This thesis deals with regularization parameter selection methods in the context of Tikhonov-type regularization with Poisson distributed data, in particular the reconstruction of images, as well as with the identification of the volatility surface from observed option prices. In Part I we examine the choice of the regularization parameter when reconstructing an image, which is disturbed by Poisson noise, with Tikhonov-type regularization. This type of regularization is a generalization of the classical Tikhonov regularization in the Banach space setting and often called variational regularization. After a general consideration of Tikhonov-type regularization for data corrupted by Poisson noise, we examine the methods for choosing the regularization parameter numerically on the basis of two test images and real PET data. In Part II we consider the estimation of the volatility function from observed call option prices with the explicit formula which has been derived by Dupire using the Black-Scholes partial differential equation. The option prices are only available as discrete noisy observations so that the main difficulty is the ill-posedness of the numerical differentiation. Finite difference schemes, as regularization by discretization of the inverse and ill-posed problem, do not overcome these difficulties when they are used to evaluate the partial derivatives. Therefore we construct an alternative algorithm based on the weak formulation of the dual Black-Scholes partial differential equation and evaluate the performance of the finite difference schemes and the new algorithm for synthetic and real option prices.

This thesis deals with regularization parameter selection methods in the context of Tikhonov-type regularization with Poisson distributed data, in particular the reconstruction of images, as well as with the identification of the volatility surface from observed option prices. In Part I we examine the choice of the regularization parameter when reconstructing an image, which is disturbed by Poisson noise, with Tikhonov-type regularization. This type of regularization is a generalization of the classical Tikhonov regularization in the Banach space setting and often called variational regularization. After a general consideration of Tikhonov-type regularization for data corrupted by Poisson noise, we examine the methods for choosing the regularization parameter numerically on the basis of two test images and real PET data. In Part II we consider the estimation of the volatility function from observed call option prices with the explicit formula which has been derived by Dupire using the Black-Scholes partial differential equation. The option prices are only available as discrete noisy observations so that the main difficulty is the ill-posedness of the numerical differentiation. Finite difference schemes, as regularization by discretization of the inverse and ill-posed problem, do not overcome these difficulties when they are used to evaluate the partial derivatives. Therefore we construct an alternative algorithm based on the weak formulation of the dual Black-Scholes partial differential equation and evaluate the performance of the finite difference schemes and the new algorithm for synthetic and real option prices.

2014年保險法第146條之4修正，增列保險業依保險法規定投資於國內證券市場上市或上櫃買賣之外幣計價股權或債券憑證之投資金額，可不計入其國外投資限額。本研究探討台灣壽險業投資外幣計價國際債券不納入國外投資限額對於台幣公債市場籌資之影響，並分析壽險業投資國際債券之贖回風險。 主要研究結果發現：(1)開放投資國際債券後，壽險業資金運用增加國外投資，但減持公債及國庫券。依據統計分析，顯示壽險業資金運用於國外投資佔比大幅增加時，除專案運用及公共投資外，其餘項目之佔比皆減少。其中台灣公債及國庫券佔比與國外投資佔比呈高度負相關。(2)壽險業對公債需求程度影響國庫籌資之成本，需求程度越低，國庫長天期籌資成本越高。透過複迴歸模型分析發現，壽險持券比(即壽險業持有公債餘額佔公債未償還餘額比例)越低，30年期公債殖利率越高。 透過本文模型，投資國際債券時，應考量可贖回國際債券相對公債之加碼、閉鎖期、國際債券再度發行之可能性與未來市場利率可能走低之幅度。以投資30年期債券為例，當可贖回國際債券相對公債之加碼減少，閉鎖期縮短，利率走跌幅度增加時，贖回風險將增加。因可贖回國際債券之高利率僅為收益率錯覺，利率走低時之贖回風險將抵銷此高利率。依據2014年至2016年債券市場資訊，本文模型評估投資人提前贖回風險為52.45bps。 / The 2014 amendment of Article 146-4 of Insurance Act extending the overseas investment ceiling to the value for foreign currency denominated listed or over-the-counter certificates of domestic stocks or bonds that are invested in by insurance enterprises in accordance with provisions of Insurance Act. This paper investigates the impact of funding in Taiwan government bond market under the overseas investment ceiling with the exclusion of international bonds investment in Taiwan life insurance industry, and analyzes the call risk of international bonds. The main results show that: (1)After the 2014 amendment of Article 146-4 of Insurance Act, foreign investments are increasing, while government bonds holdings are decreasing in investment portfolio of life insurance industry. Based on statistical analysis, as the ratio of foreign investments surging, only the ratio of authorized projects or public investment is increasing, others are falling. Especially, the ratio of government bonds and that of foreign investment are strongly negative correlation. (2) Demand of government bond of life insurance industry has impact on the funding cost of Taiwan government. The lower the demand, the higher the funding cost. Through multiple regression model, the result shows, the lower the bond holding ratio of life insurance industry, the higher the yield of Taiwan 30-year government bonds. According to the model in this paper, spread between callable international bond and government bond, lock-up period, the probability of re-issuance in international bond market, and the downtrend of interest rate should be all considered when investing in international bonds. The high yield of callable international bond is yield illusion to investors,since is largely offset by call risk. According to the model with bond market data between 2014 to 2016, the assessment of call risk is 52.45bps.

Submitted by Daniel Loureiro da Silva (danloux@hotmail.com) on 2018-01-26T14:38:47Z No. of bitstreams: 1 Opção em Debêntures.pdf: 1582205 bytes, checksum: 6007183ecaa2252dd077e1fb1902f4f9 (MD5) / Approved for entry into archive by GILSON ROCHA MIRANDA (gilson.miranda@fgv.br) on 2018-01-26T18:34:06Z (GMT) No. of bitstreams: 1 Opção em Debêntures.pdf: 1582205 bytes, checksum: 6007183ecaa2252dd077e1fb1902f4f9 (MD5) / Made available in DSpace on 2018-01-30T13:02:17Z (GMT). No. of bitstreams: 1 Opção em Debêntures.pdf: 1582205 bytes, checksum: 6007183ecaa2252dd077e1fb1902f4f9 (MD5) Previous issue date: 2017-12-28 / On June 24, 2011, Law 12.431/11 was introduced, which granted a reduction of the Income Tax rate on bonds issued with the purpose of financing infrastructure projects. Since then, the so-called "Infrastructure Bonds" have increased their importance as a funding mechanism to companies in this sector. Since the publication of the Resolution of the Central Bank of Brazil (Bacen) No. 4.476, of 4 November 2016, it was allowed to include early redemption clauses at the discretion of the issuing company. Considering the peculiarities of these bonds, the methodologies already developed for pricing call options on bonds are not applicable. In this way, this work intends to develop a methodology for pricing the existence of an early redemption clause in an Infrastructure Bond covered by Law 12.431/11, considering the term structure of Brazilian interest rates and the costs of issuing bonds. / Em 24.06.2011, foi lançada a Lei n° 12.431/11, que concedeu uma redução da alíquota de Imposto de Renda (IR) em debêntures emitidas com o objetivo de financiar projetos de infraestrutura. Desde então, as chamadas “Debêntures de Infraestrutura” têm aumentado a sua importância como mecanismo de captação das empresas desse setor. A partir da publicação da Resolução do Banco Central do Brasil (Bacen) nº 4.476, de 11.04.2016, foi permitida a inclusão de cláusulas de regaste antecipado a exclusivo critério da empresa emissora. Tendo em vista as peculiaridades desses títulos, as metodologias já desenvolvidas para precificação de opções de recompra em debêntures não são aplicáveis. Desta forma, o trabalho proposto pretende desenvolver uma metodologia para precificar a existência de uma cláusula de resgate antecipado em uma debênture de infraestrutura abrangida pela Lei n° 12.431/11, considerando a estrutura a termo das taxas de juros brasileiras e os custos de emissão de debêntures.

Esta dissertação aplica a regularização por entropia máxima no problema inverso de apreçamento de opções, sugerido pelo trabalho de Neri e Schneider em 2012. Eles observaram que a densidade de probabilidade que resolve este problema, no caso de dados provenientes de opções de compra e opções digitais, pode ser descrito como exponenciais nos diferentes intervalos da semireta positiva. Estes intervalos são limitados pelos preços de exercício. O critério de entropia máxima é uma ferramenta poderosa para regularizar este problema mal posto. A família de exponencial do conjunto solução, é calculado usando o algoritmo de Newton-Raphson, com limites específicos para as opções digitais. Estes limites são resultados do princípio de ausência de arbitragem. A metodologia foi usada em dados do índice de ação da Bolsa de Valores de São Paulo com seus preços de opções de compra em diferentes preços de exercício. A análise paramétrica da entropia em função do preços de opções digitais sínteticas (construídas a partir de limites respeitando a ausência de arbitragem) mostraram valores onde as digitais maximizaram a entropia. O exemplo de extração de dados do IBOVESPA de 24 de janeiro de 2013, mostrou um desvio do princípio de ausência de arbitragem para as opções de compra in the money. Este princípio é uma condição necessária para aplicar a regularização por entropia máxima a fim de obter a densidade e os preços. Nossos resultados mostraram que, uma vez preenchida a condição de convexidade na ausência de arbitragem, é possível ter uma forma de smile na curva de volatilidade, com preços calculados a partir da densidade exponencial do modelo. Isto coloca o modelo consistente com os dados do mercado. Do ponto de vista computacional, esta dissertação permitiu de implementar, um modelo de apreçamento que utiliza o princípio de entropia máxima. Três algoritmos clássicos foram usados: primeiramente a bisseção padrão, e depois uma combinação de metodo de bisseção com Newton-Raphson para achar a volatilidade implícita proveniente dos dados de mercado. Depois, o metodo de Newton-Raphson unidimensional para o cálculo dos coeficientes das densidades exponenciais: este é objetivo do estudo. Enfim, o algoritmo de Simpson foi usado para o calculo integral das distribuições cumulativas bem como os preços do modelo obtido através da esperança matemática. / This study aims at applying Maximum Entropy Regularization to the Inverse Problem of Option Pricing suggested by Neri and Schneider in 2012. They pointed out that the probability density that solves such problem in the case of calls and digital options could be written as piecewise exponentials on the positive real axis. The limits of these segments are the different strike prices. The entropy criteria is a powerful tool to regularize this ill-posed problem. The Exponential Family solution set is calculated using a Newton-Raphson algorithm, with specific bounds for the binary options. These bounds obey the no-arbitrage principle. We applied the method to data from the Brazilian stock index BOVESPA and its call prices for different strikes. The parametric entropy analysis for "synthetic" digital prices (constructed from the no-arbitrage bounds) showed values where the digital prices maximizes the entropy. The example of data extracted on the IBOVESPA of January 24th 2013, showed slippage from the no-arbitrage principle when the option was in the money: such principle is a necessary condition to apply the maximum entropy regularization to get the density and modeled prices. When the condition is fulfilled, our results showed that it is possible to have a smile-like volatility curve with prices calculated from the exponential density that fit well the market data. In a computational modelling perspective, this thesis enabled the implementation of a pricing method using the maximum entropy principle. Three well known algorithms were used in that extent. The bisection alone, then a combined bisection with Newton-Raphson to recover the implied volatility from market data. Thereafter, the one dimensional Newton-Raphson to calculate the coefficients of the exponential densities: purpose of the study. Finally the Simpson method was used to calculate integrals of the cumulative distributions and the modeled prices implied by the expectation.

Esta dissertação aplica a regularização por entropia máxima no problema inverso de apreçamento de opções, sugerido pelo trabalho de Neri e Schneider em 2012. Eles observaram que a densidade de probabilidade que resolve este problema, no caso de dados provenientes de opções de compra e opções digitais, pode ser descrito como exponenciais nos diferentes intervalos da semireta positiva. Estes intervalos são limitados pelos preços de exercício. O critério de entropia máxima é uma ferramenta poderosa para regularizar este problema mal posto. A família de exponencial do conjunto solução, é calculado usando o algoritmo de Newton-Raphson, com limites específicos para as opções digitais. Estes limites são resultados do princípio de ausência de arbitragem. A metodologia foi usada em dados do índice de ação da Bolsa de Valores de São Paulo com seus preços de opções de compra em diferentes preços de exercício. A análise paramétrica da entropia em função do preços de opções digitais sínteticas (construídas a partir de limites respeitando a ausência de arbitragem) mostraram valores onde as digitais maximizaram a entropia. O exemplo de extração de dados do IBOVESPA de 24 de janeiro de 2013, mostrou um desvio do princípio de ausência de arbitragem para as opções de compra in the money. Este princípio é uma condição necessária para aplicar a regularização por entropia máxima a fim de obter a densidade e os preços. Nossos resultados mostraram que, uma vez preenchida a condição de convexidade na ausência de arbitragem, é possível ter uma forma de smile na curva de volatilidade, com preços calculados a partir da densidade exponencial do modelo. Isto coloca o modelo consistente com os dados do mercado. Do ponto de vista computacional, esta dissertação permitiu de implementar, um modelo de apreçamento que utiliza o princípio de entropia máxima. Três algoritmos clássicos foram usados: primeiramente a bisseção padrão, e depois uma combinação de metodo de bisseção com Newton-Raphson para achar a volatilidade implícita proveniente dos dados de mercado. Depois, o metodo de Newton-Raphson unidimensional para o cálculo dos coeficientes das densidades exponenciais: este é objetivo do estudo. Enfim, o algoritmo de Simpson foi usado para o calculo integral das distribuições cumulativas bem como os preços do modelo obtido através da esperança matemática. / This study aims at applying Maximum Entropy Regularization to the Inverse Problem of Option Pricing suggested by Neri and Schneider in 2012. They pointed out that the probability density that solves such problem in the case of calls and digital options could be written as piecewise exponentials on the positive real axis. The limits of these segments are the different strike prices. The entropy criteria is a powerful tool to regularize this ill-posed problem. The Exponential Family solution set is calculated using a Newton-Raphson algorithm, with specific bounds for the binary options. These bounds obey the no-arbitrage principle. We applied the method to data from the Brazilian stock index BOVESPA and its call prices for different strikes. The parametric entropy analysis for "synthetic" digital prices (constructed from the no-arbitrage bounds) showed values where the digital prices maximizes the entropy. The example of data extracted on the IBOVESPA of January 24th 2013, showed slippage from the no-arbitrage principle when the option was in the money: such principle is a necessary condition to apply the maximum entropy regularization to get the density and modeled prices. When the condition is fulfilled, our results showed that it is possible to have a smile-like volatility curve with prices calculated from the exponential density that fit well the market data. In a computational modelling perspective, this thesis enabled the implementation of a pricing method using the maximum entropy principle. Three well known algorithms were used in that extent. The bisection alone, then a combined bisection with Newton-Raphson to recover the implied volatility from market data. Thereafter, the one dimensional Newton-Raphson to calculate the coefficients of the exponential densities: purpose of the study. Finally the Simpson method was used to calculate integrals of the cumulative distributions and the modeled prices implied by the expectation.

基金經理人與投資人間的代理問題起源於兩者目標的不一致，前者要求個人薪酬財富的極大化，後者要求投資組合價值的極大化。造成目標歧異的原因有兩個：其一是在資訊不對稱的環境下，投資人無法觀察到經理人投資組合是否真正從投資人利益角度出發，因此引發了經理人的道德危險；其二是在競爭激烈的基金產業□，年度定期績效評比結果與經理人薪酬紅利多寡相連結等制度的設計，加重經理人選擇持有一個高風險投資組合的逆誘因，特別是期中累積績效較差的輸家，隨著年終總績效結算日期的接近，在自利動機的驅使下可能會透過較高的操作風險調整幅度，企圖扭轉頹勢以求反敗為勝。 本研究首先利用卡方檢定、t檢定與Logistic迴歸模式，實證國內基金經理人是否具有自利性風險調整的行為傾向，接著從契約設計的觀點，以理論模式推導命題的方式，探討三種不同型態的誘因費契約對於抑制經理人自利性風險調整行為的作用，並間接利用問卷調查的方式來驗證其效果。研究的結果發現： 1.國內基金經理人不管是贏家或輸家，在越接近年終總績效的結算時，都會偏向選擇一個高風險的操作水準，同時，上述特性在非外資型投信公司所發行的基金、新基金、小規模基金、資淺經理人所操盤的基金特別明顯。 2.基金投資人對於季等短期績效的過分重視，是導致國內經理人操作風險調整幅度偏高的主要原因之一。 3.純粹誘因費契約以及只加上「上限條款」設計的契約，這兩種契約都無法抑制經理人冒高風險的傾向；而純粹誘因費契約加上「懲罰條款」的設計，有抑制經理人自利性調高操作風險行為的效果。 本研究成果的貢獻主要有兩點： 1.在政策應用上，本研究提出契約條款設計的理念，對於我國擬開放勞退等大型基金委託代客操作，雙方契約該如何設計以確保投資人的權益，有相當參考的價值，本研究同時也對投信公司、基金經理人等提出相關的制度性建議。 2.在理論推導上，本研究融合一般化均衡分析法與選擇權理論的應用，將不同型態的誘因費契約化成經理人向投資人所購買的歐式買權，標的資產為經理人所持投資組合的價值，履約價格為比較基準指數的價值，執行日期為一年期的績效評比，模型導証的重點是經理人所選擇投資組合的風險程度與該類比選擇權價值的關係。 / An important question for the contracting literature is the extent to which real behavior is driven by the financial incentives contained in compensation schemes. To address this issue, (1) we use the tournament concept as the framework, and focus on the competitive nature of mutual fund environments how to affect the managers' portfolio decision-making processes. (2) we also use the Black-Scholes option pricing model as the framework, and analyze the impact on the mutual fund managers' risk-manipulation behavior of a performance-based incentive plan. Given the asymmetric information financial markets, most investors of mutual funds can not measure the funds' risk without error, thus, agent-divergent behavior may potentially arise. In a tournament reward structure, the managers' rational attempting to maximize their expected compensation may revise the risk level or alter the composition of their portfolio during the assessment period. While there will be times when such changes don't serve the best interest of funds' investors. Our research demonstrates the following results: 1. An empirical investigation of 86 open-type mutual funds during 1995 to 1998 with the methods of test, t test and the Logistic regression shows fund managers with poor performance would become aggressive and tend to increase fund volatility in the latter part of an annual assessment period. The effect is obviously clear toward the end of the year and it is involved with the investors' myopic of the assessment to the managers' performances. 2. In our model, the bonus is similar to a call option on the funds' portfolio. Three types of incentive contracts are compared. The results show that the incentive contract with penalty can reduce managers to adversely alter the risk of the portfolio they manage. It dominates the pure incentive contract and only with a ceiling incentive contract. Questionnaires investigated by fund managers will support some hypothesis.

[pt] No novo modelo para o setor elétrico brasileiro a operação do sistema é de responsabilidade do Operador Independente do Sistema enquanto que a propriedade do equipamento e os gastos com a sua manutenção são de responsabilidade do agente prestador do serviço ancilar.Desta forma, o lucro do agente prestador do serviço ancilar de potência reativa será função do custo deste serviço uma vez que a sua remuneração é previamente definida pelo agente regulador. Dentro desta ótica os riscos operacionais dos equipamentos de compensação de potência reativa, se não forem convenientemente mitigados,poderão aumentar os gastos com a manutenção e o pagamento de multa pela indisponibilidade do equipamento.Riscos operacionais são situações de sistema que estão fora da responsabilidade do agente mas que poderão reduzir o seu faturamento.Esta dissertação descreve os riscos operacionais e faz algumas sugestões para proteger o agente da redução do seu faturamento. / [en] In the new brazilian deregulated electrical network, system operation is under Independent System Operator responsibility while the ancillary service agent is the owner of the reactive power equipment and, as a consequence, is responsible for its maintenance costs. In the new model, the profit of the agent is a function of the maintenance costs because his payment is previously defined by the authorities. The operational risks of the reactive power equipments, if not conveniently mitigated, may increase the maintenance frequency and may also result in the payment of a fee due to the equipment unavailability.Operational risks are events in the electrical transmission system that are not under the responsibility of the agent but may reduce his profits. This report describes these operational risks and makes some suggestions to protect the agent against the reduction in his profits.

Optioner säljs och köps idag flitigt av många olika anledningar. En av dessa kan vara spekulation kring framtida händelser för aktiepriser där optioner har fördelar jämfört med aktier i form av en hävstångseffekt. En annan anledning för optionshandel är för att hedga (säkra) risker vilket ställer krav på att innehavet av optionen ska kompensera den negativa effekt som riskerna bidrar till. Med andra ord, om det finns en risk för ett negativt framtida scenario som man inte vill riskera att utsätta sig för så kan optioner vara rätt verktyg att använda sig av.   Risker finns idag överallt, i olika former, vilket har bidragit till att efterfrågan av optioner har ökat enormt de senaste årtiondena. Dock kan risker vara både komplexa och varierande vilket har lett till att mer komplexa optioner har utvecklats för att mätta den efterfrågan som utvecklats på marknaden. Dessa, mer komplexa optioner, kallas exotiska optioner och de skiljer sig från de vanliga europeiska och amerikanska köp- och säljoptionerna. Däribland hittar vi bland annat lookback-optioner i form av bland annat köpoptioner på maximum och kvantiloptioner vilka är två av de huvudsakliga optionerna som diskuteras i uppsatsen.   Det har länge varit känt hur man prissätter europeiska köp- och säljoptioner via Black-Scholes-Mertons modell men desto fler komplexa optioner som tillkommer på marknaden desto mer komplicerade prissättningsmodeller utvecklas. Till skillnad från europeiska köp- och säljoptioner vars utdelning beror på aktiepriset på lösendagen så är lookback-optioner beroende av aktieprisets rörelse under hela kontraktstiden. Detta medför att prissättningen av dessa beror av fler parametrar än i Black-Scholes-Mertons modell, bland annat ockupationstiden för den stokastiska process som beskriver aktiepriset, vilket bidrar till andra prissättningsmodeller.   Uppsatsen har som syfte att redogöra för modellen som används vid prissättningen av kvantiloptioner samt presentera hur deras egenskaper förhåller sig till andra typer av lookback-optioners egenskaper. Det presenteras i rapporten att kvantiloptioner liknar vissa typer av lookback-optioner, mer bestämt köpoptioner på maximum, och att kvantiloptioners egenskaper faktiskt konvergerar mot köpoptioner på maximums egenskaper då kvantilen närmar sig 1. Utifrån detta resonemang så kan det finnas fördelar i att använda kvantiloptioner snarare än köpoptioner på maximum vilket investerare bör ta i hänsyn när, och om, kvantiloptioner introduceras på marknaden. / Options are today used by investors for multiple reasons. One of these are speculation about future market movements, here ownership of options is advantageous over usual ownership of shares in the underlying stock in terms of a leverage effect. Furthermore, investors use options to hedge different kinds of risks that they are exposed to, this demands that the option compensates the possible negative effect that the risk brings to the table. In other words, if there is a risk of a future negative scenario which the investor is risk averse to, then owning specific options which neutralize this risk could be the perfect tool to use.   Risks are today seen all over the market in different shapes which have created a great demand for options over the last decades. However, since risks can be both complex and range over multiple business areas, investors have demanded more complex options which can neutralize the risk exposures. These, more complex options, are called exotic options, and they differ from the regular American and European options in the way they behave with respect to the underlying stock. Amongst these exotic options, we can find different kind of lookback options as well as quantile options which are two of the main options that are discussed in this thesis.   It has been known for a while how to price European call and put options by the Black-Scholes-Merton model. However, with more complex options also comes more complex pricing models and unlike the European options’ payoff which depend on the underlying stock price at time of maturity, the lookback option’s and quantile option’s payoff depend on the stock price movement over the total life span of the option contract. Hence, the pricing of these options depends on more variables than the classic Black-Scholes-Merton model include. One of these variables is the occupation time of the stochastic process which describes the stock price movement, this leads to a more complex and extensive pricing model than the general Black-Scholes-Merton’s model.   The objective of this thesis is to derive the pricing model that is used for quantile options and prove that the properties of quantile options are advantageous when compared to some specific lookback options, viz. call options on maximum.  It is concluded in the thesis that quantile options in fact converges to the call option on maximum for quantiles approaching 1. However, quantile options come with some different properties which potentially makes them a good substitute for the call option on maximum. This is a relevant factor for investors to consider when, and if, quantile options are introduced to the market.

In this thesis, important theories in financial mathematics will be explained and derived. These theories will later be used to value financial derivatives. An analytical formula for valuing European call and put option will be derived and European call options will be valued under the Black-Scholes partial differential equation using three different finite difference methods. The Crank-Nicholson method will then be used to value American call options and solve their corresponding free boundary value problem. The optimal exercise boundary can then be plotted from the solution of the free boundary value problem. The algorithm for valuing American call options will then be further developed to solve the stock loan problem. This will be achieved by exploiting a link that exists between American call options and stock loans. The Crank-Nicholson method will be used to value stock loans and their corresponding free boundary value problem. The optimal exit boundary can then be plotted from the solution of the free boundary value problem. The results that are obtained from the numerical calculations will finally be used to discuss how different parameters affect the valuation of American call options and the valuation of stock loans. In the end of the thesis, conclusions about the effect of the different parameters on the optimal prices will be presented. / I det här kandidatexamensarbetet kommer fundamentala teorier inom finansiell matematik förklaras och härledas. Dessa teorier kommer lägga grunden för värderingen av finansiella derivat i detta arbete. En analytisk formel för att värdera europeiska köp- och säljoptioner kommer att härledas. Dessutom kommer europeiska köpoptioner att värderas numeriskt med tre olika finita differensmetoder. Den finita differensmetoden Crank-Nicholson kommer sedan användas för att värdera amerikanska köpoptioner och lösa det fria gränsvärdesproblemet (free boundary value problem). Den optimala omvandlingsgränsen (Optimal Exercise Boundary) kan därefter härledas från det fria gränsvärdesproblemet. Algoritmen för att värdera amerikanska köpoptioner utökas därefter till att värdera lån med aktier som säkerhet. Detta kan åstadkommas genom att utnyttja ett samband mellan amerikanska köpoptioner med lån där aktier används som säkerhet. Den finita differensmetoden Crank-Nicholson kommer dessutom att användas för att värdera lån med aktier som säkerhet. Den optimala avyttringsgränsen (Optimal Exit Boundary) kan därefter härledas från det fria gränsvärdesproblemet. Resultaten från de numeriska beräkningarna kommer slutligen att användas för att diskutera hur olika parametrar påverkar värderingen av amerikanska köpoptioner, samt värdering av lån med aktier som säkerhet. Avslutningsvis kommer slutsatser om effekterna av dessa parametrar att presenteras.