EWMA, GARCH Volatilities

Let’s see the few of the volatility models based on past innovations (meaning past observations). There are few models such as moving average which gives equal weights to past observations but we will see most commonly used models in the industry, one such is EWMA meaning, Exponential Weighted Moving Average.

The general formula is first to calculate the weights,

$p = \frac{1 - \lambda}{\lambda(1 - \lambda^M)} \lambda^m$

$where, m = 1, 2.....M$

Then applying to predicting volatility,

$\sigma_{t+1} = \sqrt{p^m R^2_{t-m+1}}$

Supposing, M is large, we can formulate more generic solution, termed RiskMetrics formerly modelled by JP Morgan using the lambda values close to 1 for volatility persistence and market reactivity.

$\sigma_{t+1} = \lambda \sigma_t + (1-\lambda) R_t$

Let’s now see for the another volatility model, most widely used and industry adopted solution, GARCH, Generalised AutoRegressive (meaning current return based on past lagged observations) Conditional heteroskedasticity (meaning non constant / non-homoskedastic variance across observations).

The formula to model for GARCH volatility is given below,