A GARCH model forecasts time-varying volatility when returns show volatility clustering and volatility mean reversion. It is useful for estimating asset variances and improving portfolio inputs. section 3.2.2 section 3.0
A practical volatility-forecasting workflow:
- Download adjusted prices and compute percentage returns, dropping the initial missing value. section 3.2.2
- Scale returns by 100 so the ARCH-library optimization is more likely to converge. section 3.2.2
- Fit an arch_model and forecast a chosen horizon; interpret each output as a one-day variance forecast. section 3.2.2
- Sum the horizon’s daily variances and annualize with 252 divided by the forecast horizon before taking the square root. section 3.2.2
AAPL three-month forecast
Using adjusted AAPL prices, a 63-trading-day forecast produces declining one-day variances from 9.8718 to 9.2234. Summing and annualizing them gives a volatility forecast of 0.4810 after undoing the return scaling. section 3.2.2
Table 3.9 S&P500 returns around the stock market crash of 1987
| Date | Close | % Change |
| 1987-10-08 | 314.160004 | -1.375025 |
| 1987-10-09 | 311.070007 | -0.983574 |
| 1987-10-12 | 309.390015 | -0.540069 |
| 1987-10-13 | 314.519989 | 1.658093 |
| 1987-10-14 | 305.230011 | -2.9537 |
| 1987-10-15 | 298.079987 | -2.342504 |
| 1987-10-16 | 282.700012 | -5.15968 |
| 1987-10-19 | 224.839996 | -20.466931 |
| 1987-10-20 | 236.830002 | 5.332684 |
| 1987-10-21 | 258.380005 | 9.099355 |
| 1987-10-22 | 248.250000 | -3.920584 |
| 1987-10-23 | 248.220001 | -0.012084 |
| 1987-10-26 | 227.669998 | -8.278947 |
| 1987-10-27 | 233.190002 | 2.424564 |
| 1987-10-28 | 233.279999 | 0.038594 |
| 1987-10-29 | 244.770004 | 4.925414 |
| 1987-10-30 | 251.789993 | 2.867994 |
The 1987 S&P 500 return sequence provides a concrete view of clustered, unusually large market movements that motivate conditional-volatility models. section 3.2.2
