# Chapter 10 Modeling Daily Returns with the GARCH Model

Updated: May 12, 2021

Copyright © Eric Zivot 2016

Outline

- Motivation
- Review stylized facts of daily return data and contrast with monthly returns
- Modeling daily returns is most useful for short term risk analysis of assets and portfolios using volatility and VaR
- CER model captures most stylized facts of monthly returns but not daily returns
- Need a model for time varying volatility
- Engle’s ARCH model captures stylized facts of daily returns and he was awarded Nobel prize in economics partially for this model.

- Engle’s ARCH Model
- Start with ARCH(1) process
- State assumptions and derive properties
- Show simulated values and do reality check against actual data
- Introduce rugarch R package
- General ARCH(p) process

- Bollerslev’s GARCH(1,1) process
- No need to go beyond GARCH(1,1)
- Give assumptions and derive basic properties
- Show simulated values and do reality check against actual data
- GARCH(1,1) with Student t errors
- Show simulated values and do reality check against actual data

- Maximum Likelihood Estimation
- Overview of the technique of MLE
- Illustration with CER model and GARCH(1,1)
- Numerical optimization
- Asymptotic properties
- Calculating standard errors

- Maximum likelihood estimation of ARCH and GARCH models
- GARCH(1,1) log-likelihood function

- Forecasting Conditional Volatility from GARCH(1,1)
- Forecasting algorithm

- Multiperiod

- Conditional VaR
- unconditional vs. conditional VaR
- 1-day ahead VaR forecast
- h-day ahead VaR forecast

In chapter 5, it was shown that daily asset returns have some features in common with monthly asset returns and some not. In particular, daily returns have empirical distributions with much fatter tails than the normal distribution and daily returns are not independent over time. Absolute and squared returns are positively autocorrelated and the correlation dies out very slowly. Daily volatility appears to be autocorrelated and, hence, predictable. In this chapter we present a model of daily asset returns, Robert Engle’s (ARCH) model, that can capture these stylized facts that are specific to daily returns. The ARCH model is one of the most important models in the field of financial econometrics, and its creator Robert Engle won the Nobel Prize in Economics in part for his work on the ARCH model and its variants.

The R packages used in this chapter are **IntroCompFinR**, **rugarch**
and **xts**. Make sure these package are installed and loaded
before running the R examples in this chapter.

- Why do we care about time varying volatility? Volatility is a standard measure of asset risk and if volatility varies over time then asset risk also varies over time. In periods of low volatility, asset risk is low and in periods of high volatility asset risk is high. Time varying volatility also impacts the risk measure value-at-risk (VaR), which gives the dollar loss that could occur over a period of time with a given probability. Volatility is an important driver for the price of call and put options as well as other derivative securities. That volatility is predictable has important implications for accurate pricing models for options and derivatives.
- Why do we care about the predictability of volatility? We can forecast future values of volatility. This allows us to forecast VaR on an asset or portfolio.