Volatility Clustering

Volatility Clustering

A well-known stylized fact in financial time series: large price moves tend to cluster together. Periods of high volatility are followed by more high volatility.

Log Returns

For a price series prices, the log return at time t is:

r[t] = ln(prices[t+1] / prices[t])

Squared Returns

Squared returns are a proxy for volatility:

r[t]² 

High squared return = high volatility at that point.

Volatility Cluster Ratio

The fraction of returns with squared return exceeding a threshold:

cluster_ratio = count(r[t]² > threshold) / n

Task

Implement:

• squared_returns(prices) → list of squared log returns
• volatility_cluster_ratio(prices, threshold) → fraction of high-volatility observations