Quantitative Analysis - Hull, Chapter 14

How do you get the weights for the GARCH(1,1)?

This is an iterative process where the the lower of the attached two equations need to be maximized. The value that is used to maximize the equation is then used to back out the weights.

What GARCH(1,1) equation is used for forecasting volatility?

* When the current volatility estimate is greater than the long-run average, the volatility term structure that is forecasted is downward sloping and vice versa.

What is the formula for updating covariance using the EWMA approach?

* Where X and Y are the current standard deviations (volatilities).

What is the ARCH(m) (Autoregressive Conditional Heteroskedasticity) model ?

A model that weighs the volatility observations based on the timing of the observations and also gives weight to the long run volatility. More recent observations receive more weight. * The w is the long run variance along with the weight associated with it.

What is the formula for variance according to Hull?

Given that the sample period is large (252 days) along with the number of observations being large (252 for daily), he makes 2 simplifying assumptions: 1) The average daily return is assumed to be zero (mean = 0) because that's close to what we see, and; 2) The denominator (m-1) is replaced with m (no df adjustment). This gives rise to a more simplified formula for variance.

What is the GARCH(1,1) (Generalized Autoregressive Conditional Heteroskedasticity) Model?

It assigns weights to the 3 different components of variance. Components: 1) Long-run average variance rate 2) Percentage change (continuous) in price between today and the prior day. 3) Prior day's variance. * Beta is sort of a decay rate. * Mean Reversion: This model assumes that the mean reverts to the long term average rate.

What is the EWMA (Exponentially Weighted Moving Average) Model?

It's a model for which older values of change receive exponentially smaller weight. But it's still a weighted average variance model. * Where "U squared for n-1" (last term) is just the percentage change (continuous) in the price of the product. * The lower the values of the A looking symbol (Lambda), the higher the weight assigned to more recent observations. * Weight for the long-run average variance rate is 0.

What is a maximum likelihood method?

One that chooses values for the parameters that maximize the chance (or likelihood) of the data occurring.

What is the assumption underlying the GARCH(1,1) model?

That volatility changes with the passage of time. If volatility in one period is high, chances of tomorrow's volatility being high are also high. A well working GARCH model should remove autocorrelation.