För att prissätta optioner är val av modell en viktig fråga. I denna kandidatuppsats beskrivs både Black & Scholes modell och SABR-modellen. Förstnämnda modell är enklare än SABR-modellen men bygger på antaganden som inte stämmer överens med verkligheten. Den ger heller inte någon explicit formel för den implicita volatiliteten och predikterar inte heller på ett korrekt sätt fenomenet volatility smile vilket observeras på marknaden. Syftet med uppsatsen är att utvärdera prestandan hos SABR-modellen och användarvänligheten, samt att undersöka lite av teorin bakom modellen och vissa av dess egenskaper. Till grund för beräkningarna ligger datamaterial hämtat från Nasdaq OMX Nordic. Enligt mina beräkningar är resultatet att SABR-modellen endast presterar marginellt bättre än Black & Scholes-modellen. Dock kan även små förbättringar spela stor roll i dessa sammanhang.

Diffusion equations arise in areas such as fluid mechanics, cellular biology, weather forecasting, electronics, mechanical engineering, atomic physics, environmental science, medicine, etc. This dissertation considers equations of this type that arise in mathematical finance. For over 40 years traders in financial markets around the world have used Black-Scholes equations for valuing financial options. These equations need to be solved quickly and accurately so that the traders can make prompt and accurate investment decisions. One way to do this is to use parallel numerical algorithms. This dissertation develops and evaluates algorithms of this kind that are based on the Laplace transform, numerical inversion algorithms and finite difference methods. Laplace transform-based algorithms have faced a legitimate criticism that they are ill-posed i.e. prone to instability. We demonstrate with reference to the Black-Scholes equation, contrary to the received wisdom, that the use of the Laplace transform may be used to produce reasonably accurate solutions (i.e. to two decimal places), in a fast and reliable manner when used in conjunction with standard PDE techniques. To set the scene for the investigations that follow, the reader is introduced to financial options, option pricing and the one-dimensional and two-dimensional linear and nonlinear Black-Scholes equations. This is followed by a description of the Laplace transform method and in particular, four widely used numerical algorithms that can be used for finding inverse Laplace transform values. Chapter 4 describes methodology used in the investigations completed i.e. the programming environment used, the measures used to evaluate the performance of the numerical algorithms, the method of data collection used, issues in the design of parallel programs and the parameter values used. To demonstrate the potential of the Laplace transform based approach, Chapter 5 uses existing procedures of this kind to solve the one-dimensional, linear Black-Scholes equation. Chapters 6, 7, 8, and 9 then develop and evaluate new Laplace transform-finite difference algorithms for solving one-dimensional and two-dimensional, linear and nonlinear Black-Scholes equations. They also determine the optimal parameter values to use in each case i.e. the parameter values that produce the fastest and most accurate solutions. Chapters 7 and 9 also develop new, iterative Monte Carlo algorithms for calculating the reference solutions needed to determine the accuracy of the LTFD solutions. Chapter 10 identifies the general patterns of behaviour observed within the LTFD solutions and explains them. The dissertation then concludes by explaining how this programme of work can be extended. The investigations completed make significant contributions to knowledge. These are summarised at the end of the chapters in which they occur. Perhaps the most important of these is the development of fast and accurate numerical algorithms that can be used for solving diffusion equations in a variety of application areas.

<p>För att prissätta optioner är val av modell en viktig fråga. I denna kandidatuppsats</p><p>beskrivs både Black & Scholes modell och SABR-modellen. Förstnämnda modell är</p><p>enklare än SABR-modellen men bygger på antaganden som inte stämmer överens med</p><p>verkligheten. Den ger heller inte någon explicit formel för den implicita volatiliteten</p><p>och predikterar inte heller på ett korrekt sätt fenomenet volatility smile vilket</p><p>observeras på marknaden.</p><p>Syftet med uppsatsen är att utvärdera prestandan hos SABR-modellen och</p><p>användarvänligheten, samt att undersöka lite av teorin bakom modellen och vissa av</p><p>dess egenskaper. Till grund för beräkningarna ligger datamaterial hämtat från Nasdaq</p><p>OMX Nordic.</p><p>Enligt mina beräkningar är resultatet att SABR-modellen endast presterar marginellt</p><p>bättre än Black & Scholes-modellen. Dock kan även små förbättringar spela stor roll i</p><p>dessa sammanhang.</p>

Mémoire numérisé par la Direction des bibliothèques de l'Université de Montréal.

The famous Black-Scholes formula provided the first mathematically sound mechanism to price financial options. It is based on the assumption, that daily random stock returns are identically normally distributed and hence stock prices follow a stochastic process with a constant volatility. Observed prices, at which options trade on the markets, don¡¯t fully support this hypothesis. Options corresponding to different strike prices trade as if they were driven by different volatilities. To capture this so-called volatility smile, we need a more sophisticated option-pricing model assuming that the volatility itself is a random process. The price we have to pay for this stochastic volatility model is that such models are computationally extremely intensive to simulate and hence difficult to fit to observed market prices. This difficulty has severely limited the use of stochastic volatility models in the practice. In this project we propose to overcome the obstacle of computational complexity by executing the simulations in a massively parallel fashion on the graphics processing unit (GPU) of the computer, utilizing its hundreds of parallel processors. We succeed in generating the trillions of random numbers needed to fit a monthly options contract in 3 hours on a desktop computer with a Tesla GPU. This enables us to accurately price any derivative security based on the same underlying stock. In addition, our method also allows extracting quantitative measures of the riskiness of the underlying stock that are implied by the views of the forward-looking traders on the option markets.

O objetivo deste trabalho visou à apresentação de uma abordagem multidisciplinar para a gestão tributária com vistas à maximização do valor da firma em contraponto à abordagem tradicional do planejamento tributário que visa à minimização dos tributos. As estratégias para a minimização dos tributos aumentam o custo em outras dimensões que devem ser analisadas para posicionar a gestão tributária na gestão estratégica da empresa. Para se efetuar a análise da gestão tributária muldisciplinar, propõe-se a utilização do Framework de Scholes e Wolfson, que se baseia em três focos: todas as partes, todos os tributos e todos os custos. Dessa forma é possível não limitar os ângulos de visão, mas sim criar uma sistematização para a gestão tributária multidisciplinar. Como recurso metodológico utilizou-se a pesquisa bibliográfica e análise de casos de estudos que contemplam o uso de instrumentos financeiros derivativos, frequentemente utilizados em planejamentos tributários e que envolvem volumes expressivos de recursos. Restou evidenciado, neste trabalho, que a gestão tributária deve ser realizada de modo integrado na empresa, analisando todos os itens propostos pelo Framework de Scholes e Wolfson, assim, muito mais que minimização de tributos, obtém-se a maximização do valor da empresa / The aim of this paper is to present a multidisciplinary approach for the tax management aiming to maximize the value of the company in relation to the traditional approach of tax planning aiming to minimize taxation. Strategies for minimizing taxes increase the cost in other dimensions, which must be analyzed in order to position the tax management in the company\'s strategic management. In order to conduct the analysis of multidisciplinary tax management, the usage of Scholes and Wolfson\'s Framework is proposed. This framework is based on three focuses: all the parties, all the taxes and all the costs. In this manner, it is possible not to limit the points of view, but to create systematization for the multidisciplinary tax management. As methodological resource, bibliographic research and the analysis of case studies contemplating the usage of derivate financial instruments commonly used in tax planning and which encompass great volume of resources were used. It could be concluded, in this paper, that tax management must be conducted in an integrated manner in the company, analyzing all the items proposed by Scholes and Wolfson\'s Framework. By doing so, it is possible not only to minimize the taxes, but also maximize the company\'s value.

This diploma thesis deals with problem of option pricing with stochastic volatility. At first, the Black-Scholes model is derived and then its biases are discussed. We explain shortly the concept of volatility. Further, we introduce three pricing models with stochastic volatility- Hull-White model, Heston model and Stein-Stein model. At the end, these models are reviewed.

Dans une première partie, après avoir rappelé les principales caractéristiques statistiques des séries financières, en particulier l'existence de corrélations non linéaires à longue portée et d'une asymétrie fortement persistante, nous mettons en évidence la pertinence des processus multifractals pour la modélisation de ces faits stylisés. Les constructions récemment proposées dans la littérature demeurent cependant exclusivement symétriques et nous montrons comment introduire de l'asymétrie dans ces modèles sans sacrifier leurs propriétés d'échelle. Il est alors possible de rendre compte du phénomène de smile de volatilité. Dans une deuxième partie, nous proposons une méthode numérique pour la valorisation et la couverture d'options en marché incomplet. Notre algorithme peut en outre être généralisé sans difficulté pour tenir compte d'autres imperfections du marché comme les frais de transaction.

<p>Abstract</p><p>Title: Options Based on CO2 Emissions: A Comparison with Traditional Options</p><p>Seminar date: 2009-06-17</p><p>Course: Bachelor thesis in business administration, 15 ECTS</p><p>Authors: Gustaf Kristiansson, Martin Nilsson</p><p>Instructor: Bengt Kjellgren</p><p>Key words: Black & Scholes, Certified Emission Reductions, emission markets, European Union Allowances, options, pricing</p><p>Purpose: This study intends to compare traditional options with the CO2 based instruments EUAs and CERs options in the fields of pricing, cap and trade, political influence, economical effects and market function.</p><p>Methodology: A combined research methodology is used in this study, which includes both a quantitative and a qualitative approach. A deductive research approach is brought out over the whole study.</p><p>Theoretical perspectives: The theoretical framework is based upon previous empirical research concerning the fields in this study. The Black & Scholes formula for option pricing has a central position.</p><p>Empirical foundation: Market data has been used to analyse the field of pricing. Interviews have been conducted with actors on the European emission trading market for a further understanding of cap and trade, political influence, economical effects and market function.</p><p>Conclusions: We have in this research identified that the CO2 based market differs from the financial market when it comes to political decisions and price fluctuation. We have also identified that the CO2 based market is not mature enough for a complete internationalisation.</p> / En formell presentation utfördes ej pga utlandsstudier.

<p>Den 19 februari 2004 presenterade IASB IFRS 2. IFRS 2 innebär att företagen skall hantera personaloptioner som en kostnad i resultaträkningen. I författningen regleras ej vilken värderingsmodell som skall användas vid värderingen av optionerna. På grund av detta kan redovisningen påverkas på olika vis beroende på vilken optionsvärderingsmodell som används av bolaget. Uppsatsens syfte är att fastställa vilka effekter olika optionsvärderingsmodeller har på bolagens värdering av personaloptioner och även beskriva vilka antaganden av indata i modellen som görs av bolagen och undersöka om dessa antaganden är teoretiskt korrekta. Uppsatsen grundas på en fallstudie av tio bolag som har aktiva optionsprogram som redovisas enligt IFRS 2. Vid jämförelsen av optionsvärderingsmodellerna så sätts bolagens indata in i två olika värderingsmodeller och beräkningar görs med dessa värden för att kunna se hur värdena på optionerna varierar mellan dessa optionsvärderingsmodeller. Eftersom volatiliteten är en mycket viktig faktor i värderingsmodellerna och påverkar det teoretiska värdet i betydande utsträckning så görs även en undersökning som skall ge en bild av hur bolagen har kommit fram till sitt volatilitetsantagande och som även utreder om detta värde är teoretiskt korrekt. Av de ursprungliga 102 bolagen hittades 35 bolag (34 %) som har personaloptioner som redovisas enligt IFRS 2. Av dessa 35 bolag hittades endast tio bolag (28,57 %) med fullständig information av dess indata. Efter min estimering av den historiska volatiliteten, fem år bakåt i tiden, så är den i genomsnitt 216,43 % högre än bolagens antagna volatilitet och optionsvärdet i genomsnitt 179,53 % högre än bolagens beräknade värde. Använder vi istället min estimering av den historiska volatiliteten, av ett år tillbaka i tiden, så hamnar vi på ett lägre värde, 41,39 % högre än bolagens beräknade verkliga värde och ett beräknat optionsvärde som är 39,13 % högre än bolagens. Skillnaderna i den riskfria räntan var oerhört stora, från 2,00 % till 4,30 %, en skillnad på 215 %. Skillnaden i beräknat verkligt värde skiljer sig minimalt mellan Black & Scholes modellen och binominalmodellen. Medeldifferensen mellan de tio analyserade bolagen var endast 0,0059 %, dvs. en knappt mätbar skillnad.</p> / <p>19th February 2004 the IASB introduced the IFRS 2. IFRS 2 means that the corporations must handle employee stock options as a cost in the income statement. It is not regulated in the constitution which model that should be used to value the options. Because of this the income statement can be affected by which model that is used by the corporation. The purpose of this paper is to confirm which affects different valuing models has to the value of the corporations employee options and even describe which assumptions of the in data that are used by the corporations and examine if this assumptions are theoretical correct. The paper is based on a case study of ten corporations who has employee stock option programs which are accounted according to the IFRS 2. At the comparing of the option valuing models the corporations in data is placed into the two models and calculations are made to see if the values vary between the models. Because that volatility is a very important factor in the option valuing models and affects the theoretical value at a significant extent an examination which is mean to give a picture of how the corporations has create the volatility assumption and that even will study if this value is correct theoretical value. Only 35 of the original 102 (34 %) corporations had employee stock options which are accounted according to the IFRS 2. Only 10 of these 35 (28,57 %) had a complete information about the in data. After my estimation of the historical volatility, five years back in time, my estimated volatility was in average 216,43 % higher than what the corporations had estimated and the average value of the options was 179,53 % higher. An estimation of the historical volatility one year back in time I get a lower value of the volatility that is in average 41,39 % higher than the corporations estimation and a option price that is 39,13 % higher. The differences of the risk free interest rate are huge, from 2,00 % to 4,30 %, a difference at 215 %. The calculated variation between the Black & Scholes model and the binomial method is tiny. The average difference between the ten examined corporation are only 0,0059 %, i.e. a hardly noticeable difference.</p>