jumps. The theory was equalized and the notation was standardized, becoming this document a reference document about this subject. The log-normality with constant volatility is not accepted by the market. Academics search consistent models with the same analytical capabilities like Black-Scholes? model which can support the observed leptokurtosis or asymmetry of the financial daily log-returns behavior. The jump models are an alternative to these issues. The Kou?s model was evaluated and this one consists of two parts: the first part being continuous and following a geometric Brownian motion and the second being a jump process with its jump intensity defined by a double exponential distribution. The model backtesting showed a better predictive performance of the Kou´s model against other models. However, there are some handicaps regarding to the parameters calibration and hedging.

Made available in DSpace on 2016-04-28T20:19:21Z (GMT). No. of bitstreams: 1 2003 - Tereza de Jesus Ramos da Silva.pdf: 1614568 bytes, checksum: 0272fd51b3a45e19c0bbf132f65427e4 (MD5) Previous issue date: 2003-03-13 / Options in future markets is a theme still with little exploration by the studious, concerned to practical work published, mainly in Brazilian Literature. In this work it has been tried to show the importance of volatility in pricing of options when applied to Black & Scholes Model. A first analysis was taken, a study of derivatives, defined as titles which values depend on other more basic variables, where the four more used types were concieved: forward contracts, swap contracts, the future and the options ones. A future contract consists in the obrigation of buying and selling an active in certain future time for a certain price. There are two types of options: call and put. A call gives its holders the right to buy an active in a settled date for a certain price and one put, the right to sell. We ve emphasized the options markets, defining a future option as an option about a future contract. The partners of these markets are the hedgers, the speculators and the arbitrators. The options have been defined inside the money, on money and out of money, which shows as differential an intrinsic value definition. The basic factors that affect the price of options about futures are, besides the volatility of future price, the future price itself, the price of exercise and the time for due date. The volatility is a very important factor, because it measures the variation of future prices with the time. It has been presented an equation of Black to the use in pricing of future contracts and it has been used the Arabic Coffee as the farming commodity object of our investigation. The calculus of premium, defined as price paid by the buyer of the options contract, has been calculated according to the results found in the application of the three methods of volatility calculus: the historic volatility which is the one that uses historic series of prices; the volatility by moving average which is the one found through an aritimetic average in a certain period of time (n days of negotiation and the implicit volatility that is an estimation of volatility which is on the level with market price. Chapter 3 has analysed all the calculus process, through treatment of data in annexes 04 to 12, describing all the mechanism used in preparing these spread sheets. From the results found, it has been concluded that the best way to be used in option pricing in Black & Scholes Model is the one of implicit volatility. / Op??es em mercados futuros ? um tema ainda pouco explorado pelos estudiosos, no que concerne a publica??o de trabalhos pr?ticos, principalmente na literatura brasileira. Neste trabalho procurou-se mostrar a import?ncia da volatilidade dentro da precifica??o de op??es quando aplicada ao modelo de Black & Scholes. Fez-se em uma primeira an?lise, um estudo dos derivativos, definidos como t?tulos cujos valores dependem de outras vari?veis mais b?sicas, onde foram conceituados os quatro tipos mais utilizados: os contratos a termo, os contratos de swaps, os futuros e os de op??es. Um contrato futuro consiste na obriga??o de comprar ou vender um ativo em certa ?poca futura por determinado pre?o. H? dois tipos de op??es: call e put. Uma call d? ao seu detentor o direito de comprar um ativo em determinada data por pre?o certo e uma put, o direito de vender. Enfatizamos os mercados de op??es, definindo uma op??o futura como uma op??o sobre um contrato futuro. Os participantes desses mercados s?o os hedgers, os especuladores e os arbitradores. Foram definidas as op??es dentro do dinheiro, no dinheiro e fora do dinheiro, que apresenta como diferencial a defini??o de valor intr?nseco. Os fatores b?sicos que afetam o pre?o das op??es sobre futuros s?o, al?m da volatilidade do pre?o futuro, o pr?prio pre?o futuro, o pre?o de exerc?cio e o tempo para o vencimento. A volatilidade ? um fator muito importante, pois ela mede a oscila??o dos pre?os futuros com o tempo. Foi apresentada a equa??o de Black quando para a utiliza??o na precifica??o de contratos futuros e utilizou-se o caf? ar?bica como a commodity agropecu?ria objeto da nossa investiga??o. O c?lculo do pr?mio, definido como o pre?o pago pelo comprador do contrato de op??es, foi calculado segundo os resultados encontrados quando da aplica??o dos tr?s m?todos de c?lculo da volatilidade: a volatilidade hist?rica que ? a que se utiliza de s?ries hist?ricas dos pre?os; a volatilidade por m?dia m?vel que ? aquela encontrada atrav?s de uma m?dia aritm?tica no lapso de um per?odo certo(n dias de negocia??o e a volatilidade impl?cita que ? uma estimativa de volatilidade que se iguala ao pre?o de mercado. O cap?tulo 3 analisou todo o processo de c?lculo, atrav?s do tratamento dos dados constantes dos anexos 04 a 12, descrevendo todo o mecanismo utilizado na confec??o dessas planilhas. A partir dos resultados encontrados, concluiu-se que o melhor m?todo a ser utilizado na precifica??o de op??es no modelo de Black & Scholes ? o da volatilidade impl?cita.