Illustrates an example of GARCH

Question

Illustrates an example of GARCH.

Answer 1

The simplest GARCH family member is GARCH (1,1) wherein the variance, v_n, of stock returns at time step n is modelled through

v_n = (1 - α - β)w₀ + βv_n-1 + αv_n-1B²_n-1,

Here w₀ is the long-term variance, α and β are positive parameters, along with α + β< 1, and B_n are independent Brownian motions, which is, random numbers drawn by a normal distribution. The newest variance, v_n, can therefore be thought of like a weighted average of the latest variance, the long-term average and the latest square of returns.