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The diploma thesis describes portfolio management with proportional transaction costs. The main aim is to describe using of shadow prices to find the optimal Markov policies keeping the proportion of the investor's wealth invested in the risky asset within the corresponding interval in order to maximize the long run geometric growth rate. On the real market, the investor must pay transaction costs when he buys/sells shares. In the diploma thesis we transform these prices into the shadow price; when trading in the shadow price there are no transaction costs. The solution itself is based on Itô formula and the martingal theory. The prices of shares are modeled as geometric Brownian motion. Powered by TCPDF (www.tcpdf.org)

In the presented thesis we study three methods of pricing European currency barrier options. With help of these methods we value selected barrier options with underlying asset EUR/CZK. In the first chapter we introduce the basic definitions from the world of financial derivatives and we describe our data. In the second chapter we deal with the classical model based on geometric Brownian motion of underlying asset and we prove a theorem of valuating Up-In-barrier option in this model. In the third chapter we introduce a model with stochastic volatility, the Heston model. We calibrate this model to market data and we use it to value our barrier options. In the last chapter we describe a jump diffusion model. Again we calibrate this jump diffusion model to market data and price our barrier options. The aim of this thesis is to decribe and to compare different methods of valuating barrier options. 1

[pt] A escolha de uma carreira é uma das decisões mais importantes na vida de uma pessoa, e é feita em um ambiente repleto de incertezas em relação ao futuro. Este trabalho analisa o aspecto financeiro da escolha entre uma carreira numa empresa privada e uma carreira em um órgão público, com ingresso por meio de um concurso. A análise pelo tradicional fluxo de caixa descontado apresenta uma série de limitações por não captar aspectos como a incerteza e a flexibilidade da tomada de decisão. Assim é aplicada uma abordagem segundo a teoria das Opções Reais, que se mostra mais adequada a este caso, pois permite que a flexibilidade de escolha seja modelada e considerada na escolha de carreira de um indivíduo. Neste estudo, os ganhos em uma empresa privada são modelados por meio de um processo estocástico enquanto a carreira pública tem um valor determinístico. Existe flexibilidade de data em relação ao ingresso na carreira pública, porém esta decisão é irreversível. Os resultados sugerem que a opção de ingressar na carreira pública pode ter valor significativo em relação à carreira privada. / [en] Choosing a career is one of the most important decisions in a person s life, and is done in an environment full of uncertainties about the future. This study analyzes the financial aspect of a career choice between a private company and a career in the government, with admission through a contest. The analysis through the traditional discounted cash flow would bring a lot of limitations, not capturing aspects such as uncertainty and flexibility of decision making. So real options theory approach is applied, which appears more appropriate in this case because it allows the flexibility of choice to be modeled and considered in the choice of an individual s career. In this study earnings in a private company are modeled through a stochastic process while public career has a deterministic value. There is flexibility regarding the date of entry into public career, but this decision is irreversible. The results suggest that the option of joining the public career may have significant value in relation to private career.

[pt] O objetivos desta dissertação é fazer uma análise de investimentos usando a teoria das Opções Reais de uma planta GTL. Está análise é a mais indicada, pois se verificam várias flexibilidades nesta planta em relação aos inputs (pode ser usado mais de um produto como matéria- prima) e em relação aos outputs (existem várias combinações possíveis de produção). Torna-se de grande importância neste caso saber calcular o valor destas opções e verificar se vale a pena ou não a construção de uma planta que possa usar como matéria prima mais de um produto e/ou que possa produzir mais de uma possível combinação de produção. A construção de uma planta que possua a possibilidade de trocar de insumo e/ou trocar a combinação de produção só será viável caso o valor criado pela flexibilidade seja maior do que o custo necessário para implementá-la (investimento adicional e custos operacionais extras). Sendo assim, o objetivo desta dissertação é calcular até quanto a Petrobras estaria disposta a pagar para ter uma planta que possua a opção de swicth use dos inputs e/ou outputs, o valor que ela teria que investir para usufruir desta flexibilidade, e através da diferença entre estes valores verificar se vale a pena ou não a construção da planta com flexibilidade de input e/ou output. / [en] The objective of this dissertation is to do a analysis of investiment using the real option theory for the plant GTL. This analysis is the best because there are many flexibilities in this plant in relation the inputs (the plant can operate with several inputs) and in relation the outputs (there are many possible combination of production). In this case is very important to know how to calculate the value of these options and to verify if it is worthwhile or not the construction of a plant that could use two inputs and/or is able to procuce several possible combinations of production. The construction of the plant that change the input abd /or can changer the production combination is viable if the value created by flexibility is large than the necessary cost to implement its (additional investiment and extra operational costs). So, the objective of this dissertation is to calculate until hen Petrobras would be avaible to pay in order to have a plant that has the option of swicth use of inputs and/or outputs, the value it would have to invest to use this flexibility, and through the difference between these values verify if is worthwhile or not the construction of the plant with the flexibility of input and/or output.

[pt] Esta dissertação procura analisar através de um estudo de caso, as alternativas de desenvolvimento de um campo de petróleo já descoberto, mas ainda não explorado, utilizando a Teoria das Opções Reais. A partir deste estudo, será possível avaliar uma alternativa de desenvolvimento da produção de dois poços de petróleo, que serão explorados no futuro, dependendo das condições de mercado e das informações técnicas geradas pela produção inicial do campo. A dissertação tem como principal objetivo comparar os resultados das incertezas de mercado no preço do petróleo representadas pelos processos estocásticos, o Movimento Geométrico Browniano e o Processo de Reversão à Média com Saltos, para determinação da ferramenta gerencial denominada Curva de Gatilho. / [en] This dissertation search for to analyze through a study of case, the alternatives of development of a oil field already discovered, but not yet exploited, using the Theory of the Real Options. From this study, it will be possible to evaluate an alternative of development of the production of two wells, that will be explored in the future, depending on the market conditions and of the technical informations generated for the initial production of the field. The dissertation has as mean objective to compare the results of the uncertainties of market in the price of oil rerepresented by Estocastic Processes, the Geometric Browniano Movement and the Process of Mean Reversion with Jumps, for determination of the management tool named of Trigger.

>Magister Scientiae - MSc / We present various methods of pricing Asian options. The methods include Monte Carlo simulations designed using control and antithetic variates, numerical solution of partial differential equation and using lower bounds.The price of the Asian option is known to be a certain risk-neutral expectation. Using the Feynman-Kac theorem, we deduce that the problem of determining the expectation implies solving a linear parabolic partial differential equation. This partial differential equation does not admit explicit solutions due to the fact that the distribution of a sum of lognormal variables is not explicit. We then solve the partial differential equation numerically using finite difference and Monte Carlo methods.Our Monte Carlo approach is based on the pseudo random numbers and not deterministic sequence of numbers on which Quasi-Monte Carlo methods are designed. To make the Monte Carlo method more effective, two variance reduction techniques are discussed.Under the finite difference method, we consider explicit and the Crank-Nicholson’s schemes. We demonstrate that the explicit method gives rise to extraneous solutions because the stability conditions are difficult to satisfy. On the other hand, the Crank-Nicholson method is unconditionally stable and provides correct solutions. Finally, we apply the pricing methods to a similar problem of determining the price of a European-style arithmetic basket option under the Black-Scholes framework. We find the optimal lower bound, calculate it numerically and compare this with those obtained by the Monte Carlo and Moment Matching methods.Our presentation here includes some of the most recent advances on Asian options, and we contribute in particular by adding detail to the proofs and explanations. We also contribute some novel numerical methods. Most significantly, we include an original contribution on the use of very sharp lower bounds towards pricing European basket options.

The Turtle Trading Rules was a successful trend-following trading strategy for commodities in the 1980s but has lost recognition in recent days. The strategy revolved around rules for entering and exiting trades as well as position sizing for each trade. The rules was based on the fundamental aim to capture market trends while at the same time maintaining a controlled risk exposure. This thesis aims to revise the Turtle Trading Rules, here applied on major market indexes, and to examine its viability through different financial metrics. This is done by first implementing the aforementioned trading strategy to the indexes, and later by synthesizing market data through Geometric Brownian Motions. The latter primarily to examine how the strategy perform in different financial environments, what market traits favor the strategy, and to complement the previous examination without altering the core principles of the Turtle Trading Rules. The results suggest that the revised rules for major market indexes is not viable. This because of the poor return, the highest achieved 20-year return and average annual return was 25.1 % and 1.4% respectively (without taking trading fees into account). Furthermore, the strategy applied on synthetic data suggests that favorable traits are highly cyclical data with low volatility, which seldom is the case for real financial time series. The results further indicate that the main issue lies in the rules not being able to distinguish noise from actual entry and exit triggers. Moreover, the drawdown further suggest that it is the exit rather than the entry rules that are to blame for the poor performance during the cycle of a trade. / Turtle Trading-reglerna var en framgångsrik trendföljande handelsstrategi för råvaror på 1980- talet men har sedan dess tappat i populäritet. Strategin kretsade kring regler för inträde och utträde ur ordrar samt kring positionsstorleken för varje order. Reglerna byggde på det grundläggande syftet att fånga marknadstrender och samtidigt upprätthålla en kontrollerad riskexponering. Den här avhandlingen syftar till att revidera Turtle Trading-reglerna, som här tillämpas på utvalda marknadsindex, och att undersöka dess lönsamhet genom olika finansiella mått. Detta görs genom att först implementera den tidigare nämnda handelsstrategin till indexen, och senare genom att syntetisera marknadsdata genom geometriska brownska rörelser (Geometric Brownian Motions). Det senare för att i första hand undersöka hur strategin fungerar i olika finansiella miljöer, vilka marknadsdrag som gynnar strategin, och för att komplettera den tidigare granskningen utan att ändra grundprinciperna i Turtle Trading-reglerna. Resultaten tyder på att de reviderade reglerna för marknadsindexen inte är tillräckligt lönsamma. Detta på grund av den låga avkastningen, den högst observerade 20-årsavkastningen och den genomsnittliga årliga avkastningen var 25,1 % respektive 1,4% (utan hänsyn till handelsavgifter). Dessutom antyds när strategin tillämpas på syntetiska data att tydliga cykliska variationer samt låg volatilitet är fördelaktiga egenskaper, vilket sällan är fallet för reella finansiella tidsserier. Resultatet indikerade vidare att grundproblemet ligger i att reglerna inte kan skilja brus i datan från faktiska inträde och exit triggers. Dessutom tyder drawdown-graferna på att det är exit snarare än inträdesreglerna som är orsaken till det dåliga resultatet i utförandet av köp- och säljprocessen.

>Magister Scientiae - MSc / Warrant pricing has become very crucial in the present market scenario. See, for example, M. Hanke and K. Potzelberger, Consistent pricing of warrants and traded options, Review Financial Economics 11(1) (2002) 63-77 where the authors indicate that warrants issuance affects the stock price process of the issuing company. This change in the stock price process leads to subsequent changes in the prices of options written on the issuing company's stocks. Another notable work is W.G. Zhang, W.L. Xiao and C.X. He, Equity warrant pricing model under Fractional Brownian motion and an empirical study, Expert System with Applications 36(2) (2009) 3056-3065 where the authors construct equity warrants pricing model under Fractional Brownian motion and deduce the European options pricing formula with a simple method. We study this paper in details in this mini-thesis. We also study some of the mathematical models on warrant pricing using the Black-Scholes framework. The relationship between the price of the warrants and the price of the call accounts for the dilution effect is also studied mathematically. Finally we do some numerical simulations to derive the value of warrants.

Parametric models are used to understand dynamical systems and predict its future behavior. It is difficult to estimate the model’s parametric values since there are usually many parameters and they are highly correlated. The aim of this project is to apply the method of indirect inference and the two stage approach to estimate the drift and volatility parameters of a Geometric Brownian Motion. This was first done by estimating the parameters of a known Geometric Brownian process. Then, the Coca-Cola Company’s stock was used for a five-year forecast to study the estimators’ predictive power. The two stage approach struggles when the data does not truly follow a Geometric Brownian Motion, but when it does it produces highly efficient and accurate estimates. The method of indirect inference produces better estimates, than the two stage approach,for data that deviates from a Geometric Brownian Motion.Therefore, it is preferable to use indirect inference over two stage approach for stock price forecasting. / Parametriska modeller används för attförstå dynamiska system och förutspå dess framtida beteende.Det är utmanande att skatta modellens parametriska värdeneftersom det vanligtvis finns många parametrar och de är oftastarkt korrelerade. Målet med detta projekt är att tillämpametoderna indirect inference och two stage approach för attskatta drivnings- och volatilitetsparametrarna av en geometriskBrownsk rörelse. Först skattades parametrarna av en kändGeometrisk Brownsk rörelse. Sedan användes The Coca-ColaCompanys aktie i syfte att studera estimatorernas förmåga attförutspå en femårig period. Two stage approach fungerar dåligtför data som inte helt följer en geometrisk Brownsk rörelse, mennär datan gör det är skattningarna noggranna och effektiva.Indirect inference ger bättre skattningar än two stage approachnär datan inte helt följer en geometrisk Brownsk rörelse. Därförär indirect inference att föredra för aktieprognoser. / Kandidatexjobb i elektroteknik 2022, KTH, Stockholm