Transcript Estimating Volatilities and Correlations, Chapter 15

Estimating
Volatilities and
Correlations
Chapter 21
2015/7/17
1
Standard Approach to
Estimating Volatility
 Define sn as the volatility per day on 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), an unbiased estimate is
m
1
2
s n2 
(
u

u
)

m  1 i 1 n i
1 m
u   un  i
m i 1
2015/7/17
2
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
1 m 2
s  i 1 un i
m
2
n
2015/7/17
3
Weighting Scheme
Instead of assigning equal weights to the
observations we can set
s  i 1  i u
m
2
n
2
n i
where
m

i 1
2015/7/17
i
1
4
ARCH(m) Model
In an ARCH(m) model we also assign some
weight to the long-run variance rate, V:
s  V  i 1  i u
m
2
n
2
n i
where
m
   i  1
i 1
2015/7/17
5
EWMA Model
 In an exponentially weighted moving
average model, the weights assigned to the
u2 decline exponentially as we move back
through time.
2015/7/17
6
 Especially, if
 i 1   i
1  1  1 2 
s  (1   )(u
2
n
2
n 1
 u
 1,  1  (1   )
2
n2
 u
2
2
n 3

)
s n21  (1   )(un2 2  un23   2un2 4  )
s  s
2
n
2
n 1
 (1   )u
2
n 1
m
 s n2  (1   )  i 1un2i   ms 02
i 1
2015/7/17
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
 JP Morgan use  = 0.94 for daily volatility
forecasting
2015/7/17
8
GARCH (1,1)
In GARCH (1,1) we assign some weight to
the long-run average variance rate
s  V  s
2
n
2
n 1
 bu
2
n 1
Since weights must sum to 1
    b 1
2015/7/17
9
GARCH (1,1) continued
Setting w  V the GARCH (1,1) model is
s  w  s
and
2
n
2
n 1
 bu
2
n 1
w
V 
1   b
2015/7/17
10
Example
 Suppose
s  0.000002  013
. u
2
n
2
n 1
 086
. s
2
n 1
 the long-run variance rate is 0.0002 so that
the long-run volatility per day is 1.4%
2015/7/17
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
0000002
.
 013
.  00001
.
 086
.  0000256
.
 000023336
.
The new volatility is 1.53% per day
2015/7/17
12
GARCH (p,q)
p
q
i 1
j 1
s n2  w  i un2i   b j s n2 j
2015/7/17
13
Mean Reversion
 The GARCH(1,1) is equivalent to a model
where the variance V follows the
stochastic process
dV  a(VL  V )dt  Vdz
 GARCH(1,1) incorporates mean reversion,
EWMA does not. When w is negative,
GARCH(1,1) is not stable, and we should
use EWMA.
2015/7/17
14
Other Models
 We can design GARCH models so that the
weight given to ui2 depends on whether ui is
positive or negative.
2015/7/17
15
Maximum Likelihood Methods
In maximum likelihood methods we choose
parameters that maximize the likelihood of
the observations occurring
2015/7/17
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
9
p(1  p)
 We maximize this to obtain a maximum
likelihood estimate: p=0.1
2015/7/17
17
Example 2（Suppose the variance is
constant)
Estimate the variance of observations from a
normal distribution with mean zero
 1
 ui2  
exp



 2v  
i 1  2 v
n
Maximize:

ui2 

 ln(v )  v 
i 1 

1 n 2
v   ui
n i 1
n
or:
This gives:
2015/7/17
18
Application to GARCH
We choose parameters that maximize

u 

 ln(vi )  
vi 
i 1 
n
2015/7/17
2
i
19
Excel Application (Table 21.1, page 477)
 Start with trial values of w, , and b
 Update variances
 Calculate

ui2 

 ln( vi )  
vi 
i 1 
m
 Use solver to search for values of w, , and b that
maximize this objective function
 Important note: set up spreadsheet so that you are
searching for three numbers that are the same
order of magnitude (See page 478)
2015/7/17
20
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.
2015/7/17
21
How Good is the Model?
 We compare the autocorrelation of the ui’s
with the autocorrelation of the ui/si
 The Ljung-Box statistic tests for
autocorrelation
2015/7/17
22
Forecasting Future Volatility
A few lines of algebra shows that
E[s n2k ]  V  (  b ) k (s n2  V )
The variance rate for an option expiring on
day m is
1 m1
2
E
s
 n k
m k 0

2015/7/17

23
Forecasting Future Volatility
continued (equation 19.4, page 473)
Def ine
1
a  ln
 b
The volatility per annum f or a T - day option is


1  e  aT
V (0)  VL 
252VL 
aT


2015/7/17
24
Volatility Term Structures (Table 21.4)
 The GARCH (1,1) suggests that, when calculating vega,
we should shift the long maturity volatilities less than the
short maturity volatilities
 Impact of 1% change in instantaneous volatility for
Japanese yen example:
Option Life
(days)
10
30
50
100
500
Volatility
increase (%)
0.84
0.61
0.46
0.27
0.06
2015/7/17
25
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
2015/7/17
26
Correlations continued
Under GARCH (1,1)
covn = w +  un-1vn-1+b covn-1
2015/7/17
27
Positive semi-definite Condition
A variance-covariance matrix, W, is internally
consistent if the positive semi-definite
condition
w Ww  0
T
for all vectors w is satisfied.
2015/7/17
28
Example
The variance covariance matrix
 1

 0

 0.9
0
1
0.9
0.9

0.9

1
is not internally consistent
2015/7/17
29