In this bachelor thesis, methods to model financial time series, like the DAX index, are introduced. In the beginning, basic concepts of financial time series, like the stylized facts and white noise or the martingale difference are defined. Following, there is the introduction to the ARMA process, which stands for ""autoregressive moving average"". ""Autoregressive"" meaning that the observation X(t) depends on the previous observations X(t-i) and ""moving average"" implying that the mean is calculated as a moving average depending on the previous residuals. In this section, the focus is on the main properties of this process, the causality and invertibility, the prediction and fitting these models to data. Furthermore the AR(p) and the MA(q) process are explained. Here, it is important to consider the autocorrelation function. Completing the section the important task of predicting financial time series in the analysis of financial time series is shown. When modelling financial time series we can observe that the volatility, i.e. the conditional standard deviation, is not constant but changing with time. This phenomenon is known as heteroskedasticity. In order to find a model considering this phenomenon, the ARCH model was introduced by Robert Engle in 1982. That model made such an impact on financial econometrics that he was awarded the Nobel Prize in 2003. The ARCH model gives more weight to the recent observation and less to information that happened a long time ago. Following the modelling of changing volatility with conditionally heteroskedastic models, called GARCH models, is defined. There the conditional variance not only depends on the history of the financial time series but also on its own history. This model was introduced in 1986 by Tim Bollerslev. These models are considered in Section 4. Essential is the modelling of data with these models and the prediction of the mean and the variance. Furthermore we realized a GARCH(1,1)-model fitting to DAX-Data in R (http://www.r-project.org/index.html, visited on July 5, 2010). Then, it is shown how GARCH models can be used in risk management, for example in the calculation of risk measures like the Value-at-Risk. As an application there is a one step ahead prediction for different observations, based on a GARCH(1,1) model and an ARMA(1,1)-GARCH(1,1) model. Concluding, we use these predictions to obtain the VaR estimated each day for the next day.     «

In this bachelor thesis, methods to model financial time series, like the DAX index, are introduced. In the beginning, basic concepts of financial time series, like the stylized facts and white noise or the martingale difference are defined. Following, there is the introduction to the ARMA process, which stands for ""autoregressive moving average"". ""Autoregressive"" meaning that the observation X(t) depends on the previous observations X(t-i) and ""moving average"" implying that the mean is ca...     »