The GARCH model was introduced by Engle [1982], Engle and Bollerslev [1986],
and used in countless subsequent papers. The equations are easier to understand
in the formulation given in Zumbach [2004]

with the three parameters σ∞,w∞,τ, and with μ = exp(-δt ∕ τ). In this form,
the GARCH(1,1) process is written with one “internal variable” σ_{1} that
can be interpreted as a historical volatility measured by an exponential
moving average at the time horizon τ. The variance for the next time
step σeff^{2}(t) is given by the mean term, plus the difference between the
historical variance and the mean variance, weighted by 1 - w∞. In the
second form, the next-step variance is a convex combination of the mean
variance σ∞^{2} and the historical variance σ_{
1}^{2}. With these equations, the three
parameters have a direct interpretation in terms of the properties of the time
series.

Usually, the GARCH volatility equation is written in the equivalent form

(4)

as it appears in the original article. The 3 parameters can be translated between
both forms by using α_{0} = σ∞^{2}(1 - μ)w∞, α_{1} = (1 - w∞)(1 - μ) and
β_{1} = μ. Although equivalent, the form (4) for the volatility equation has two
drawbacks. First, it is less amenable to an intuitive understanding of
the different terms. Second, the natural generalisation of Eq. 4 leads to
the GARCH(p,q) form. Most studies find that GARCH(p,q) does not
improve much over GARCH(1,1) for resonably small values of p and q.
On the other hand, in the form 3, the GARCH equations lead to the
introduction of more components σ_{1},σ_{2},σ_{3},..., with increasing time horizons
τ_{1},τ_{2},τ_{3},.... With a few components, a large span of time intervals can be
covered.

The process equations are affine for the variance, because the parameter σ∞^{2} (or
α_{0}) appears as an additive constant. This constant is important, as it fixes the
mean level for the volatility.

The correlations for the GARCH(1,1) process can be computed analytically; they
decay exponentially fast with a characteristic time τcorr = -δt ∕ ln(μcorr) with
μcorr = α_{1} + β_{1} = μ + (1 -w∞)(1 -μ). The k-steps variance forecast formula can
be expressed as

(5)

showing explicitely the exponential mean reversion toward σ∞^{2} for large time
intervals Δt.

The parameters for the simulations are

σ∞ = 0.11

w∞ = 0.1.

τ = 1 day

This corresponds to α_{1} = 0.00187, β_{1} = 0.99792 and a decay for the correlation of
10.0 day. The innovations have a Student distribution with 3.3 degree of freedom.
The simulation time corresponds to 200 years with a time increment δt = 3
minutes.

References

R. F. Engle. Autoregressive conditional heteroskedasticity with estimates
of the variance of U. K. inflation. Econometrica, 50:987–1008, January
1982.

Robert F. Engle and Tim Bollerslev. Modelling the persistence of
conditional variances. Econometric Reviews, 5:1–50, 1986.

Gilles Zumbach. Volatility processes and volatility forecast with long
memory. Quantitative Finance, 4:70–86, 2004.