Most of the derivative pricing models are developed in the jump diffusion models, and many literatures assume those jumps are diversifiable. However, we find many risk cannot be avoided through diversification. In this paper, we extend the Jarrow and Yildirim model to consider the existence of systematic jump risk in nominal interest rate, real interest rate and inflation rate to derive the no-arbitrage condition by using Esscher transformation. In addition, this study also derives the value of TIPS and TIPS European call option. Furthermore, we use the econometric theory to decompose TIPS market price volatility into a continuous component and a jump component. We find the jump component contribute most of the TIPS market price volatility. In addition, we also use the TIPS yield index to obtain the systematic jump component and systematic continuous component to find the systematic jump beta and the systematic continuous beta. The results show that the TIPS with shorter time to maturity are more vulnerable to systematic jump risk. In contrast, the individual TIPS with shorter time to maturity is more vulnerable to systematic jump. Finally, the sensitive analysis is conducted to detect the impacts of jumps risk on the value of TIPS European call option.

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.

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.

Chen, Chang, Wen and Lin (2013)提出馬可夫調控跳躍過程模型(MMJDM)描述股價指數報酬率，布朗運動項、跳躍項之頻率與市場狀態有關。然而，利率並非常數，本論文以狀態轉換模型配適零息債劵之動態過程，提出狀態轉換下的利率與具跳躍風險的股票報酬之二維模型(MMJDMSI)，並以1999年至2013年的道瓊工業指數與S&P 500指數和同期間之一年期美國國庫劵價格為實證資料，採用EM演算法取得參數估計值。經由概似比檢定結果顯示無論道瓊工業指數還是S&P 500指數，狀態轉換下利率與跳躍風險之股票報酬二維模型更適合描述報酬率。接著，利用Esscher轉換法推導出各模型下的股價指數之歐式買權定價公式，再對MMJDMSI模型進行敏感度分析以評估模型參數發生變動時對於定價公式的影響。最後，以實證資料對各模型進行模型校準及計算隱含波動度，結果顯示MMJDMSI在價內及價外時定價誤差為最小或次小，且此模型亦能呈現出波動度微笑曲線之現象。 / To model asset return, Chen, Chang, Wen and Lin (2013) proposed Markov-Modulated Jump Diffusion Model (MMJDM) assuming that the Brownian motion term and jump frequency are all related to market states. In fact, the interest rate is not constant, Regime-Switching Model is taken to fit the process of the zero-coupon bond price, and a bivariate model for interest rate and stock index return with regime-switching and dependent jump risks (MMJDMSI) is proposed. The empirical data are Dow Jones Industrial Average and S&P 500 Index from 1999 to 2013, together with US 1-Year Treasury Bond over the same period. Model parameters are estimated by the Expectation-Maximization (EM) algorithm. The likelihood ratio test (LRT) is performed to compare nested models, and MMJDMSI is better than the others. Then, European call option pricing formula under each model is derived via Esscher transformation, and sensitivity analysis is conducted to evaluate changes resulted from different parameter values under the MMJDMSI pricing formula. Finally, model calibrations are performed and implied volatilities are computed under each model empirically. In cases of in-the-money and out-the-money, MMJDMSI has either the smallest or the second smallest pricing error. Also, the implied volatilities from MMJDMSI display a volatility smile curve.

為了改進Black-Scholes模式的實證現象，許多其他的模型被建議有leptokurtic特性以及波動度聚集的現象。然而對於其他的模型分析的處理依然是一個問題。在本論文中，我們建議使用馬可夫跳躍擴散過程，不僅能整合leptokurtic與波動度微笑特性，而且能產生波動度聚集的與長記憶的現象。然後，我們應用Lucas的一般均衡架構計算選擇權價格，提供均衡下當跳躍的大小服從一些特別的分配時則選擇權價格的解析解。特別地，考慮當跳躍的大小服從兩個情況，破產與lognormal分配。當馬可夫跳躍擴散模型的馬可夫鏈有兩個狀態時，稱為轉換跳躍擴散模型，當跳躍的大小服從lognormal分配我們得到選擇權公式。使用轉換跳躍擴散模型選擇權公式，我們給定一些參數下研究公式的數值極限分析以及敏感度分析。 / To improve the empirical performance of the Black-Scholes model, many alternative models have been proposed to address the leptokurtic feature of the asset return distribution, and the effects of volatility clustering phenomenon. However, analytical tractability remains a problem for most of the alternative models. In this dissertation, we propose a Markov jump diffusion model, that can not only incorporate both the leptokurtic feature and volatility smile, but also present the economic features of volatility clustering and long memory. Next, we apply Lucas's general equilibrium framework to evaluate option price, and to provide analytical solutions of the equilibrium price for European call options when the jump size follows some specific distributions. In particular, two cases are considered, the defaultable one and the lognormal distribution. When the underlying Markov chain of the Markov jump diffusion model has two states, the so-called switch jump diffusion model, we write an explicit analytic formula under the jump size has a lognormal distribution. Numerical approximations of the option prices as well as sensitivity analysis are also given.