Enter (introduced by Tim Bollerslev in 1986). A GARCH(1,1) model—the industry workhorse—uses only three parameters to capture volatility dynamics:
This is where (Autoregressive Conditional Heteroskedasticity) and its big brother GARCH (Generalized ARCH) come to save the day.
Adds a term that activates only when the previous shock was negative. $$\sigma_t^2 = \alpha_0 + \alpha_1 \epsilon_t-1^2 + \gamma \epsilon_t-1^2 I_t-1 + \beta_1 \sigma_t-1^2$$ (Where $I_t-1$ is 1 if $\epsilon_t-1 < 0$). arch models
Consider daily returns of Tesla stock:
Suggests that the variance is based on previous information (time- information). Enter (introduced by Tim Bollerslev in 1986)
This article provides a deep dive into ARCH models, explaining their mechanics, applications, extensions, and practical implementation in the 2026 financial landscape. What are ARCH Models?
Asset returns often have higher probabilities of extreme observations compared to normal distributions. $$\sigma_t^2 = \alpha_0 + \alpha_1 \epsilon_t-1^2 + \gamma
[ \sigma_t^2 = \omega + \alpha_1 \epsilon_t-1^2 + \alpha_2 \epsilon_t-2^2 + ... + \alpha_q \epsilon_t-q^2 ]
Proposed by Bollerslev in 1986, GARCH(p, q) extends ARCH by including lagged variance terms, enabling the model to capture longer-lasting volatility patterns with fewer parameters. 2. EGARCH (Exponential GARCH)
# 3. Fit the model # 'update_freq=0' suppresses output during iteration results = model.fit(update_freq=5)
The ARCH model has a flaw: you often need many lags (q can be large) to capture the long memory of volatility. That’s a lot of parameters to estimate.