Last modified: January 24, 2026
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Financial Time Series Models
Financial series (prices, returns, exchange rates) often look very different from the classical stationary Gaussian assumptions. Common features include:
- Heavy tails (extreme events occur more often than normal theory predicts).
- Asymmetry (returns can be skewed).
- Volatility clustering (large moves tend to follow large moves).
- Serial dependence in variance, even when raw returns have weak autocorrelation.
Log Returns
If $P_t$ is the price at time $t$, the log return is:
$$ Z_t = \log(P_t) - \log(P_{t-1}) $$
Working with returns instead of prices often produces a more stable series for modeling.
ARCH and GARCH
ARCH-type models describe changing variance over time by making variance depend on past shocks.
ARCH($p$):
$$ Z_t = \sqrt{h_t} \, \epsilon_t, \quad \epsilon_t \sim IID\,N(0,1) $$
$$ h_t = \alpha_0 + \sum_{i=1}^{p} \alpha_i Z_{t-i}^2 $$
GARCH($p, q$):
$$ h_t = \alpha_0 + \sum_{i=1}^{p} \alpha_i Z_{t-i}^2 + \sum_{j=1}^{q} \beta_j h_{t-j} $$
with $\alpha_0 > 0$ and $\alpha_i, \beta_j \ge 0$.
Synthetic Volatility Example
The plot below shows a synthetic series with volatility clustering and the corresponding conditional volatility.
These models are foundational for risk management, option pricing, and measuring time-varying uncertainty in financial markets.