In this thesis, we consider the pricing problem of an American put option. We introduce a new market model for the evolution of the underlying asset price. Our model adds a new parameter to the well known Heston model. Hence we name our model the extended Heston model. To solve the American put pricing problem we adapt the idea developed by Fouque et al. (2000) to derive the asymptotic formula. We then connect the idea developed by Medvedev and Scaillet (2010) to provide an asymptotic solution for the leading order term P0. We do numerical analysis to gain insight into the accuracy and validity of our asymptotic approximation formula.

The Heston stochastic volatility model is one extension of the Black-Scholes model which describes the money markets more accurately so that more realistic prices for derivative products are obtained. From the stochastic differential equation of the underlying financial product a partial differential equation (p.d.e.) for the value function of an option can be derived. This p.d.e. can be solved with the finite difference method (f.d.m.). The stability and consistency of the method is examined. Furthermore a boundary condition is proposed to reduce the numerical error. Finally a non uniform structured grid is derived which is fairly optimal for the numerical result in the most interesting point. / Das stochastische Volatilitaetsmodell von Heston ist eines der Erweiterungen des Black-Scholes-Modells. Von der stochastischen Differentialgleichung fuer den unterliegenden Prozess kann eine partielle Differentialgleichung fuer die Wertfunktion einer Option abgeleitet werden. Es wird die Loesung mittels Finiter Differenzenmethode untersucht (Konsistenz, Stabilitaet). Weiterhin wird eine Randbedingung und ein spezielles nicht-uniformes Netz vorgeschlagen, was zu einer starken Reduzierung des numerischen Fehlers der Wertfunktion in einem ganz bestimmten Punkt fuehrt.

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 Heston stochastic volatility model is one extension of the Black-Scholes model which describes the money markets more accurately so that more realistic prices for derivative products are obtained. From the stochastic differential equation of the underlying financial product a partial differential equation (p.d.e.) for the value function of an option can be derived. This p.d.e. can be solved with the finite difference method (f.d.m.). The stability and consistency of the method is examined. Furthermore a boundary condition is proposed to reduce the numerical error. Finally a non uniform structured grid is derived which is fairly optimal for the numerical result in the most interesting point. / Das stochastische Volatilitaetsmodell von Heston ist eines der Erweiterungen des Black-Scholes-Modells. Von der stochastischen Differentialgleichung fuer den unterliegenden Prozess kann eine partielle Differentialgleichung fuer die Wertfunktion einer Option abgeleitet werden. Es wird die Loesung mittels Finiter Differenzenmethode untersucht (Konsistenz, Stabilitaet). Weiterhin wird eine Randbedingung und ein spezielles nicht-uniformes Netz vorgeschlagen, was zu einer starken Reduzierung des numerischen Fehlers der Wertfunktion in einem ganz bestimmten Punkt fuehrt.

Introduction: This thesis presents a python implementation and evaluation of the Heston-Queue-Hawkes (HQH) model, a recent jump-diffusion model for pricing options. The model is capable of tracking options for a wide range of different underlying assets. The model is expected to perform better on Fourier-based fast pricing algorithms such as the COS Method, however in this thesis we’ll only look at Monte Carlo solvers for the HQH model. The type of option studied in this master’s thesis is European options, however, the implementation could be extended to other types of options.  Methodology: The methodology for evaluating the HQH model (in this paper) involves the use of a custom Monte Carlo simulation implemented in Python. The Monte Carlo method enables simulating multiple scenarios and provides reliable results across a variety of situations, making it an appropriate tool for evaluating the model's performance.  Evaluation: The HQH model is evaluated on ease of implementation in python and it’s general ability to reflect different market phenomena such as volatility in price movements.  Improvement: This thesis also investigates the possibility of improving the model or adding corrections, parameters, readjustments, or the like to the model to improve results. The aim is to enhance the model's usefulness, and this evaluation seeks to identify potential improvements.  Worth noting: The goal of this thesis is to align with the research interests of financial institutions and provide a practical, applied approach to evaluating options pricing models. The research presented in this thesis aims to mirror the type of projects that a company like Visigon may be requested to undertake by a bank (and engineering work in general). Additionally, the findings and methodology developed in this thesis aims to inform and contribute to future research in options pricing models which may help markets perform better.

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.

Este trabalho analisa três estratégias de hedge de opção, buscando identificar a importância da escolha da estratégia para a obtenção de um bom desempenho do hedge. O conceito de hedge é analisado de forma retrospectiva e uma teoria geral de hedge é apresentada. Em seguida são descritos alguns estudos comparativos de desempenho de estratégias de hedge de opção e suas metodologias de implementação. Para esta análise comparativa são selecionadas três estratégias de hedge de opção de compra do tipo européia: a primeira utiliza o modelo Black-Scholes-Merton de precificação de opções, a segunda utiliza uma solução de programação dinâmica para hedge dinâmico multiperiódico e a terceira utiliza um modelo GARCH para precificação de opções. As estratégias são comentadas e comparadas do ponto de vista de suas premissas teóricas e por meio de testes comparativos de desempenho. O desempenho das estratégias é comparado sob uma perspectiva dinâmicamente ajustada, multiperiódica e autofinanciável. Os dados para comparação de desempenho são gerados por simulação e o desempenho é avaliado pelos erros absolutos médios e erros quadráticos médios, resultantes na carteira de hedge. São feitas ainda considerações a respeito de alternativas de estimação e suas implicações no desempenho das estratégias. / This work analyzes three option hedging strategies, to identify the importance of choosing a strategy in order to achieve a good hedging performance. A retrospective analysis of the concept of hedging is conducted and a general hedging theory is presented. Following, some comparative papers of hedging performance and their implementation methodologies are described. For the present comparative analysis, three hedging strategies for European options have been selected: the first one based on the Black-Scholes-Merton model for option pricing, the second one based on a dynamic programming solution for dynamic multiperiod hedging and the third one based on a GARCH model for option pricing. The strategies are compared under their theoric premisses and through comparative performance testes. The performances of the strategies are compared under a dynamically adjusted multiperiodic and self-financing perspective. Data for performance comparison are generated by simulation and performance is evaluated by mean absolute errors and mean squared errors resulting on the hedging portfolio. An analysis is also done regarding estimation approaches and their implications over the performance of the strategies.