Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
17.1
Estimating Volatilities
and Correlations
Chapter 17
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
17.2Standard Approach to
Estimating Volatility
? Define sn as the volatility per day between
day n-1 and day n,as estimated at end of day
n-1
? Define Si as the value of market variable at
end of day i
? Define ui= ln(Si/Si-1)
s n n i
i
m
n i
i
m
m
u u
u
m
u
2 2
1
1
1
1
1
?
?
?
?
?
?
?
?
?
?
( )
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
17.3
Simplifications Usually Made
? Define ui as (Si-Si-1)/Si-1
? Assume that the mean value of ui is
zero
? Replace m-1 by m
This gives
s n n iimm u2 211? ???
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
17.4
Weighting Scheme
Instead of assigning equal weights to
the observations we can set
s ?
?
n i n ii
m
i
i
m
u
2 2
1
1
1
?
?
??
?
?
?
w h e r e
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
17.5
ARCH(m) Model
In an ARCH(m) model we also assign
some weight to the long-run variance
rate,VL:
?
?
?
? ?
????
????s
m
i
i
m
i iniLn
uV
1
1
22
1
w h e r e
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
17.6
EWMA Model
? In an exponentially weighted moving
average model,the weights assigned to
the u2 decline exponentially as we move
back through time
? This leads to
2
1
2
1
2 )1(
?? ????s?s nnn u
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
17.7
Attractions of EWMA
? Relatively little data needs to be stored
? We need only remember the current
estimate of the variance rate and the
most recent observation on the market
variable
? Tracks volatility changes
? RiskMetrics uses ? = 0.94 for daily
volatility forecasting
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
17.8
GARCH (1,1)
In GARCH (1,1) we assign some weight
to the long-run average variance rate
Since weights must sum to 1
? ? ? ? b ?1
2
1
2
1
2
?? bs?????s nnLn uV
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
17.9
GARCH (1,1) continued
Setting w ? ?V the GARCH (1,1) model
is
and
b???
w?
1LV
2
1
2
1
2
?? bs???w?s nnn u
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
17.10
Example
? Suppose
? The long-run variance rate is 0.0002 so
that the long-run volatility per day is
1.4%
s sn n nu2 12 120 000002 0 13 0 86? ? ?? ?.,,
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
17.11
Example continued
? Suppose that the current estimate of the
volatility is 1.6% per day and the most
recent percentage change in the market
variable is 1%.
? The new variance rate is
The new volatility is 1.53% per day
0 000002 0 13 0 0001 0 86 0 000256 0 00023336.,,,,,? ? ? ? ?
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
17.12
GARCH (p,q)
s w ? b sn i n i j
j
q
i
p
n ju
2 2
11
2? ? ?
?
??
???
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
17.13
Other Models
? We can design GARCH models so that
the weight given to ui2 depends on
whether ui is positive or negative
? We do not have to assume that the
conditional distribution is normal
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
17.14
Variance Targeting
? One way of implementing GARCH(1,1)
that increases stability is by using
variance targeting
? We set the long-run average volatility
equal to the sample variance
? Only two other parameters then have to
be estimated
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
17.15
Maximum Likelihood Methods
? In maximum likelihood methods we
choose parameters that maximize the
likelihood of the observations occurring
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
17.16
Example 1
? We observe that a certain event
happens one time in ten trials,What is
our estimate of the proportion of the
time,p,that it happens?
? The probability of the outcome is
? We maximize this to obtain a maximum
likelihood estimate,p=0.1
9)1(10 pp ?
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
17.17
Example 2
Estimate the variance of observations from a
normal distribution with mean zero
M ax i mi z e,
or,
T hi s g i v es,
1
2 2
1
2
1
2
1
2
1
? v
u
v
v
u
v
v
n
u
i
i
n
i
i
n
i
i
n
e x p
ln ( )
??
?
?
?
?
?
?
?
?
?
?
?
? ?
?
?
?
?
?
?
?
?
?
?
?
?
?
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
17.18
Application to GARCH
We choose parameters that maximize
? ?
?
?
?
?
?
?
?
? l n ( )v uvi i
ii
n 2
1
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
17.19
How Good is the Model?
? The Ljung-Box statistic tests for
autocorrelation
? We compare the autocorrelation of the
ui2 with the autocorrelation of the ui2/si2
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
17.20
Forecasting Future Volatility
A few lines of algebra shows that
The variance rate for an option expiring
on day m is
)()(][ 22 LnkLkn VVE ?sb????s ?
? ?1 2
0
1
m E n kk
m
s ?
?
??
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
17.21
Volatility Term Structures
? The GARCH (1,1) model allows us to
predict volatility term structures changes
? It suggests that,when calculating vega,
we should shift the long maturity
volatilities less than the short maturity
volatilities
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
17.22
Correlations
Define ui=(Ui-Ui-1)/Ui-1 and vi=(Vi-Vi-1)/Vi-1
Also
su,n,daily vol of U calculated on day n-1
sv,n,daily vol of V calculated on day n-1
covn,covariance calculated on day n-1
The correlation is covn/(su,n sv,n)
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
17.23
Correlations continued
Under GARCH (1,1)
covn = w + ? un-1vn-1+b covn-1
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
17.24
Positive Finite Definite
Condition
A variance-covariance matrix,W,is
internally consistent if the positive semi-
definite condition
for all vectors w
w wT W ? 0
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
17.25
Example
The variance covariance matrix
is not internally consistent
1 0 0 9
0 1 0 9
0 9 0 9 1
.
.
.,
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