C12 - Hypothesis Testing: GeneralReturn

Volatility of the financial time series belongs to the crucial estimated parameters in finance (e.g. in risk management, derivative pricing). It is well known, that volatility varies in time, so that new approaches of volatility modeling have appeared. In this paper two models of the conditional heteroskedasticity - fractionally integrated GARCH (FIGARCH) and EWMA are presented. These models are illustrated on the daily historical returns of stock index PX and index BUX. Standard tests of normality, autocorrelation and conditional heteroskedasticity are applied to these log-return time series and before estimating the models, which confirm a usability of the conditional heteroskedasticity models. Empirical results of the Rescale Range analysis (R/S) indicate a long memory in the volatility process of PX index and the first 40 autocorrelations of the square log-returns show their hyperbolic decay. The volatility models are estimated by quasi-maximum likelihood method with Student's t-distribution and used to the calculation of the 1-day 95% and 99% Value at Risk values. Finally, the validity of the models is verified by Kupiec's test, TUFF and Christoffersen's test. These tests demonstrate, that the FIGARCH model is a suitable alternative to the EWMA model in the Value at Risk calculation.

Estimation of volatility of financial time series plays a crucial role in pricing derivatives. Volatility is often estimated from historical data; however, it is well known that volatility varies in time. We propose a method to choose a suitable length of historical data to estimate contemporary volatility. The method is based on adaptation of a procedure used in statistical quality control - a hypothesis, that data contains a changepoint of volatility, is tested and if the test gives a positive answer, the changepoint is estimated. Then, a period of data where no changepoint is statistically significant is used to estimate contemporary volatility. The approach is illustrated on an analysis of CZK/EUR exchange rates.