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Continuous processes with quadratic varaition
Svoboda, Miroslav ; Dostál, Petr (advisor) ; Dvořák, Jiří (referee)
The work is devoted to the properties of the continuous random processes with a compact index set that are having finite quadratic variation. In the thesis we define the stochastic Riemannn integral and then follow a development of a theory leading to deriving of Ito formula. The terms, concretely quadratic variation and Ito's formula and in the process are introduced using the konvergence in probability for the continuous random processes. The applied part of the thesis, starting in chapter 6, is considering an investor trading on the stock market. Using the Ito formula we will show that both the Black-Sholes and the bachelier models are modelling the fair price of the European call vanilla option, when the price of the share on the market is modelled by. Powered by TCPDF (www.tcpdf.org)
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Black-Scholes models of option pricing
Čekal, Martin ; Maslowski, Bohdan (advisor) ; Beneš, Viktor (referee)
Title: Black-Scholes Models of Option Pricing Author: Martin Cekal Department: Department of Probability and Mathematical Statistics Supervisor: prof. RNDr. Bohdan Maslowski, DrSc., Charles University in Prague, Faculty of Mathematics and Physics, Department of Probability and Mathematical Statistics. Abstract: In the present master thesis we study a generalization of Black-Scholes model using fractional Brownian motion and jump processes. The main goal is a derivation of the price of call option in a fractional jump market model. The first chapter introduces long memory and its modelling by discrete and continuous time models. In the second chapter fractional Brownian motion is defined, appropriate stochastic analysis is developed and we generalize the notion of Lévy and jump processes. The third chapter introduces fractional Black-Scholes model. In the fourth chapter, tools developed in the second chapter are used for the construction of jump fractional Black-Scholes model and derivation of explicit formula for the price of european call option. In the fifth chapter, we analyze long memory contained in simulated and empirical time series. Keywords: Black-Scholes model, fractional Brownian motion, fractional jump process, long- memory, options pricing.
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Black-Scholes models of option pricing
Čekal, Martin
Title: Black-Scholes Models of Option Pricing Author: Martin Cekal Department: Department of Probability and Mathematical Statistics Supervisor: prof. RNDr. Bohdan Maslowski, DrSc., Charles University in Prague, Faculty of Mathematics and Physics, Department of Probability and Mathematical Statistics. Abstract: In the present master thesis we study a generalization of Black-Scholes model using fractional Brownian motion and jump processes. The main goal is a derivation of the price of call option in a fractional jump market model. The first chapter introduces long memory and its modelling by discrete and continuous time models. In the second chapter fractional Brownian motion is defined, appropriate stochastic analysis is developed and we generalize the notion of Lévy and jump processes. The third chapter introduces fractional Black-Scholes model. In the fourth chapter, tools developed in the second chapter are used for the construction of jump fractional Black-Scholes model and derivation of explicit formula for the price of european call option. In the fifth chapter, we analyze long memory contained in simulated and empirical time series. Keywords: Black-Scholes model, fractional Brownian motion, fractional jump process, long- memory, options pricing.
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Valuation of financial derivatives
Matušková, Radka ; Hurt, Jan (advisor) ; Zichová, Jitka (referee)
In the present thesis we deal with several possible approaches to financial de- rivatives pricing. In the first part, we introduce the basic types of derivatives and the methods of trading. Furthermore, we present several models for the valuati- on of specific financial derivative, i.e. options. Firstly we describe Black-Scholes model in detail, which considers that the development of the underlying asset price is governed by Wiener process. Following are the jumps diffusion models that are extension of the Black-Scholes model with jumps. Then we get to jump models, which are based on Lévy processes. Finally, we will deal with the model, which considers that the development of the underlying asset price is governed by fractional Brownian motion with Hurst's coefficient greater than 1/2. All models are suplemented with sample examples. 1
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Implementation of Statistical Functions Using HLS
Šinaľ, Peter ; Martínek, Tomáš (referee) ; Dvořák, Milan (advisor)
The aim of this thesis was to design and implement selected statistical functions used in technical analysis. I focused on moving averages, Black-Schles model for calculating option prices and Indicator Delta. These functions are through HLS transformed into an appropriate description for programmable FPGA. During the transformation process, emphasis is on low latency and resource consumption. Created solutions demonstrate the potential of HLS. They show complexity of the technical analysis and hardware requirements. Achieved results show high accuracy in the simulations. Deviation from the reference value is approximately 6,615*10e-3. The results also indicate thet that reducing latency does not necessarily cause an increase in the consumption of resources on the chip.
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A comparison of the Black-Scholes model with the Heston model
Obhlídal, Jiří ; Málek, Jiří (advisor) ; Fičura, Milan (referee)
The thesis focuses on methods of option prices calculations using two different pricing models which are Heston and Black-Scholes models. The first part describes theory of these two models and conlcudes with a comparison of the risk-neutral measures of these two models. In the second part, the relations between input parameters and the option price generated by these models are clarified. This part ends up with an analysis of the market data and it answers the question which model predicts better.
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Stochastic equations and numerical solution of pricing option model
Janečka, Adam ; Jablonský, Josef (advisor) ; Pelikán, Jan (referee)
In the present work, we study the topic of stochastic differential equations, their numerical solution and solution of models for pricing of options which follow from stochastic differential equations using the Itô calculus. We present several numerical methods for solving stochastic differential equations. These methods are then implemented in MATLAB and we investigate their properties, especially their convergence characteristics. Furthermore, we formulate two models for pricing of European call options. We solve these models using a variant of the spectral collocation method, again in MATLAB.
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Option pricing with stochastic volatility
Bartoň, Ľuboš ; Málek, Jiří (advisor) ; Witzany, Jiří (referee)
This diploma thesis deals with problem of option pricing with stochastic volatility. At first, the Black-Scholes model is derived and then its biases are discussed. We explain shortly the concept of volatility. Further, we introduce three pricing models with stochastic volatility- Hull-White model, Heston model and Stein-Stein model. At the end, these models are reviewed.
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Option Pricing and Variance Gamma Process
Moravec, Radek ; Málek, Jiří (advisor) ; Paholok, Igor (referee)
The submitted work deals with option pricing. Mathematical approach is immediately followed by an economic interpretation. The main problem is to model the underlying uncertainities driving the stock price. Using two well-known valuation models, binomial model and Black-Scholes model, we explain basic principles, especially risk neutral pricing. Due to the empirical biases new models have been developped, based on pure jump process. Variance gamma process and its special symmetric case are presented.
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The analysis of particular models of credit risk
Sedlárová, Michala ; Málek, Jiří (advisor) ; Polák, Michal (referee)
The main aim of my final thesis is to familiar reader with different ways of measuring credit risk by means of particular structural models of credit risk. This issue has been already described by foreign authors. Though, neither Czech nor Slovak economists have been deeply involved in this topic so far. For this reason, I have decided to focus on those models and both describe them as well as put them into the practice. My final thesis gradually focus on individual detailed model description in each chapter in following sequence: Credit Metrics, Black-School model, Merton model, KMV, Credit Grades. Moreover, it also targets model's construction as well as practical application. Regarding practical model's application, Black-School model is applied on IBM and KMV on Kraft Foods Company. Admittedly, that proves the fact that structural models are not only theoretical models, but also practical models applyable on real companies. Finally, I will compare all above mentioned models in selected parameters.
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