I am looking into how to measure volatility, and I am not sure if I have confused myself too much in my research. So now I really need your help. So please either confirm my understanding of volatility, or else correct me.
The thing I am struggling with is conceptualizing that volatility isn't observable.
For example, to evaluate a GARCH model's performance in predicting the volatility, one way to do that would be to estimate the difference between the forecast by GARCH and the actual volatility by some evaluation function like MSE (mean squared error). However, the actual volatility, even though it is in the past, ie. ex post, isn't observable.
The volatility (even ex post) isn't observable because, well it can't be. It's a measurement including two observables at at least two separate times. What time intervals would you choose to describe the actual volatility?
Let's say we are looking into the volatility of Apple's stock AAPL. We have forecasted the volatility of a specific day t to be a value x. We now want to know the true volatility. Would the true volatility of day t be given by taking all the transactions throughout the day and take the square root of the variance? It is just a proxy for the volatility. Including all the trades of AAPL in a single day would mean a higher volatility than the actual volatility because of the bid/ask spread.
I am not sure though, if there wasn't a bid/ask spread, would taking all the observations into account (realized volatility) generate the actual ex post volatility?
Hope someone can clarify things for me. Thank you in advance!