Modelling Volatility By Functional Data Analysis:an Empirical Application To The Chinese Stock Market

Nowadays in researches of financial market, one of the widely concerned point is the implicit volatility of returns with various underlying assets, from which only indirect information including patterns and direction of variation can be derived, for the price is assumed to be a random variable or process. There are plenty of methods and models based on volatility modelling in history such as random variation model and GARCH models which have heavy assumptions in model and parameters. On the one hand, the innovation of IT leads the rapid development of financial high-frequency data type, with which we may apply novel statistical methods to extract and model volatility. The interval between data, on the other hand, is also shrinking to make it reasonable to adopt functional data analysis. Containing a series of methods such as B-spline basis expansion, local linear regression, functional principal component analysis, and functional regression model, functional data analysis assumes slightly on model and fits the area of financial hi-frequency data analysis. Our analysis base on one of the stocks listing on SHANGHAI STOCK EXCHANGE, utilizing the absolute returns rate of CNPC (601857). We describe the process of volatility extraction. Firstly, discrete data will be transformed into continuous functions. Secondly, volatility trajectory of each sample is derived with the combination of both principal component functions and principal component scores along with the meaning of each principal component function explained. Finally, functional auto-regression model of first order is considered to make a future prediction with past information. All parameters are estimated by equivalent OLS methods. We also contribute a volatility extraction comparison between spline basis expansion and local linear regression. The result shows that functional data analysis provides accurate time-varying patterns of intra-day data, especially in directions along the time axis, and support strongly in volatility modelling and future prediction.