In this dissertation we focus on two points in the study of financial statistics, volatility estimation and option pricing. In the first chapter we briefly introduce the relevant terms and concepts as well as a general literature review for the diffusion and GARCH model and realized volatility. The second chapter investigates the convergence speed of the GARCH European option price to its bivariate diffusion limit price, particularly under the EGARCH(1,1) specification. Viewed as a discrete-time approximation, a properly constructed GARCH model weakly converges to a bivariate diffusion as the time interval between observations shrinks to zero. Thus, the difference in European option prices must also approach zero in the limit. However, the rate of convergence is unknown. Our major contribution is being the first to show that the option price convergence is at a rate near n 1/2 . In chapter three we propose a new methodology to estimate the integrated volatility matrix using intra-day high frequency data. The novel feature of our method is to combine regularization and sparsity to extend the usual realized volatility type of estimation of one or few assets to a much larger number of assets, usually on the order of sample size. Asymptotic analyses reveal that significant improvement is gained for the estimators. Chapter four is devoted to numerical studies. We conduct extensive simulation studies for complex price and volatility models with large p , the number of assets. The simulation studies demonstrate that the proposed estimators perform well for finite p and sample size. We apply the method to real high-frequency data sets in chapter five. Finally, we end this dissertation by drawing some concluding remarks and comments for future research in chapter six.
