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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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Pricing Options Using Monte Carlo Simulation
Dutton, Ryan ; Dědek, Oldřich (advisor) ; Červinka, Michal (referee)
Monte Carlo simulation is a valuable tool in computational finance. It is widely used to evaluate portfolio management rules, to price derivatives, to simulate hedging strategies, and to estimate Value at Risk. The purpose of this thesis is to develop the mathematical foundation and an algorithmic structure to carry out Monte Carlo simulation to price a European call option, investigate Black-Scholes model to look into the parallel between Monte Carlo simulation and Black-Scholes model, provide a solution for Black-Scholes model using Lognormal distribution of a stock price rather than solving Black-Scholes original partial differential equation, and finally compare the results of Monte Carlo simulation with Black- Scholes closed-form formula. Author's contribution can be best described as developing the mathematical foundation and the algorithm for Monte Carlo simulation, comparing the simulation results with the Black-Scholes model, and investigating how path-dependent options can be implemented using simulation when closed-form formulas may not be available. JEL Classification C02, C6, G12, G17 Keywords Monte Carlo simulation, Option pricing, Black-Scholes model Author's e-mail ryandutton4@gmail.com Supervisor's e-mail oldrich.dedek@fsv.cuni.cz
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Stochastic Models in Financial Mathematics
Waczulík, Oliver ; Hurt, Jan (advisor) ; Večeř, Jan (referee)
Title: Stochastic Models in Financial Mathematics Author: Bc. Oliver Waczulík Department: Department of Probability and Mathematical Statistics Supervisor: doc. RNDr. Jan Hurt, CSc., Department of Probability and Mathe- matical Statistics Abstract: This thesis looks into the problems of ordinary stochastic models used in financial mathematics, which are often influenced by unrealistic assumptions of Brownian motion. The thesis deals with and suggests more sophisticated alternatives to Brownian motion models. By applying the fractional Brownian motion we derive a modification of the Black-Scholes pricing formula for a mixed fractional Bro- wnian motion. We use Lévy processes to introduce subordinated stable process of Ornstein-Uhlenbeck type serving for modeling interest rates. We present the calibration procedures for these models along with a simulation study for estima- tion of Hurst parameter. To illustrate the practical use of the models introduced in the paper we have used real financial data and custom procedures program- med in the system Wolfram Mathematica. We have achieved almost 90% decline in the value of Kolmogorov-Smirnov statistics by the application of subordinated stable process of Ornstein-Uhlenbeck type for the historical values of the monthly PRIBOR (Prague Interbank Offered Rate) rates in...
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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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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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Pricing Options Using Monte Carlo Simulation
Dutton, Ryan ; Dědek, Oldřich (advisor) ; Červinka, Michal (referee)
Monte Carlo simulation is a valuable tool in computational finance. It is widely used to evaluate portfolio management rules, to price derivatives, to simulate hedging strategies, and to estimate Value at Risk. The purpose of this thesis is to develop the mathematical foundation and an algorithmic structure to carry out Monte Carlo simulation to price a European call option, investigate Black-Scholes model to look into the parallel between Monte Carlo simulation and Black-Scholes model, provide a solution for Black-Scholes model using Lognormal distribution of a stock price rather than solving Black-Scholes original partial differential equation, and finally compare the results of Monte Carlo simulation with Black- Scholes closed-form formula. Author's contribution can be best described as developing the mathematical foundation and the algorithm for Monte Carlo simulation, comparing the simulation results with the Black-Scholes model, and investigating how path-dependent options can be implemented using simulation when closed-form formulas may not be available. JEL Classification C02, C6, G12, G17 Keywords Monte Carlo simulation, Option pricing, Black-Scholes model Author's e-mail ryandutton4@gmail.com Supervisor's e-mail oldrich.dedek@fsv.cuni.cz
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Pricing financial derivatives
Chudáček, Petr ; Hurt, Jan (advisor) ; Dostál, Petr (referee)
This bachelor thesis deals with selected methods of pricing of fi- nancial derivatives. It begins with introduction to financial derivatives, simple methods of pricing them and establishing terminology. It follows with summary of mathematical definitions and theorems necessary for deriving selected models for option pricing. In chapter dealing with diffusion models, there are introduced Black-Scholes Model, Binomial Model, and CEV model. The following chapters deal with Merton's Jump-Diffusion Model, i.e., a diffusion model enriched with jumps, and Variance-Gamma Model as the representative of (pure) jump models. This thesis is interspersed with numerical 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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Stochastic Models in Financial Mathematics
Waczulík, Oliver ; Hurt, Jan (advisor) ; Večeř, Jan (referee)
Title: Stochastic Models in Financial Mathematics Author: Bc. Oliver Waczulík Department: Department of Probability and Mathematical Statistics Supervisor: doc. RNDr. Jan Hurt, CSc., Department of Probability and Mathe- matical Statistics Abstract: This thesis looks into the problems of ordinary stochastic models used in financial mathematics, which are often influenced by unrealistic assumptions of Brownian motion. The thesis deals with and suggests more sophisticated alternatives to Brownian motion models. By applying the fractional Brownian motion we derive a modification of the Black-Scholes pricing formula for a mixed fractional Bro- wnian motion. We use Lévy processes to introduce subordinated stable process of Ornstein-Uhlenbeck type serving for modeling interest rates. We present the calibration procedures for these models along with a simulation study for estima- tion of Hurst parameter. To illustrate the practical use of the models introduced in the paper we have used real financial data and custom procedures program- med in the system Wolfram Mathematica. We have achieved almost 90% decline in the value of Kolmogorov-Smirnov statistics by the application of subordinated stable process of Ornstein-Uhlenbeck type for the historical values of the monthly PRIBOR (Prague Interbank Offered Rate) rates in...
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