Various volatility models.
Appearance of the indicator.
Realized volatility (RV) summarizes the price range, in which the price moved for a given period of time. Realized volatility is calculated by the formula:
$$ RV_x = \sum_{i=0}^{x}(high_i-low_i) $$
where $high$ is the maximum of the bar, $low$ is the minimum of the bar, $x$ is the specified period of time.
The traditional volatility calculation involves calculating the standard deviation of returns, which is based on the average return. However, when the price of an asset demonstrates a trend movement, the average return can significantly differ from zero, and changing the length of the time window used for calculation can lead to artificially high volatility values. To address this problem, a local volatility model has been developed that calculates the standard deviation of differences between successive asset prices rather than their returns. This gives an indication of how much price changes from one tick to the next, regardless of the overall trend.
The GARCH (generalized autoregressive conditional heteroskedasticity) model is a statistical model used to predict the volatility of a financial asset. This model accounts for fluctuations in volatility over time, recognizing that volatility can change in a heteroskedastic (i.e., non-constant variance) manner and can be affected by past events.
The main advantage of GARCH models is their ability to model volatility as changing over time, which makes them particularly suitable for analyzing financial markets where volatility is not constant. In GARCH models, volatility at a given point in time depends on both past values of volatility itself and past perturbations (innovations) in the data. This allows the models to account for conditional heteroskedasticity, a phenomenon where the distribution of a variable is unstable over time.
GARCH-in-Mean (GARCH-in-Mean, GARCH-M) was proposed by Engle et al. in 1987. In this case, we are not talking about a special model for conditional variance. It is about using conditional variance as one of the factors in a regression model for the risk premium.
The general formula of the GARCH-M model is:
$$ y_{t}= α + f(σ^2_t) - E|f(σ^2_{t+1})| + u_t $$
Where:
$σ^2_{t}$ is the conditional variance at time $t$ (i.e., volatility squared), $α$ is a coefficient representing the effect of the lagged error squared on the conditional variance.
In the context of financial forecasting, the GARCH model is used to estimate the future volatility of an asset.