Chapter 2: FAQs 165
would like to know what it is going to be in the future,
if not precisely then perhaps know its future expected
value. This requires a model.
The simplest popular model assumes that we can get
an estimate for volatility over the nextNdays, in the
future, by looking at volatility over the previousNdays,
the past. Thismoving windowvolatility is initially
appealing but suffers from the problem that if there
was a one-off jump in the stock price it will remain in
the data with the same weight for the nextNdays and
then suddenly drop out. This leads to artificially inflated
volatility estimates for a while. One way around this is
to use the second most popular volatility model, the
exponentially weighted moving average(EWMA). This
takes the form
vn=βvn− 1 +(1−β)R^2 n− 1 ,whereβis a parameter between zero and one, and the
Rs are the returns, suitably normalized with the time
step. This models the latest variance as a weighted
average between the previous variance and the lat-
est square of returns. The largerβthemoreweightis
attached to the distant past and the less to the recent
past. This model is also simple and appealing, but it has
one drawback. It results in no time structure going into
the future. The expected variance tomorrow, the day
after, and every day in the future is just today’s vari-
ance. This is counterintuitive, especially at times when
volatility is at historical highs or lows.
And so we consider the third simplest model,
vn=(1−α−β)w 0 +βvn− 1 +αR^2 n− 1 ,the GARCH(1, 1) model. This adds a constant, long-term
variance, to the EWMA model. The expected variance,k
time steps in the future, then behaves like
E[vn+k]=w 0 +(vn−w 0 )(α+β)n.