0.1 Realized Volatility
Where GARCH and stochastic volatility models infer latent volatility from daily returns, realized volatility (RV) instead measures volatility nonparametrically from high-frequency (intraday) data, exploiting quadratic-variation theory from Stochastic Calculus.
Partition day \(t\) into \(n\) intraday intervals with log returns \(r_{t,i}\), \(i=1,\dots,n\). The realized variance is \[ RV_t = \sum_{i=1}^n r_{t,i}^2. \] If the log-price follows an Itô process \(d\log S_u = \mu_u\,du + \sigma_u\,dW_u\), then as \(n\to\infty\) (sampling frequency \(\to 0\)), \(RV_t \xrightarrow{p} \int_{t-1}^t \sigma_u^2\,du\) — the quadratic variation/integrated variance over the day. RV is thus a consistent, model-free estimator of latent integrated volatility.
Practical issues. - Microstructure noise: at very high frequencies, bid-ask bounce and discreteness bias \(RV_t\) upward; addressed via sparse/optimal sampling, realized kernels, or pre-averaging estimators. - Jumps: \(RV_t\) estimates total quadratic variation, including price jumps. Bipower variation (using products of adjacent absolute returns) is robust to jumps and estimates only the continuous component, so \(RV_t - BV_t\) isolates the jump contribution.
Use in forecasting. Realized volatility is far more informative and less noisy than squared daily returns as a volatility proxy, and is the input series for reduced-form volatility-forecasting models such as the HAR-RV model, as well as for evaluating/backtesting GARCH and stochastic volatility forecasts.