Estimation and modeling of volatility for option valuation

This thesis will outline not only the methods, but also illustrate the properties attributed to various estimates of stock market volatility when it is presumed to be a fixed quantity over time. The purpose of studying this parameter is to place a fair value upon European call options that have such underlying financial assets. The intension is to produce estimators which are generated from all the available information, that means from direct observation of the asset\u27s price as well as options placed upon the asset in the market, while at the same time obtaining the asymptotic properties which are desired. Under the simple assumption of constant volatility, the large sample behavior of such an estimator can be determined. ^ This thesis will also address the more complex models where it is presumed that the volatility does vary with time. If no jumps are present within the high frequency financial data gathered, it is shown that the GARCH structure attributed to the financial time series will result in a Heterogenous Autoregressive pattern for the integrated volatilities. Therefore, the realized volatility measures, which attempt to estimate these integrated quantities, will share this Heterogeneous AR structure. Furthermore, this modeling can be extended into higher dimensions and can be used to describe the movements of the realized covariation matrix.