Volatility Modeling

Volatility Modeling

Core Concepts

EWMA (Exponentially Weighted Moving Average)

A simple volatility model that gives more weight to recent observations. RiskMetrics popularized this approach with a standard decay factor.

sigma^2_t = lambda * sigma^2_{t-1} + (1 - lambda) * r^2_{t-1}

where:

• lambda = decay factor (RiskMetrics standard: 0.94 for daily, 0.97 for monthly)
• r_{t-1} = return in period t-1 (typically demeaned, but for daily returns the mean is often assumed to be zero)
• sigma^2_{t-1} = previous period's variance estimate

Properties:

• Assigns exponentially decaying weights to past squared returns.
• Effective window is approximately 1/(1 - lambda) observations. For lambda = 0.94, effective window is approximately 17 days.
• No mean reversion: the model is equivalent to IGARCH (integrated GARCH) where alpha + beta = 1. Volatility shocks persist indefinitely.
• Simple to implement and requires only one parameter.

GARCH(1,1)

The Generalized Autoregressive Conditional Heteroskedasticity model adds a constant term that induces mean reversion in volatility.

sigma^2_t = omega + alpha * r^2_{t-1} + beta * sigma^2_{t-1}