Analyzing volatility

GARCH Models in R

Kris Boudt

Professor of finance and econometrics

About the instructor

Kris Boudt
- PhD in financial risk forecasting
- Use GARCH models to win by not losing (much)
R package rugarch of Alexios Ghalanos.

Kris Boudt

Calculating returns

Relative financial gains and losses, expressed in terms of returns

$$ R_{t} = \frac{P_{t} - P_{t-1}}{P_{t-1}} $$

Function CalculateReturns in PerformanceAnalytics

# Example in R for daily S&P 500 prices (xts object)
library(PerformanceAnalytics)
SP500returns <- CalculateReturns(SP500prices)

Daily S&P 500 returns

Properties of daily returns:

The average return is zero
Return variability changes through time

Standard deviation = measure of return variability.

Synonym: Return volatility.

Greek letter $\sigma_t$.

Distribution of daily returns per year

How to estimate return volatility

Function sd() computes the (daily) standard deviation:

sd(sp500ret)

0.01099357

Corresponding formula for $\hat \sigma$ computed using $T$ daily returns:

$$ \hat \sigma = \sqrt{\frac{1}{T-1} \sum_{t=1}^T (R_t - \hat \mu )^2},$$

where $\hat \mu$ is the mean return.

Annualized volatility

sd(sp500ret) is daily volatility
Annualized volatility = $\sqrt{252}$ $\times$ daily volatility

# Compute annualized standard deviation
sqrt(252) * sd(sp500ret)

0.1745175

Distribution of daily returns per year

Rolling volatility estimation

Rolling estimation windows :
Window width? Multiple of 22 (trading days).

Function chart.RollingPerformance()

library(PerformanceAnalytics)

chart.RollingPerformance(R = sp500ret,
                         width = 22,
                         FUN = "sd.annualized",
                         scale = 252, 
                         main = "Rolling 1 month volatility")

Rolling window volatility

About GARCH models in R

Estimation of $\sigma_t$ requires time series models, like GARCH.

Let's refresh the basics of computing rolling standard deviations in R

GARCH Models in R