Download Lecture on Volatility

Volatility
Chapter 10
1
Motivation:
• Until the early 80s econometrics had focused
almost solely on modeling the means of series,
i.e. their actual values. Recently however we
have focused increasingly on the importance of
volatility, its determinates and its effects on
mean values.
Definition of Volatility
• Suppose that Si is the value of a variable on day
i. The volatility per day is the standard
deviation of ln(Si /Si-1)
• Normally days when markets are closed are
ignored in volatility calculations
• The volatility per year is 252 times the daily
volatility
• Variance rate is the square of volatility
3
Standard 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) 及以下公式
m
1
2
 n2 
(
u

u
)

m  1 i 1 n  i
1 m
u   un  i
m i 1
4
Simplifications Usually Made in
Risk Management
• Define ui as (Si−Si-1)/Si-1
• Assume that the mean value of ui is zero
daily - short near 0
• Replace m-1 by m
(high frequent)
1 m 2
2
This gives s n = å un-i
m i=1
5
Implied Volatilities
• Of the variables needed to price an
option the one that cannot be observed
directly is volatility
• We can therefore imply volatilities from
market prices and vice versa
6
VIX Index: A Measure of the Implied
Volatility of the S&P 500
7
Linear Regression –the Workhorse
of Financial Modeling
• General form for the linear regression of Yt on
X1t , X 2 t ,..., X kt is
Yt 1  β 0  β1X1t  β 2 X 2 t  ...  β k X kt  ε t 1
ε t ~ N(0, σ )
2
• Variance of the error term is constant over time. This
property is called conditional homoscedasticity
Distinction between the Conditional
and Unconditional Variance
• The unconditional variance is just the
standard measure of the variance
var(x) =E(x -E(x))2
• The conditional variance is the measure
of our uncertainty about a variable given
a model and an information set .
cond var(x) =E(x-E(x|  ))2
this is the true measure of uncertainty
Modeling conditional means and
variances
•If  is N(0, 1), and Y = a + b,
othen the mean is E(Y) = a
oVar(Y ) = b2.
•To model the conditional mean of Yt given
X t  (X1t , X 2 t ,..., X kt )
owrite Yt as the conditional mean plus white noise
Yt  E t 1[Yt | X t 1 ]  σε t
10
Variation in the Conditional
Variance
•To allow a non-constant conditional variance in the
model, multiply the white noise term by the
conditional standard deviation. This product is
added to the conditional mean as in the previous
slide.
Yt  E t 1[Yt | X t 1 ]  σ t ε t
• t must be non-negative since it is a standard
deviation
11
Stylized Facts of asset returns
• Thick tails, they tend to be leptokurtic
• Volatility clustering, Mandelbrot, ‘large
changes tend to be followed by large changes of
either sign’
• Leverage Effects, refers to the tendency for
changes in stock prices to be negatively
correlated with changes in volatility.
• Forcastable events, volatility is high at regular
times such as news announcements or other
expected events, or even at certain times of day,
e.g. less volatile in the early afternoon.
Volatility Clustering (read books)
14
Heavy Tails
• Daily exchange rate changes are not normally
distributed
– The distribution has heavier tails than the normal
distribution
– It is more peaked than the normal distribution
• This means that small changes and large
changes are more likely than the normal
distribution would suggest
• Many market variables have this property,
known as excess kurtosis
15
Normal and Heavy-Tailed
Distribution
16
Are Daily Changes in Exchange
Rates Normally Distributed?
Real World (%)
>1 SD
>2SD
>3SD
>4SD
>5SD
>6SD
25.04
5.27
1.34
0.29
0.08
0.03
Normal Model (%)
31.73
4.55
0.27
0.01
0.00
0.00
17
ARCH and GARCH
18
References
The classics:
• Engle, R.F. (1982), Autoregressive Conditional
Heteroskedasticity with Estimates of the Variance of
U.K.
• Bollerslev, T.P. (1986), Generalized Autoregresive
Conditional Heteroscedasticity.
Introduction/Reviews:
• Bollerslev T., Engle R. F. and D. B. Nelson (1994),
ARCH Models
• Engle, R. F. (2001), GARCH 101: The Use of
ARCH/GARCH Models in Applied Econometrics.
Weighting Scheme
Instead of assigning equal weights to the
observations when calculating volatility
we can use different weights i for each
observation
2

u
i 1 i n i
m
 
2
n
where
m

i 1
i
1
20
ARCH(q) Model
Engle(1982)
Auto-Regressive Conditional Heteroscedasticity
Yt  βX t  σ t ε t where ε t ~ N(0,1)
σ  γVL  i 1 α i (σ t i ε t i )
q
2
t
2
where
q
γ  αi  1
i 1
In an ARCH(q) model we also assign some
weight to the long-run variance rate, VL:
21
Note: as we are dealing with a variance
  VL and both   0 and  i  0 all i
Even though the errors may be serially
uncorrelated they are not independent,
there will be volatility clustering and fat
tails.
22
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 (yesterday+
σ  λσ
2
t
2
t 1
 (1  λ)(Yt  E t 1[Yt | X t 1 ])
23
2
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
innovation to the market variable
• Tracks volatility changes
• RiskMetrics uses  = 0.94 for daily
volatility forecasting
24
GARCH(p,q)
Bollerslev (1986)
In empirical work with ARCH models high q
is often required, a more parsimonious
representation is the Generalised ARCH
model VL= average of LT variance
p
σ t  ω  α i σ ε
2
i 1
2
2
t i t 1
q
  β jσ
2
tj
j1
ω  γVL
25
GARCH (1,1)
In GARCH (1,1) we assign some weight
to the long-run average variance rate
σ 2t  ω ασ 2t 1ε 2t 1  βσ 2t 1
ω  γVL
ω
VL 
1  α β
Since weights must sum to 1
 +  +  =1
26
Example
• Suppose
σ  0.000002  0.13 σ ε  0.86 σ
2
t
2
2
t 1 t 1
2
t 1
• The long-run variance rate is 0.0002 so
that the long-run volatility per day is
1.4%
27
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
0000002
.
 013
.  00001
.
 086
.  0000256
.
 000023336
.
The new volatility is 1.53% per day
28
Independence vs. Zero
Correlation
• Independence implies zero correlation but not
vice versa
• GARCH processes are good examples
• Dependence of the conditional variance on the
past is the reason the process is not
independent
• Independence of the conditional mean on the
past is the reason that the process is
uncorrelated
Other Models
• Many other GARCH models have been
proposed
• For example, we can design a GARCH
models so that the weight given to i2
depends on whether i is positive or
negative
30
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
31
Maximum Likelihood Methods
• All parameters in GARCH models are
simply estimated by maximum likelihood
using the same basic likelihood function,
assuming normality
T
log( L)   ( log(  t )   t /  t )
2
2
2
i 1
• In maximum likelihood methods we
choose parameters that maximize the
likelihood of the observations occurring
32
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
p(1  p)
9
• We maximize this to obtain a maximum
likelihood estimate: p=0.1
33
Example 2
Estimate the variance of observations from a
normal distribution with mean zero
Observations are: u1 , u2 ,........, um
The variance is denoted by 
The likelihood that ui is observed is given by
the probability density function of the
normal distribution:
  ui2 
1

exp 
2v
 2v 
34
The likelihood of n observations occurring
in the order in which they are observed is:
 1
  ui2 

Maximize :  
exp 
i 1  2v
 2v 
n

ui2 
or :   ln( v)  
v 
i 1 
1 n 2
This gives : v   ui
n i 1
n
35
Application to GARCH (1,1)
We choose parameters that maximize

u 

 ln(vi )  
vi 
i 1 
n
2
i
36
Forecasting and Persistence with
GARCH(1,1) model
σ 2t 1  ω ασ 2tε 2t  βσ 2t  (1  α β) VL  ασ 2tε 2t  βσ 2t
σ 2t 1  VL  α(σ 2tε 2t  VL )  β(σ 2t  VL )
Taking expectation at time t
E t (σ 2t 1  VL )  (α  β) E t [σ 2t  VL ]
By repeated substitutions:
E t (σ2t  j )  VL  (α β) j (σ2t  VL )
As j→∞, the forecast reverts to the unconditional variance: ω/(1-α-β).
When α+β=1, as in EWMA model, the expected future variance
rate equals to today’s variance rate.