Introduction

The Generalized Autoregressive Conditional Heteroskedasticity (GARCH) model is a powerful tool used to estimate and forecast volatility in financial markets. Developed by Robert F. Engle in the 1980s, the GARCH (1,1) model accounts for the clustering of volatility observed in financial time series data. In this chapter, we delve into the mechanics of the GARCH (1,1) model, its components, and how it can be applied to estimate volatility for risk assessment and prediction.

Components of the GARCH $(1,1)$ Model

The GARCH $(1,1)$ model consists of two main components: the autoregressive component (AR) and the moving average of past squared shocks (MA). The model assumes that volatility is a function of both past squared shocks and past conditional variances.

1) Autoregressive Component (AR): The autoregressive component captures the persistence of volatility shocks. It accounts for the fact that recent volatility shocks tend to influence future volatility. Mathematically, the AR component can be represented as:

$$\sigma_t^2=\omega+\alpha \cdot \varepsilon_{t-1}^2+\beta \cdot \sigma_{t-1}^2$$

Where:

• $\sigma_t^2$ is the conditional variance at time $t$.
• $\omega$ is the constant term.
• $\alpha$ is the coefficient of the autoregressive term.
• $\varepsilon_{t-1}^2$ is the squared residual at time $t-1$.
• $\beta$ is the coefficient of the moving average of squared shocks term.
• $\sigma_{t-1}^2$ is the conditional variance at time $t-1$.

2) Moving Average Component (MA): The moving average component accounts for the persistence of past squared shocks. It reflects the idea that past volatility shocks continue to impact future volatility. The MA component can be viewed as a form of volatility persistence.

Applying the GARCH $(1,1)$ Model

To apply the GARCH $(1,1)$ model, you need historical financial data and a software package capable of estimating model parameters. The steps include: 1) Preprocess the data: Clean and organize the financial time series data.
2) Choose an appropriate lag length: Determine the number of lagged terms to include in the model.
3) Estimate the model parameters: Use optimization techniques to estimate the values of $\omega$, $\alpha$ , and $\beta$.
4) Evaluate model diagnostics: Check for model adequacy and assess the goodness-of-fit.
5) Forecast future volatility: Use the estimated GARCH $(1,1)$ model to forecast volatility for upcoming periods.

Example: Suppose you have daily stock price returns data for a specific asset. By applying the GARCH $(1,1)$ model, you can estimate the asset’s conditional volatility. Let’s say you find $\omega=0.0001$, $\alpha=0.15$, and $\beta=0.80$. Using these values, you can forecast the asset’s future volatility, enabling you to make informed risk management decisions.