# What is the difference between conditional volatility and realized volatility?

I am working on conditional volatility and realized volatility but the difference between these two measures is not clear to me.

Can anybody explain how these two volatilities are related? Does the realized vola affect the conditional vola? I have read some papers which use particular models to model conditional vola and sometimes call it conditional vola and sometimes just vola. Does that mean that e.g. a Garch model can be used to model volatility or do I use Garch to model conditional volatility?

Conditional volatility is the volatility of a random variable given (i.e. conditioning on) some extra information. E.g. in the GARCH model the conditional volatility is conditioned on past values of itself and of model errors (see below). Unconditional volatility is the "general" volatility of a random variable when there is no extra information (no conditioning).

Realized volatility is the empirical unconditional variance over a given time period. E.g. if 5-minute returns on a stock price are collected over a trading day, their empirical variance can be called realized volatility ("realized" in the sense that it has already been measured). Recall that variance is a property of the data generating process that is unobservable and can only be measure with imperfect precision from the data.

Does that mean that e.g. a Garch model can be used to model volatility or do I use Garch to model conditional volatility?

The GARCH($s,r$) model looks like this: \begin{aligned} r_t &= \mu_t + u_t, \\ u_t &= \sigma_t \varepsilon_t, \\ \sigma_t^2 &= \omega + \sum_{i=1}^s\alpha_i u_{t-i}^2 + \sum_{j=1}^r\beta_j \sigma_{t-j}^2, \\ \varepsilon_t &\sim i.i.d.(0,1). \end{aligned} It specifies the conditional distribution of a random variable $r_t$ conditional on past values of the model errors $u_t$, conditional variances $\sigma^2_{t-j}$ and whatever other variables that determine the conditional mean $\mu_t$.

The unconditional variance $\sigma_t$ of the error term $u_t$ is given by $\frac{\omega}{1-\sum_{i=1}^s\alpha_i u_{t-i}^2 + \sum_{j=1}^r\beta_j \sigma_{t-j}^2}$, while the unconditional variance of $r_t$ is generally more messy as it also involves $\mu_t$ which may be arbitrarily complicated. (However, the conditional mean is often taken to be as simple as $\mu_t=0$ in which case the conditional and unconditional variances of $r_t$ equal those of $u_t$.)

I have read some papers which use particular models to model conditional vola and sometimes call it conditional vola and sometimes just vola.

I think "conditional" is sometimes omitted for brevity if it is supposed to be clear from the context. Otherwise it is mentioned explicitly to avoid ambiguity.

The main point of GARCH theories is that volatility is always changing and that it is therefore a latent ("not directly observable") variable. The underlying volatility at a point in time is called the conditional volatility at that particular moment and is modeled by various GARCH-type equations.

The historical or realized volatilities on the other hand are the observed volatilities during specific intervals of time, the historical is usually computed from daily data and the realized from more high frequency data (such as 10 minute data).

But even a realized volatility will be estimated over an INTERVAL of time, even if short, and is therefore different from the conditional volatility which is instantaneous.

The idea behind the early "realized volatility" papers was in fact to come closer to the conditional volatility by sampling the underlying time series at higher frequency, but knowing that you can never get to the ultimate goal of "seeing" conditional (instantaneous) volatility.

• So "conditional volatility at time t" versus "realized volatility in the interval $[t_0,t_1]$" – Alex C Dec 11 '16 at 21:17
• The true underlying volatility (formally characterized by variance, standard deviation, expected absolute deviation or the like), being a feature of the data generating process, is never observable, and that is not specific to GARCH models. What can be observed are some empirical measurements of volatility such as realized volatility. I think this is an important distinction that is missing in the answer. Also, the term conditional is due to conditioning on some variables (such as past volatility in GARCH models), so the emphasis on a particular time point is misplaced, IMHO. – Richard Hardy Dec 14 '16 at 20:27