Asymptotic Theory for the Sample Autocorrelation Function and the Extremes of Stochastic Volatility

1 Asymptotic theory for the sample autocorrelation and the extremes of stochastic volatility On a personal level, I am very grateful to Professor Thomas Mikosch for accepting to supervise my Ph.D. studies and for his help and support. Staff and Faculty of the Department of Mathematical Sciences at Copenhagen University helped me a lot since I arrived in Denmark, and they were very friendly and supportive with me. The Egyptian Culture and Educational Bureau in Berlin gave me a special Social and Fellow Support for me and my family. A special thank goes to Jesper Lund Pedersen for helping me in translating the summary to Danish. Finally, I will forever be grateful to the lecturers at the Department whose courses I enjoyed. A special thank goes to Professor Martin Jacobsen for his advice and help. In particular, I would like to thank my wife for her strong personal support. She chose to be away from Egypt and her parents in the last 4 years in order to be close to me. ii Summary This thesis is concerned with one of the nonlinear financial time series models, the stochastic volatility model. For financial time series, nonlinear time series models are better suited to describe their behavior. In contrast to linear time series models such as ARMA, the investigation of nonlinear models is still work in progress. We start by collecting some of the standard properties of a stochastic volatility process. Under mild assumptions, the stochastic volatility sequence is a strictly stationary ergodic martingale difference sequence. Using this fact, we derive the central limit theorem for the sample mean of this sequence. Another property of the stochastic volatility model is strong mixing. This fact is helpful to establish a central limit theorem for the sample variance of this sequence. In both cases, under the assumption of finite variance innovations, the limiting distribution is a Gaussian distribution. In contrast to this case, under the assumption of infinite variance stable innovations, these estimators have a limiting distribution which is not Gaussian and rather unfamiliar. The autocorrelation function is an important tool in time series analysis. We study its estimator, the sample autocorrelation function and its limit Gaussian distribution via a multivariate central limit theorem. Another important characterization of a time series is provided by its spectral density. We estimate the spectral density of a stochastic volatility process in some heavy– and light–tailed cases by …