ARCH Model – Introduction

For decades, economists assumed risk (variance) was constant. In 1982, Robert Engle introduced the ARCH (AutoRegressive Conditional Heteroskedasticity) model, which allowed variance to change over time. He won the Nobel Prize (2003) for this.

Breaking Down the Acronym

1. AutoRegressive: The current value depends on past values.
2. Conditional: Based on past information.
3. Heteroskedasticity: Changing Variance (skedasticity = variance).

So, ARCH means "Modeling changing variance based on past errors."

The Logic

The ARCH model says: "The volatility today depends on the squared magnitude of the shock (return) yesterday."

If yesterday had a huge price move (positive or negative), today's variance will be high. This captures Volatility Clustering.

The Formula (ARCH(1))

sigma_t^2 = alpha_0 + alpha_1 * epsilon_(t-1)^2

• sigma_t^2: Calculated Variance for today.
• alpha_0: A constant base variance.
• epsilon_(t-1)^2: Squared residual (shock) from yesterday.
• alpha_1: How sensitive volatility is to new information.

Limitations of ARCH

1. High Order Needed: To model real markets, you often need an ARCH(10) or ARCH(15) model, meaning you look back 15 days. This creates too many parameters to estimate.
2. Solvability: It's hard to ensure variance stays positive if parameters are negative (which they shouldn't be).

These limitations led to the development of GARCH (Generalized ARCH).

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