Comparing GARCH models

This is a summary of the article Volatility Forecasting I: GARCH Models by Reider (2009). The examples and graphs are created using the R package rugarch.

According to Reider, there are three main uses of forecasting volatility: risk management, asset allocation, and for taking bets on future volatility. My focus is on the last one, taking bets on future volatility.

Stylized facts about volatility in financial time series:

• Squared returns ares positively autocorrelated
• Volatility is characterized by spikes that eventually return to mean
• Returns exhibit excess kurtosis

Choosing a model 

If the models have the same number of paramters, we can simply compare the maximum value of their likelihood functions.

In this example, I'm using daily returns from the index FTSE 100 between 2004 and 2013.

Returns:

Square returns:

The conditional volatility from a GARCH(1,1) model compared to absolute returns:

The likelihood function of the GARCH(1,1) model is 8470.921. If I want to compare that to a model with more paramters (say a GARCH(2,1), then I need to compare the AIC.

AIC
GARCH(1,1): -6.4915
GARCH(2,1): -6.4934

Since the GARCH(2, 1) has a lower AIC, it produces a higher maximum likelihood after penalizing for the increase in parameters. I.e. it is the better model according to AIC. Reider writes that Bayes Information Criterion does a better job of penalizing for an increase in the number of paramters. When we compare this measure, the GARCH(1,1) is better.

BIC
GARCH(1,1): -6.4780
GARCH(2,1): -6.4777

Another test that can be performed is to test if the new residuals are i.i.d. This can be done using the Portmanteau test.

Testing GARCH models for volatilty trading

From reider:

To test whether the GARCH-type models can be used as a volatility trading strategy, it must be compared with implied volatilities. If, for example, our forecast for realized volatility is higher than the implied volatility, we could buy the option and delta hedge it to capture this di erence. If we are correct and the realized volatility turns out to be higher than the implied volatility, our pro ts from gamma trading the option should exceed the costs of the option.

And some final thoughts for future investigations:

A relatively recent working paper by Ahoniemi (2006) /.../ tries to forecast changes in the VIX, which is a weighted average of put and call volatility for several strikes and two near-dated expirations. No attempt was made to forecast implied volatilities of longer-dated options. She uses an ARMA(1,1) model to forecast changes in the VIX, but augments it with GARCH forecasts as well as several exogenous variables.

Source: http://cims.nyu.edu/~almgren/timeseries/Vol_Forecast1.pdf