  In an era when derivatives is getting popular, risk management has gradually
become the core content of modern finance. In order to study how to accurately
estimate the volatility of the S&P 500 index, after introducing the theoretical
background of several methods, this paper uses the historical volatility
method, GARCH model method and implied volatility method to estimate the real
volatility respectively. At the same time, two ways of adjusting the estimation
window, rolling and increasing, are also considered. The unbiased test and
goodness of fit test are used to evaluate these methods. The empirical result
shows that the implied volatility is the best estimator of the real volatility.
The rolling estimation window is recommended when using the historical
volatility. On the contrary, the estimation window is supposed to be increased
when using the GARCH model.

Mathematical Finance