Pricing American options has always been problematic due to its early exercise characteristic. As no closed-form analytical solution for any of the widely used models exists, many numerical approximation methods have been proposed and studied. In this thesis, we investigate the Least-Square Monte Carlo Simulation (LSMC) method of Longstaff & Schwartz for pricing American options under the two-dimensional Heston model. By conducting extensive numerical experimentation, we put the LSMC to test and investigate four different continuation functions for the LSMC. In addition, we consider investigating seven different combination of Heston model parameters. We analyse the results and select the optimal continuation function according to our criteria. Then we uncover and study the early exercise boundary foran American put option upon changing initial volatility and other parameters of the Heston model.

The calibration of model parameters is a crucial step in the process of valuation of complex derivatives. It consists of choosing the model parameters that correspond to the implied market data especially the call and put prices. We discuss in this thesis the calibration strategy for the Heston model, one of the most used stochastic volatility models for pricing complex derivatives. The main problem with this model is that the asset price does not have a known probability distribution function. Thus we use either Fourier expansions through its characteristic function or Monte Carlo simulations to have access to it. We hence discuss the approximation induced by these methods and elaborate a calibration strategy with a focus on the choice of the objective function and the choice of inputs for the calibration. We assess that the put option prices are a better input than the call prices for the optimization function. Then through a set of experiments on simulated put prices, we find that the sum of squared error performs better choice of the objective function for the differential evolution optimization. We also establish that the put option prices where the Black Scholes delta is equal to 10\%, 25\%, 50\% 75\% and 90\% gives enough in formations on the implied volatility surface for the calibration of the Heston model. We then implement this calibration strategy on real market data of Eurostoxx50 Index and observe the same distribution of errors as in the set of experiments. / Kalibreringen av modellparametrar är ett viktigt steg i värderingen av komplexa derivat. Den består av att välja modellparametrar som motsvarar de implicita marknadsdata, särskilt köp- och säljpriserna. I denna avhandling diskuterar vi kalibreringsstrategin för Hestonmodellen, en av de mest använda modellerna för stokastisk volatilitet för prissättning av komplexa derivat. Huvudproblemet med denna modell är att tillgångspriset inte har en känd sannolikhetsfördelningsfunktion. Därför använder vi antingen Fourier-expansioner genom dess karakteristiska funktion eller Monte Carlo-simuleringar för att få tillgång till den. Vi diskuterar därför den approximation som dessa genereras av dessa metoder och utarbetar en kalibreringsstrategi med fokus på valet av målfunktion och valet av indata för kalibreringen. Vi bedömer att säljoptionspriserna är en bättre input än samtalspriserna för differentialutvecklingsoptimeringsfunktionen. Genom flera experiment med simulerade säljpriser finner vi sedan att summan av kvadratfel ger bättre val av objektivfunktionen för differentialutvecklingsoptimering. Vi konstaterar också att säljoptionspriserna där Black Scholes deltat är lika med 10\%, 25\%, 50\%, 75\% och 90\% ger tillräcklig information om den implicita volatilitetsytan för kalibrering av Hestonmodellen. Vi tillämpar sedan denna kalibreringsstrategi på verkliga marknadsdata för Eurostoxx50-indexet och observerar samma felfördelning som i experimenten.

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

The aim of this diploma thesis is to analyze and implement selected option pricing models using statistical software. The first chapter introduces theoretical basics of options as financial instruments ideal for hedging and speculation. The second chapter constitutes the core part of this thesis since it unveils theoretical concepts of risk-neutral pricing and at the same time analyze some basic, as well as highly sophisticated option pricing models. In addition, each model is accompanied by a practical example of their effective implementation. The final chapter characterize the most widely used option trading strategies and defines the ideal expected market development linked to each strategy.

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.

This thesis investigated Stochastic Runge–Kutta Lawson (SRKL) schemes and their application to the Heston model. Two distinct SRKL discretization methods were used to simulate a single asset’s dynamics under the Heston model, notably the Euler–Maruyama and Midpoint schemes. Additionally, standard Monte Carlo and variance reduction techniques were implemented. European and Asian option prices were estimated and compared with a benchmark value regarding accuracy, effectiveness, and computational complexity. Findings showed that the SRKL Euler–Maruyama schemes exhibited promise in enhancing the price for simple and path-dependent options. Consequently, integrating SRKL numerical methods into option valuation provides notable advantages by addressing challenges posed by the Heston model’s SDEs. Given the limited scope of this research topic, it is imperative to conduct further studies to understand the use of SRKL schemes within other models.

資本市場之系統性風險加劇時，對於利率變動型人壽保險所持有之區隔資產將出現大幅波動，進而影響保險公司之清償能力，本研究透過建立區隔資產負債表之隨機模型，檢視系統性風險下對於人壽保險業違約風險之變化，並透過敏感度分析找出對違約風險影響最大的因子。 本研究依據利率變動型壽險之現金流量建立公司之資產負債模型，預期建立Heston (1993)模型描述標的資產的隨機波動過程，相較於以往Black-Scholes (1973)模型更能反映真實的市場波動。本研究藉由資產與負債的變化，衡量保險公司違約風險，同時分析影響違約風險之各項因子，包含解約、死亡與資產配置策略之關聯性。本研究結果顯示，宣告利率、評價時間長度及資產配置策略等皆會影響保險公司之違約風險及其破產幅度。 / When systemic risk of capital markets exacerbates, the segment assets that held by interest sensitive life insurance policies will fluctuate widely and affect insurer's solvency. This paper considers the problem of valuating the default risk of the life insurers under systematic risk, by constructing a stochastic model of segment balance sheet. In this paper, we establish insurer's asset-liability model on the basis of interest sensitive life insurance policies' cash flow.In particular, we use Heston(1993) model to simulate stochastic process of assets, which is better reflect market volatility than Black-Scholes(1973) model in reality. And moreover, by means of the variation on asset and liability, this study evaluating the default risk of life insurers and analyze the factors affect default risk, like the correlation between surrender, death and asset allocation. And using the result of sensitivity analysis to determine which factor is more important, like guaranteed rate, time period of valuation and so on.

Les titres financiers sont souvent modélisés par des équations différentielles stochastiques (ÉDS). Ces équations peuvent décrire le comportement de l'actif, et aussi parfois certains paramètres du modèle. Par exemple, le modèle de Heston (1993), qui s'inscrit dans la catégorie des modèles à volatilité stochastique, décrit le comportement de l'actif et de la variance de ce dernier. Le modèle de Heston est très intéressant puisqu'il admet des formules semi-analytiques pour certains produits dérivés, ainsi qu'un certain réalisme. Cependant, la plupart des algorithmes de simulation pour ce modèle font face à quelques problèmes lorsque la condition de Feller (1951) n'est pas respectée. Dans ce mémoire, nous introduisons trois nouveaux algorithmes de simulation pour le modèle de Heston. Ces nouveaux algorithmes visent à accélérer le célèbre algorithme de Broadie et Kaya (2006); pour ce faire, nous utiliserons, entre autres, des méthodes de Monte Carlo par chaînes de Markov (MCMC) et des approximations. Dans le premier algorithme, nous modifions la seconde étape de la méthode de Broadie et Kaya afin de l'accélérer. Alors, au lieu d'utiliser la méthode de Newton du second ordre et l'approche d'inversion, nous utilisons l'algorithme de Metropolis-Hastings (voir Hastings (1970)). Le second algorithme est une amélioration du premier. Au lieu d'utiliser la vraie densité de la variance intégrée, nous utilisons l'approximation de Smith (2007). Cette amélioration diminue la dimension de l'équation caractéristique et accélère l'algorithme. Notre dernier algorithme n'est pas basé sur une méthode MCMC. Cependant, nous essayons toujours d'accélérer la seconde étape de la méthode de Broadie et Kaya (2006). Afin de réussir ceci, nous utilisons une variable aléatoire gamma dont les moments sont appariés à la vraie variable aléatoire de la variance intégrée par rapport au temps. Selon Stewart et al. (2007), il est possible d'approximer une convolution de variables aléatoires gamma (qui ressemble beaucoup à la représentation donnée par Glasserman et Kim (2008) si le pas de temps est petit) par une simple variable aléatoire gamma. / Financial stocks are often modeled by stochastic differential equations (SDEs). These equations could describe the behavior of the underlying asset as well as some of the model's parameters. For example, the Heston (1993) model, which is a stochastic volatility model, describes the behavior of the stock and the variance of the latter. The Heston model is very interesting since it has semi-closed formulas for some derivatives, and it is quite realistic. However, many simulation schemes for this model have problems when the Feller (1951) condition is violated. In this thesis, we introduce new simulation schemes to simulate price paths using the Heston model. These new algorithms are based on Broadie and Kaya's (2006) method. In order to increase the speed of the exact scheme of Broadie and Kaya, we use, among other things, Markov chains Monte Carlo (MCMC) algorithms and some well-chosen approximations. In our first algorithm, we modify the second step of the Broadie and Kaya's method in order to get faster schemes. Instead of using the second-order Newton method coupled with the inversion approach, we use a Metropolis-Hastings algorithm. The second algorithm is a small improvement of our latter scheme. Instead of using the real integrated variance over time p.d.f., we use Smith's (2007) approximation. This helps us decrease the dimension of our problem (from three to two). Our last algorithm is not based on MCMC methods. However, we still try to speed up the second step of Broadie and Kaya. In order to achieve this, we use a moment-matched gamma random variable. According to Stewart et al. (2007), it is possible to approximate a complex gamma convolution (somewhat near the representation given by Glasserman and Kim (2008) when T-t is close to zero) by a gamma distribution.

Les titres financiers sont souvent modélisés par des équations différentielles stochastiques (ÉDS). Ces équations peuvent décrire le comportement de l'actif, et aussi parfois certains paramètres du modèle. Par exemple, le modèle de Heston (1993), qui s'inscrit dans la catégorie des modèles à volatilité stochastique, décrit le comportement de l'actif et de la variance de ce dernier. Le modèle de Heston est très intéressant puisqu'il admet des formules semi-analytiques pour certains produits dérivés, ainsi qu'un certain réalisme. Cependant, la plupart des algorithmes de simulation pour ce modèle font face à quelques problèmes lorsque la condition de Feller (1951) n'est pas respectée. Dans ce mémoire, nous introduisons trois nouveaux algorithmes de simulation pour le modèle de Heston. Ces nouveaux algorithmes visent à accélérer le célèbre algorithme de Broadie et Kaya (2006); pour ce faire, nous utiliserons, entre autres, des méthodes de Monte Carlo par chaînes de Markov (MCMC) et des approximations. Dans le premier algorithme, nous modifions la seconde étape de la méthode de Broadie et Kaya afin de l'accélérer. Alors, au lieu d'utiliser la méthode de Newton du second ordre et l'approche d'inversion, nous utilisons l'algorithme de Metropolis-Hastings (voir Hastings (1970)). Le second algorithme est une amélioration du premier. Au lieu d'utiliser la vraie densité de la variance intégrée, nous utilisons l'approximation de Smith (2007). Cette amélioration diminue la dimension de l'équation caractéristique et accélère l'algorithme. Notre dernier algorithme n'est pas basé sur une méthode MCMC. Cependant, nous essayons toujours d'accélérer la seconde étape de la méthode de Broadie et Kaya (2006). Afin de réussir ceci, nous utilisons une variable aléatoire gamma dont les moments sont appariés à la vraie variable aléatoire de la variance intégrée par rapport au temps. Selon Stewart et al. (2007), il est possible d'approximer une convolution de variables aléatoires gamma (qui ressemble beaucoup à la représentation donnée par Glasserman et Kim (2008) si le pas de temps est petit) par une simple variable aléatoire gamma. / Financial stocks are often modeled by stochastic differential equations (SDEs). These equations could describe the behavior of the underlying asset as well as some of the model's parameters. For example, the Heston (1993) model, which is a stochastic volatility model, describes the behavior of the stock and the variance of the latter. The Heston model is very interesting since it has semi-closed formulas for some derivatives, and it is quite realistic. However, many simulation schemes for this model have problems when the Feller (1951) condition is violated. In this thesis, we introduce new simulation schemes to simulate price paths using the Heston model. These new algorithms are based on Broadie and Kaya's (2006) method. In order to increase the speed of the exact scheme of Broadie and Kaya, we use, among other things, Markov chains Monte Carlo (MCMC) algorithms and some well-chosen approximations. In our first algorithm, we modify the second step of the Broadie and Kaya's method in order to get faster schemes. Instead of using the second-order Newton method coupled with the inversion approach, we use a Metropolis-Hastings algorithm. The second algorithm is a small improvement of our latter scheme. Instead of using the real integrated variance over time p.d.f., we use Smith's (2007) approximation. This helps us decrease the dimension of our problem (from three to two). Our last algorithm is not based on MCMC methods. However, we still try to speed up the second step of Broadie and Kaya. In order to achieve this, we use a moment-matched gamma random variable. According to Stewart et al. (2007), it is possible to approximate a complex gamma convolution (somewhat near the representation given by Glasserman and Kim (2008) when T-t is close to zero) by a gamma distribution.

Submitted by Cristiane Shirayama (cristiane.shirayama@fgv.br) on 2011-06-03T16:41:12Z No. of bitstreams: 1 66080100251.pdf: 1071566 bytes, checksum: 633248672cb6ac94f704bfeda06b29d3 (MD5) / Approved for entry into archive by Vera Lúcia Mourão(vera.mourao@fgv.br) on 2011-06-03T16:46:36Z (GMT) No. of bitstreams: 1 66080100251.pdf: 1071566 bytes, checksum: 633248672cb6ac94f704bfeda06b29d3 (MD5) / Approved for entry into archive by Vera Lúcia Mourão(vera.mourao@fgv.br) on 2011-06-03T17:00:17Z (GMT) No. of bitstreams: 1 66080100251.pdf: 1071566 bytes, checksum: 633248672cb6ac94f704bfeda06b29d3 (MD5) / Made available in DSpace on 2011-06-03T18:49:55Z (GMT). No. of bitstreams: 1 66080100251.pdf: 1071566 bytes, checksum: 633248672cb6ac94f704bfeda06b29d3 (MD5) Previous issue date: 2011-02-11 / Nos últimos anos, o mercado brasileiro de opções apresentou um forte crescimento, principalmente com o aparecimento da figura dos High Frequency Traders (HFT) em busca de oportunidades de arbitragem, de modo que a escolha adequada do modelo de estimação de volatilidade pode tornar-se um diferencial competitivo entre esses participantes. Este trabalho apresenta as vantagens da adoção do modelo de volatilidade estocástica de Heston (1993) na construção de superfície de volatilidade para o mercado brasileiro de opções de dólar, bem como a facilidade e o ganho computacional da utilização da técnica da Transformada Rápida de Fourier na resolução das equações diferenciais do modelo. Além disso, a partir da calibração dos parâmetros do modelo com os dados de mercado, consegue-se trazer a propriedade de não-arbitragem para a superfície de volatilidade. Os resultados, portanto, são positivos e motivam estudos futuros sobre o tema. / In recent years, the Brazilian option market has grown considerable, especially with the emergence of the High Frequency Traders (HFT) in search of arbitrage opportunities, so that the appropriate choice of a volatility estimation model should become a competitive differentiator among these participants. This paper presents the advantages of adopting the Heston stochastic volatility model on the construction of the volatility surface for the Brazilian US Dollar option market, as well as the easiness and the computational gain by applying the Fast Fourier Transform technique on the models differential equations resolution. Furthermore, from calibration of the model parameters to market data, it is possible to bring the no-arbitrage property to the volatility surface. The results, therefore, are positive and motivate further studies on the subject.