Estitnating
Volatilities
and
Correlations
In
this chapter we explain how historical data can be used to produce estimates of the
current and future levels of volatilities and correlations. The chapter is relevant both to
the calculation of value at risk using the modelbuilding approach and to the valuation
of derivatives. When calculating value at risk, we are most interested in the current
levels of volatilities and correlations because we are assessing possible changes in the
value of a portfolio over a very short period of time. When valuing derivatives, forecasts
of volatilities and correlations over the whole life of the derivative are usually required.
The chapter considers models with imposing names such as exponentially weighted
moving average (EWMA), autoregressive conditional heteroscedasticity (ARCH), and
generalized autoregressive conditional heteroscedasticity (GARCH). The distinctive
feature of the models is that they recognize that volatilities and correlations are not
constant. During some periods, a particular volatility or correlation may be relatively
low, whereas during other periods it may be relatively high. The models attempt to keep
track of the
variation~
in
the volatility or correlation through time.
19.1
ESTIMATING VOLATILITY
Define
(J1l
as the volatility of a market variable on day
n,
as estimated at the end of
day
12

L The square of the volatility,
(J~,
on day
12
is the
variance rate.
We described
the standard approach to estimating
from historical data in Section 13.4. Suppose
that the value of the market variable at the end of day
i
is
Si'
The variable
lli
is defined
as the continuously compounded return during day
i
(between the end of day
iI
the end of day i):
S·
=
In'
SiI
An unbiased estimate of the variance rate per day,
observations on the
using the most recent
I1l
(19.1)
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CHAPTER 19
where
it
is the mean of the
lliS:
1
//I
=
Llllli
111
i=l
For the purposes ofmonitoring daily volatility, the formula in equation (19.1) is usually
changed in a number of ways:
1.
lli
is defined as the percentage change in the market variable between the end of
day
iI and the end ofday
i,
so that:!
(19.2)
2.
is assumed to be zero?
3.
111

1 is replaced by
111.
3
These three changes make very little difference to the estimates that are calculated, but
they allow us to simplify the formula for the variance rate to
1
1
,,1
a;;
= 
£...J
ll;;i
111
i=!
where
is given by equation (19.2).4
(19.3)
Weighting Schemes
Equation (19.3) gives equal weight to
ll~l>
ll~2"'"
ll~_I/1'
Our objective is to estimate
the current level of volatility,
all'
It
therefore makes sense to give more weight to recent
data. A model that does this is
(19.4)
The variable
(Xi
is the amount of weight given to the observation
i
days ago. The
(X's
are
positive. If we choose them so that
(Xi
<
(Xj
when
i
>
j,
less weight is given to older
observations. The weights must sum to unity, so we have
I
This is consistent with the point made in Section 18.3 about the way that volatility is defined for the
purposes of VaR calculations.
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 Spring '11
 DonBlasius
 Math, Correlation, Volatility, Maximum likelihood, GARCH, variance rate

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