Master of Science / Department of Statistics / James Neill / In financial mathematics, asset prices for European options are often modeled according to the Black-Scholes-Merton (BSM) model, a stochastic differential equation (SDE) depending on unknown parameters. A derivation of the solution to this SDE is reviewed, resulting in a stochastic process called geometric Brownian motion (GBM) which depends on two unknown real parameters referred to as the drift and volatility. For additional insight, the BSM equation is expressed as a heat equation, which is a partial differential equation (PDE) with well-known properties. For American options, it is established that asset value can be characterized as the solution to an obstacle problem, which is an example of a free boundary PDE problem. One approach for estimating the parameters in the GBM solution to the BSM model can be based on the method of maximum likelihood. This approach is discussed and applied to a dataset involving the weekly closing prices for the Dow Jones Industrial Average between January 2012 and December 2012.

The objective of this thesis is to review the two popular mathematical models of the financialderivatives market. The models are the classical Black–Scholes–Merton and the Continuoustime Markov chain (CTMC) model. We study the CTMC model which is illustrated by themathematician Ragnar Norberg. The thesis demonstrates how the fundamental results ofFinancial Engineering work in both models.The construction of the main financial market components and the approach used for pricingthe contingent claims were considered in order to review the two models. In addition, the stepsused in solving the first–order partial differential equations in both models are explained.The main similarity between the models are that the financial market components are thesame. Their contingent claim is similar and the driving processes for both models utilizeMarkov property.One of the differences observed is that the driving process in the BSM model is the Brownianmotion and Markov chain in the CTMC model.We believe that the thesis can motivate other students and researchers to do a deeper andadvanced comparative study between the two models.

The fast Fourier transform (FFT), even though it has been widely applicable in Physics and Engineering, it has become attractive in Finance as well for it’s enhancement of computational speed. Carr and Madan succeeded in implementing the FFT for pricing of an option. This project, inspired by Carr and Madan’s paper, attempts to elaborate and connect the various mathematical and theoretical concepts that are helpful in understanding of the derivation. Further, we derive the characteristic function of the risk neutral probability for the logarithmic terminal stock price. The Black-Scholes-Merton (BSM) model is also revised including derivation of the partial deferential equation and the formula. Finally, comparison of the BSM numerical implementation with and without the FFT method is done using MATLAB.

The purpose of this thesis is to compare option pricing models. We have investigated the constant volatility models Black-Scholes-Merton (BSM) and Merton’s Jump Diffusion (MJD) as well as the stochastic volatility models Heston and Bates. The data used were option prices from Microsoft, Advanced Micro Devices Inc, Walt Disney Company, and the S&P 500 index. The data was then divided into training and testing sets, where the training data was used for parameter calibration for each model, and the testing data was used for testing the model prices against prices observed on the market. Calibration of the parameters for each model were carried out using the nonlinear least-squares method. By using the calibrated parameters the price was calculated using the method of Carr and Madan. Generally it was found that the stochastic volatility models, Heston and Bates, replicated the market option prices better than both the constant volatility models, MJD and BSM for most data sets. The mean average relative percentage error for Heston and Bates was found to be 2.26% and 2.17%, respectively. Merton and BSM had a mean average relative percentage error of 6.90% and 5.45%, respectively. We therefore suggest that a stochastic volatility model is to be preferred over a constant volatility model for pricing options. / Syftet med denna tes är att jämföra prissättningsmodeller för optioner. Vi har undersökt de konstanta volatilitetsmodellerna Black-Scholes-Merton (BSM) och Merton’s Jump Diffusion (MJD) samt de stokastiska volatilitetsmodellerna Heston och Bates. Datat vi använt är optionspriser från Microsoft, Advanced Micro Devices Inc, Walt Disney Company och S&P 500 indexet. Datat delades upp i en träningsmängd och en test- mängd. Träningsdatat användes för parameterkalibrering med hänsyn till varje modell. Testdatat användes för att jämföra modellpriser med priser som observerats på mark- naden. Parameterkalibreringen för varje modell utfördes genom att använda den icke- linjära minsta-kvadratmetoden. Med hjälp av de kalibrerade parametrarna kunde priset räknas ut genom att använda Carr och Madan-metoden. Vi kunde se att de stokastiska volatilitetsmodellerna, Heston och Bates, replikerade marknadens optionspriser bättre än båda de konstanta volatilitetsmodellerna, MJD och BSM för de flesta dataseten. Medelvärdet av det relativa medelvärdesfelet i procent för Heston och Bates beräknades till 2.26% respektive 2.17%. För Merton och BSM beräknades medelvärdet av det relativa medelvärdesfelet i procent till 6.90% respektive 5.45%. Vi anser därför att en stokastisk volatilitetsmodell är att föredra framför en konstant volatilitetsmodell för att prissätta optioner.

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

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