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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!

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    $\begingroup$ By volatility the theoretician means the instantaneous volatility not in a day in an hour or a minute but in an instant. Empirically we cannot measure it for the reasons you mention: very few trades at high frequency, microstructure noise in the form of bid-ask spread, etc. In a Brownian Motion there are infinite points in a time interval no matter how small, on a Stock Exchange not so: there is a discrete trades, then a long silence lasting many milliseconds, then maybe another trade... $\endgroup$ – noob2 Apr 6 at 13:30
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Let me start from the beginning. What do you observe in financial markets? The data, the information that is given to you in as raw a form as possible, are things like bid prices, ask prices and trading volumes. That's data. Usually, people will take the mid-point of the bid and ask spread, define this as the fair value of a security and use it as a single price.

Now, if you move towards returns, you have to transform those mid-point prices: either you compute it as a ratio $R_T := p_t/p_{t-1} - 1$ or you take the difference of logarithms $r_t := ln(p_t/p_{t-1})$. Either way, strictly speaking, what you just computed is a statistic. Most people would still call this "data," but if you want to be extra Kosher, even returns aren't data. They are a transformation of data and, thus, a statistic.

Now, by volatility, we usually mean either $\sqrt{Var_t(\Pi_{\tau=1}^T (1 + R_{t+\tau}))}$ or $\sqrt{Var_t(\sum_{\tau=1}^T r_{t+\tau})}$. In other words, we'd like to know the standard deviation of returns over the period going from time $t+1$ to $T$, given information up to time $t$. From this point of view, returns are random variables and the returns you compute from prices are the realizations of those random variables. The issue is that, no matter how you put the problem, you do not observe standard deviations over compounded returns for some period of time. You cannot open up Yahoo!Finance and see that anywhere. On the other hand you can compute a statistic which would inform you about it. In a more volatile environment, returns jump around a lot more, so there is some hope that you can estimate it.

So, in essence, you do not observe volatility, but you observe the consequences of volatility. There are various ways to go about estimating volatility. If you neglect the issue of conditional non-normality (i.e., the fact that there are relatively frequent and large jumps in prices), you can obtain an estimator of quadratic variation for a given day using the sum of squared logarithmic returns taken at a much higher frequency (say, around 5 or 10 minutes returns). That's called "realized volatility." Formally, it's a frequentist estimator of integrated (think cumulated) variance and it is valid under arbitrary Ito diffusion processes. In practice, because you're neglecting jumps when you use this, the estimator is a bit polluted by other stuff, but the really Kosher thing to do tends to be extremely cumbersome.

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  • $\begingroup$ Thank you for your answer. I have marked it as solved. Just to be sure: The instantaneous volatility you can't measure, because you need two points to measure it from. But what about over a given period from close to close of two days? For an asset with either no trades or trades at the same price, so the return is zero. Why can't we say that on that period (day) the volatility was zero (given zero mean)? We look backwards so we know the true outcome of the return is zero, so why can't we say that the volatility is also zero? Is it because this is only one realization? $\endgroup$ – Anders Apr 17 at 10:20

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