On this page
A GARCH(1,1) forecast updates variance, using a recent squared return, the previous variance estimate and a constant. Take the square root only after the update if the question asks for volatility. The fitted model has a finite long-run variance when its shock and persistence weights sum to less than one.
What the update is doing
A large return, whether positive or negative, increases the next variance estimate because the return is squared. The previous estimate carries volatility clustering forward. The constant supplies an anchor that gradually pulls later forecasts toward a long-run level. GARP places valuation and risk models in its FRM study materials; its volatility forecasting paper discusses GARCH and mean reversion.
Use returns as decimals in this form. The shock weight is alpha; beta carries the previous variance. Both inputs on the right are squared quantities, so adding a percentage volatility directly to them mixes units.
An illustrative update
These invented daily figures show the arithmetic, not observed market data: omega is 0.000004, alpha is 0.10, beta is 0.80, prior variance is 0.0001 and the prior return is 0.02. Squaring the return gives 0.0004. The next variance is 0.000004 plus 0.10 times 0.0004 plus 0.80 times 0.0001, or 0.000124. Its square root is about 0.01114, so the updated daily volatility is about 1.114%.
For the same illustrative parameters, alpha plus beta is 0.90. The long-run variance is therefore 0.000004 divided by 0.10, or 0.00004; the corresponding volatility is about 0.632%. The next forecast is above that long-run level, but the model’s anchor implies a gradual decline if no further large shocks arrive. A sum at or above one does not provide this finite positive long-run variance under the usual GARCH specification.
The exam mistake to avoid
Do not report 0.000124 as volatility: it is variance. Do not call omega the long-run variance either; the denominator in the second formula matters. Finally, a negative return does not enter this basic equation as a negative shock because it is squared. When revising, write “variance” beside every intermediate result before taking the final root. The FRM programme page gives programme context, while Omni’s methodology explains its approach to study.
Sources
- FRM Study Materials · garp.org · Retrieved
- Volatility Forecasting in Futures Markets: A Statistical and Value-at-Risk Evaluation · garp.org · Retrieved
