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Log-optimal investment
Flimmel, Samuel ; Lachout, Petr (advisor) ; Kopa, Miloš (referee)
The creation of portfolio is an important and frequent task to solve in financial sector. This paper indroduces one of mathematical models used for this problem. For studied market we assume it`s logaritmic utility function and ergodic stationarity only. The low number of assumptions makes this model quite simple and clear. In this paper we describe the model and prove some of it`s features applicable for our model. First, we analyze the case of an known market distribution and suggest an algorithm for obtaining a portfolio. Later, we analyze the case of an unknown market distribution and introduce one of the suitable methods as well. Empirical distribution helps us to gain requiered results asymptotically. Finally, we study behavior of an empirical distribution method for shorter time periods on real data simulation.
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Estimation of parameters of clipped time series
Flimmel, Samuel
In some situations we cannot observe the original time series and instead, we record only binary data which express whether the values of the original series exceeded a certain threshold or not. The thesis deals with estimation of characteristics of the original series constructed from the binary (so called clipped or hard-limited) data, in particular in Gaussian ARMA models. We summarize some basic characteristics of the clipped series and describe their relation to the original ones. Some practical examples are provided as well. The estimation of parameters in AR(p) model is shown for the case of zero threshold. Using a similar approach, an estimator of the MA(1) model parameter is proposed and its properties are studied with emphasis on asymptotic variance. Subsequently, we propose an estimation procedure for AR(p) and MA(1) models with unknown (non-zero) threshold. The behaviour of our estimators is investigated in a simulation study, which provides a comparison with estimators constructed from the original data. Finally, a real data analysis is presented for an illustration. Powered by TCPDF (www.tcpdf.org)
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Volatility models in R
Vágner, Hubert ; Bašta, Milan (advisor) ; Flimmel, Samuel (referee)
This diploma thesis focuses on modeling volatility in financial time series. The main approach to modelling volatility is using GARCH models which can capture the variability of conditional volatility of time series. For modelling a conditional mean value in time series are used ARMA models. In the series there are usually not fulfilled the assumption of earnings normality, therefore, are the earnings in most cased characterized by the leptokurtic shape of distribution. The thesis introduces some more distribution types, which can be more easily used for the earnings distribution - above all the Students t distribution. The aim of the thesis in the first part is to present the topic of financial time series and description of the GARCH models including their further modification. There are used e.g. IGARCH or other models capturing asymmetric impact of shocks such as GJR-GARCH. The second part deals with generated data, where are more in detail explored the volatility models and their behavior in corresponding financial time series. The third part focuses on the volatility estimation and forecasting for the financial time series. Firstly this concerns development of stock index MICEX secondly currency pair Russian Ruble to Czech Crown and eventually price development of the Brent crude oil. The goal of the third part is to present the impacts on volatility of chosen time series applied on the example of economic sanctions against Russia after annexation of the Crimea peninsula which happened in the first quarter 2014.
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Estimation of parameters of clipped time series
Flimmel, Samuel
In some situations we cannot observe the original time series and instead, we record only binary data which express whether the values of the original series exceeded a certain threshold or not. The thesis deals with estimation of characteristics of the original series constructed from the binary (so called clipped or hard-limited) data, in particular in Gaussian ARMA models. We summarize some basic characteristics of the clipped series and describe their relation to the original ones. Some practical examples are provided as well. The estimation of parameters in AR(p) model is shown for the case of zero threshold. Using a similar approach, an estimator of the MA(1) model parameter is proposed and its properties are studied with emphasis on asymptotic variance. Subsequently, we propose an estimation procedure for AR(p) and MA(1) models with unknown (non-zero) threshold. The behaviour of our estimators is investigated in a simulation study, which provides a comparison with estimators constructed from the original data. Finally, a real data analysis is presented for an illustration. Powered by TCPDF (www.tcpdf.org)
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Estimation of parameters of clipped time series
Flimmel, Samuel ; Hudecová, Šárka (advisor) ; Hendrych, Radek (referee)
In some situations we cannot observe the original time series and instead, we record only binary data which express whether the values of the original series exceeded a certain threshold or not. The thesis deals with estimation of characteristics of the original series constructed from the binary (so called clipped or hard-limited) data, in particular in Gaussian ARMA models. We summarize some basic characteristics of the clipped series and describe their relation to the original ones. Some practical examples are provided as well. The estimation of parameters in AR(p) model is shown for the case of zero threshold. Using a similar approach, an estimator of the MA(1) model parameter is proposed and its properties are studied with emphasis on asymptotic variance. Subsequently, we propose an estimation procedure for AR(p) and MA(1) models with unknown (non-zero) threshold. The behaviour of our estimators is investigated in a simulation study, which provides a comparison with estimators constructed from the original data. Finally, a real data analysis is presented for an illustration. Powered by TCPDF (www.tcpdf.org)
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Log-optimal investment
Flimmel, Samuel ; Lachout, Petr (advisor) ; Kopa, Miloš (referee)
The creation of portfolio is an important and frequent task to solve in financial sector. This paper indroduces one of mathematical models used for this problem. For studied market we assume it`s logaritmic utility function and ergodic stationarity only. The low number of assumptions makes this model quite simple and clear. In this paper we describe the model and prove some of it`s features applicable for our model. First, we analyze the case of an known market distribution and suggest an algorithm for obtaining a portfolio. Later, we analyze the case of an unknown market distribution and introduce one of the suitable methods as well. Empirical distribution helps us to gain requiered results asymptotically. Finally, we study behavior of an empirical distribution method for shorter time periods on real data simulation.
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