Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
15.1
Estimating
Volatilities and
Correlations
Chapter 15
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
15.2Standard Approach to
Estimating Volatility (Equation 15.1)
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,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
15.3
Simplifications Usually Made
(Equation 15.4)
Define ui as (Si-Si-1)/Si-1
Assume that the mean value of ui is zero
Replace m-1 by m
This gives (MLE)
s n n iimm u2 211
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
15.4
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 he r e
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
15.5
ARCH(m) Model
In an ARCH(m) model we also assign some
weight to the long-run variance rate,V:s
n i n ii
m
i
i
m
V u
2 2
1
1
1
w her e
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
15.6
EWMA Model
(Equation 15.7)
In an exponentially weighted moving average
model,the weights assigned to the u2 decline
exponentially as we move back through time
This leads to (a special case of (15.4) with
i+1=i,0<?<1)
s?s?n n nu2 12 121( )
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
15.7
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
JP Morgan use? = 0.94 for daily volatility
forecasting
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
15.8
GARCH (1,1)
(Equation 15.8)
In GARCH (1,1) we assign some weight
to the long-run average variance rate
Since weights must sum to 1
b?1
ss bn n nV u2 12 12
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
15.9
GARCH (1,1) (continued)
Setting wV,the GARCH (1,1) model
is
and
s w?s bn n nu2 12 12
Vw? b1
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
15.10
Example
Suppose
the long-run variance rate is V=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,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
15.11
Example (continued)
Suppose that the current estimate of the
volatility is 1.6% per day and the most recent
proportional 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,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
15.12
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,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
15.13
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,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
15.14Choice between EWMA and
GARCH(1,1) (Comparison)
Similarity:
-- b in GARCH(1,1) is similar to? in EWMA -- the decay rate
-- If w=0,then GARCH(1,1) reduces to EWMA
Difference:
-- ARCH(1,1) assigns some weight to the long-run average
volatility,while EWMA does not.
-- GARCH(1,1) incorporates mean-reversion whereas EWMA
does not.
-- If the estimate of w < 0,GARCH(1,1) is not stable,
Use EWMA instead.
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
15.15
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,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
15.16
Maximum Likelihood Methods
(P,374)
–In maximum likelihood methods we
choose parameters that maximize the
likelihood of the observations occurring
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
15.17
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
10 1 9p p( )?
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
15.18
Example 2
Estimate the variance of observations from a
normal distribution with mean zero
2
1
2
1
2
1
1
M a x im iz e,L ( ) = e x p
22
or,l og L ( ) = l n( )
1
T h is giv e s,
n
i
i
n
i
i
n
i
i
u
vv
u
v
v
vu
n
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
15.19
Application to GARCH
We choose parameters that maximize
Example—The Japanese yen exchange
rate (p,375-376)
w=0.00000176,?=0.0626,b=0.8976
l n ( )v uvi i
ii
n 2
1
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
15.20
How Good is the Model?
(Table 15.2,p,378)
The Ljung-Box statistic tests for autocorrelation
If K=15,the critical value for?=5% is 25
where?k is the ACF for lag k,wk=(m-2)/(m-k)
We compare the autocorrelation of the ui’s
with the autocorrelation of the ui/si
LB=123 for ui2 => strong autocorrelation;
LB=8.2 for ui2/ si2 =>weak autocorrelation
2
1L j u n g - B o x s t i s t i c = m
K
kkk w
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
15.21
Forecasting Future Volatility
(Equation 15.13,page 379)
A few lines of algebra shows that
The variance rate for an option expiring on
day m is
E V Vn k k n[ ] ( ) ( )s? b s2 2
1 2
0
1
m E n kk
m
s?
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
15.22
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,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
15.23
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
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
15.24
Correlations (continued)
Under EWMA
covn=? covn-1+(1-?)un-1?n-1
Under GARCH (1,1)
covn= w +? un-1vn-1+b covn-1
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
15.25
Positive Finite Definite
Condition (Equation 15.16,p,384)
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,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
15.26
Example
The variance covariance matrix
is not internally consistent
1 0 0 9
0 1 0 9
0 9 0 9 1
.
.
.,
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
15.27
Assignments
15.2,15.3,15.5,15.7-15.13
Assignment Questions
Tang Yincai,? 2003,Shanghai Normal University
15.1
Estimating
Volatilities and
Correlations
Chapter 15
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
15.2Standard Approach to
Estimating Volatility (Equation 15.1)
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,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
15.3
Simplifications Usually Made
(Equation 15.4)
Define ui as (Si-Si-1)/Si-1
Assume that the mean value of ui is zero
Replace m-1 by m
This gives (MLE)
s n n iimm u2 211
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
15.4
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 he r e
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
15.5
ARCH(m) Model
In an ARCH(m) model we also assign some
weight to the long-run variance rate,V:s
n i n ii
m
i
i
m
V u
2 2
1
1
1
w her e
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
15.6
EWMA Model
(Equation 15.7)
In an exponentially weighted moving average
model,the weights assigned to the u2 decline
exponentially as we move back through time
This leads to (a special case of (15.4) with
i+1=i,0<?<1)
s?s?n n nu2 12 121( )
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
15.7
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
JP Morgan use? = 0.94 for daily volatility
forecasting
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
15.8
GARCH (1,1)
(Equation 15.8)
In GARCH (1,1) we assign some weight
to the long-run average variance rate
Since weights must sum to 1
b?1
ss bn n nV u2 12 12
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
15.9
GARCH (1,1) (continued)
Setting wV,the GARCH (1,1) model
is
and
s w?s bn n nu2 12 12
Vw? b1
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
15.10
Example
Suppose
the long-run variance rate is V=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,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
15.11
Example (continued)
Suppose that the current estimate of the
volatility is 1.6% per day and the most recent
proportional 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,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
15.12
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,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
15.13
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,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
15.14Choice between EWMA and
GARCH(1,1) (Comparison)
Similarity:
-- b in GARCH(1,1) is similar to? in EWMA -- the decay rate
-- If w=0,then GARCH(1,1) reduces to EWMA
Difference:
-- ARCH(1,1) assigns some weight to the long-run average
volatility,while EWMA does not.
-- GARCH(1,1) incorporates mean-reversion whereas EWMA
does not.
-- If the estimate of w < 0,GARCH(1,1) is not stable,
Use EWMA instead.
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
15.15
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,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
15.16
Maximum Likelihood Methods
(P,374)
–In maximum likelihood methods we
choose parameters that maximize the
likelihood of the observations occurring
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
15.17
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
10 1 9p p( )?
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
15.18
Example 2
Estimate the variance of observations from a
normal distribution with mean zero
2
1
2
1
2
1
1
M a x im iz e,L ( ) = e x p
22
or,l og L ( ) = l n( )
1
T h is giv e s,
n
i
i
n
i
i
n
i
i
u
vv
u
v
v
vu
n
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
15.19
Application to GARCH
We choose parameters that maximize
Example—The Japanese yen exchange
rate (p,375-376)
w=0.00000176,?=0.0626,b=0.8976
l n ( )v uvi i
ii
n 2
1
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
15.20
How Good is the Model?
(Table 15.2,p,378)
The Ljung-Box statistic tests for autocorrelation
If K=15,the critical value for?=5% is 25
where?k is the ACF for lag k,wk=(m-2)/(m-k)
We compare the autocorrelation of the ui’s
with the autocorrelation of the ui/si
LB=123 for ui2 => strong autocorrelation;
LB=8.2 for ui2/ si2 =>weak autocorrelation
2
1L j u n g - B o x s t i s t i c = m
K
kkk w
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
15.21
Forecasting Future Volatility
(Equation 15.13,page 379)
A few lines of algebra shows that
The variance rate for an option expiring on
day m is
E V Vn k k n[ ] ( ) ( )s? b s2 2
1 2
0
1
m E n kk
m
s?
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
15.22
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,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
15.23
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
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
15.24
Correlations (continued)
Under EWMA
covn=? covn-1+(1-?)un-1?n-1
Under GARCH (1,1)
covn= w +? un-1vn-1+b covn-1
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
15.25
Positive Finite Definite
Condition (Equation 15.16,p,384)
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,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
15.26
Example
The variance covariance matrix
is not internally consistent
1 0 0 9
0 1 0 9
0 9 0 9 1
.
.
.,
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
15.27
Assignments
15.2,15.3,15.5,15.7-15.13
Assignment Questions
