Timeseries Engineering

GARCH(1,1) with ω = 0.05, α = 0.10 and β = 0.85 has a long-run variance of ω/(1 − α − β) = 1, and a volatility shock halves in 13.5 days. Simulated returns are fat-tailed — kurtosis 3.58 against 3 for a normal — even though every day’s shock is normal. Clustering alone produces fat tails.

• GARCH(1,1) is hard to beat (Hansen & Lunde 2005), but it reacts to large moves after they happen; it does not foresee them.
• Persistence close to 1 may be an artefact of structural breaks in the data.
• Symmetric GARCH misses the leverage effect: falls raise volatility more than rises (EGARCH, GJR).

rng = np.random.default_rng(7)
omega, alpha, beta, n = 0.05, 0.10, 0.85, 20_000
long_run = omega / (1 - alpha - beta)
half_life = np.log(0.5) / np.log(alpha + beta)     # days for a volatility shock to halve
r, s2 = np.zeros(n), np.full(n, long_run)
for t in range(1, n):
    s2[t] = omega + alpha * r[t - 1] ** 2 + beta * s2[t - 1]
    r[t] = np.sqrt(s2[t]) * rng.normal()
kurt = ((r - r.mean()) ** 4).mean() / r.var() ** 2   # normal = 3

Assumes import numpy as np and import pandas as pd. Each snippet simulates or makes up its own series, as the comments say.

# Python — GARCH(1,1)
from arch import arch_model

model = arch_model(returns, vol='Garch', p=1, q=1)
result = model.fit(disp='off')
print(result.summary())
# Forecast next 5 periods of volatility
fcast = result.forecast(horizon=5)