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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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Girsanov Theorem
Navrátil, Robert ; Šnupárková, Jana (advisor) ; Maslowski, Bohdan (referee)
Girsanov Theorem Bachelor's thesis - Robert Navrátil Abstract Modern theory of probability and financial mathematics require the theory of stochastic calculus. Its foundations contain Wiener process (Brownian motion) and the integral of stochastic process with respect to another stochastic process. This thesis deals with building the mathematical theory needed to construct the stochastic integral, with the construction itself, the Girsanov Theorem and its applications. The Girsanov Theorem uses equivalent probability measure to transform Wiener process with drift to Wiener process without drift. Using the Girsanov Theorem, we change our measure to the equivalent risk neutral measure and we deduce Black-Scholes formula which estimates the prize of European call option with underlying stock asset. The stock prize is modelled using the geometric Brownian motion. Finally, we demonstrate, on real life data, how this model works and what are its outcomes. 1
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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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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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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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History of mathematical modelling on financial markets
Cigán, Martin ; Brada, Jaroslav (advisor) ; Langer, Miroslav (referee)
The main goal of this thesis is to introduce the reader to the evolution of some of the well-known mathematical models used in the valuation of investment instruments. The first chapter deals with some of the basic terms used in the following text. The next chapters introduce mathematical models, which are used to valuate stocks, bonds and derivatives. Each chapter contains also a brief description of the instrument itself and in some cases the methods used to evaluate the instruments before the introduction of models. The thesis contains a chapter on concept of portfolio due to its importance in the development of mathematical modelling in this field.
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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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