Et
Simple asset Continuously Compounded Returns -
returns
Pit-Pit-1
Conditional Volatility (Mar)
(F
1) Ut +
NitE (1 + rit) Pit -
Wit = In (Pit) In (Pit-1)
Pit
· -
>
N(M +, 07)
-
=
f (r (F+ 1) * variance is Now
time-varying *
(Pit) 1 (n) Pit -1)
+ =
In E+
-
Tit
Pit -I +
·
= +
> R W model w/ Bo = 5 Et ~ N(0 3) r+ M+ + E+
Model Outlines
-
. . , =
,
& ARMA : conditional mean
= In(Pt -In (P Var(E+ (F-1) = 0 = E(E1F -1] + = ECIr-Me21F 1] + - where Mt = ECr+ IF -1) +
② GARCH :
volatility f ((n(Pi n)( F+ )
+ = N(k + (n(p +) ,
k: 2) Let's demean r+ 50 r+ r+ =
M+ = E+
③ VAR :
vector of returns F E[r (F 13
2
k-periodandFre
·
=
+ -
Factor DCovariancematricss
:
L
Annualized Vol.
-
k-period ahead forecast variance of /P + +) on (# of
Modeling Single year
In
Asset
·
:
of m in a
Var (In (Pi + k) -
E((n(Pi + a) (F+ ]) = koz
Em /# of
* 1
f(r N(M 0 1)
near
=
F+) =
: m in a
++
+ +1 , ++
f((+ (F+ ) = N(k k 2)
↑ d f . & info ,
p . .
.
at time + ·
K-period ahead forecast of ri+k = In (P++) -
In (Pt) :
Check for non-constant
E (Ti + = (F+ ] = E[(n(Pi + 1)(F+ ] -
In(p +) volatility clustering
Autoregressive Models for returns -E
= k
·
look
o ACF of E & test Ho :
Pro
~( No time-varying volatility
returns over time I K-period ahead forecast of variance of ritk
: , r z
=
Var (ri + k =
E(ri + /F+ ]) = kot >
-
evidence of time-varying volatility if rejectH o
known
~
unknown
t I
T
Model Selection Estimators
Volatility
properties of returns Criterions /out of
sample)
* skewness degree of symmetry T
=
2
① Annualized Historical volatility k most recent days
-
F (ri -ri) over :
& Lowest MSE =
,
=
2 K1- smoother vol & less
(t Variance)
1)
as > extreme values
R
· .
② Highest =
-a & smaller MSE
own up to +
& tailed tail
= more Xtreme val . in R tail than L ③ Smallest Al = In (2) + (k+ 1) =
& Kurtosis :
weight in tails relative to middle ② Exponential Smoothing Volatility Estimator day I
① Smallest
(k)
from
(i "thickness" of
BIC : In ( +
: Algorithm starting
tails) Crews
"Leptokurtic" alphaI
Var(r) o estimate sir+
e common : ① estimate
X
.
.
-
penalty for ↑K has : =
11.
weight ②
+
&
more
= (ri - ) 33 : fatter fails than (N) Procedure
I Unknown :
of - E
+ 21 ⑤ obtain of. . . . . of
= 3 : Normal (N)
54 1) Estimate & select small set of models
3 3 :
thinner tails than (N) data ARCH(9) ' GARCH (P
modeled data
from
training 2)
Normal dist . ,
,
> If b wou Id
Y you
-
2) Predict V models ?
on
testing data Q : What is the volatility now
&
13 over-estimate events
probability of 3) Compare predictive
:
extreme
accuracy of each model
> 3 :
under-estimate probability of events
extreme using a criterion
ARCH(9) :
4)) Choose model
③ Test of Normality :
Jarque-Bera
Ho : skew : excess Kurtosis = 0 >
-
Normal distribution
Splitting Data Et Ej
+
- -
Reject if : p-value < 0 05 Var(t+ (F+ 1) = h =
① Simple
-
.
: train ,
o test
⑪ Var(
edeF
-,) = h+ = 0 E(r" (F 1] conditional
Rolling Window
= is the
③ variance
+ -
of returns
② Expanding Window :
Lei
Pred-
-
Reject Ho if :
(Pr . r+ c) E OR p-val . 1 0 05 .
- Predict T GARCH(P 9) , is similar to exponential smoothing
↑
W
If
Pit
r+
X (Pr 1) (Pr+, r+ 2) )(Pr+ 1)
(
-
Pred =
+ 1 :
> h
* unknown
r+ - ...
m
-
, -
: ,
,
Var = h
>
-
slowly declining ACF
-j
·
-
BiriBr Pred Another way of writing this (GARCH (1 1) ,
Model) :
2
(AR(p) Var( o (l Bh
period ahead B)
Bi 0 h+
Til Multiple Forecasts
= = = -
x -
+ ar + + 1
or p-value on
variance of Et is
weighted aug. of
>
-
a
Lets consider AR(I) model + Bo B, r+ Et
①o
an + +
Model 1 : =
AR(P)
1
constant/unconditional variance
lags
:
is # of #
:
where p
* ACF
slowly declining model (IB 1)
2+ ②
- 1 :
Yesterday's news
A
select from PACF plot SPACF
Stationary/mean-reverting . u
p t
PACFp3 ③ h +. 1 Yesterday's forecast
)
s :
f(r /F ) N(M + B, (r
. .
, ....
are significant &SPACFp +, ... 3 are NOT ++ + = + -
M) ,
TARCH(P P)
~ +=
Bor Birt + E
,
~
only active if E+ > 0
-
1 variri =
MBB
02 + ~N(0 0) kn ow
d
·
, i i . .
m = = h
② E + 1+ 1 ...., + - T
↓ La
EVIEWS : S E of
EFj +.j
+
. .
E(r(F 1) + -
= =
Bo regression
~
N( ,)
95 % [1 : r+ 28,/ If 30 -
Var(r+ /F + 1) ↑ when Ij is
negative
f(r+ (F + 1)
-
=
=
-
Var (5+ + -
E(F + ]) 2 Et
~
Model Chack - Unconditional Variance (Var) V () /Avg + . Variance
Test Ho : Pe +, e+ = 0 L = 0 >
-
2+ 1 2 +
, 1
Let's demean -Mt
+
r=
so
should fail
05)
oh G
model
to rejectH o if fits (i ↓
>
P-val . 0 ~ Et
-
e
upper-bonda Var)) 0
. .
ECr] E[h ]
.
V + = = = + (L .
I E
AR(1) Model
. .
(M)
Plugging into a GARCH
(11)
model:
Mean-reverting 1)
Model/Random
return
Walk (B ,
=
Var(rg
=
,
D /B 1 <1
Non-Stationary E(h + ] dE[r2] BE[h 1]
Stationary (Mean-reverting
= w + + +
. : -
f(p+ + k(F+1 N(p + kBo ko)
dES r BECn
=
#- M (B , )(r+ M) Et
+
=
,
+
- 1]
=
1
-
= w + +
E((r+ -
m) (F+ 1) = (B , ((r+ 1
-
M)
Vario =
P++ k
kBo P+ 2 ++ + 2+ + k
+ B() = w + BBo O
② B= 1
+ +
Non-stationary Wi
: = + ...
w
E(P++ /F+ ] Sample Var (r+ )
=
kBo + P+
drift param :"
Bo
1 + B111
O
③ IB / 1
if >
In of Stationary/mean-reverting
↑
controls direction order for to exist
Explosive/Non-Stationary
+
:
.
a
Boas
,
of
Model 2 : ARMA(P, C)
wandering
N+ =
Bo + ,
Birt
&i -i +
a
+i + Et Var(P ++ k -
E(P+ + k(F + ]) = ko GARCH(P 9) model where f(r) = N) . . .
-
,
J
① E+ 07 " key Assumption
:HM
~i i .
.
d .
N10 ,
② E+ t r+ - 1 ,
...,
N+ - T = 0
Sno bound ht
LE
+ +
as k + a upper +
ELr(Ft -
1) = = Bo Bjrj +- wherei t did w/ mean 0 & variance 1 (doesn't have to be N)
f(r+ (F+ 1) = N( ,) 19 % Cl .: # = zE Tests of a R W
.
.
Var(iT ==
E E E
,
=
=
Model 3 :
MA(9) where : #E+ lags
Dickey-Fuller Test : Ho :
rnit root B = 1 - R W
declining
>
A PACF
.
Slowly
-
,
Diagnostic Checking
Ha : not unit root >
- B, * 1 - not
* Select
a from ACF plot s t SACF ACFq3 R W
.
2
.
. , ..., -
are significant SACFaH , .... 3 are NOT Reject if p-value is small Let = E and n = 2 - 200 1
,
Vari
+ + Var(2 + )
=
=
# Bo +
is
① Et ~i i . .
d .
N(0 , 8) G ARCH(P , 2) model is correct if Var(2 ) + = /
EIB 1. : 125
Test : look & ACF of If which tests Ho :
P2F + ,
= 0 >
- No time-varying
volatility
·
corr(t+, m+ - 1) = . .. = corr(r+, r+ q) = 0
>
- If fail to reject Ho ->no time-varying volatility >
-
GARCH (P , 4)
cour(r+, r 1) corr(t+, r T) 0 model
good
·
19+
is
+ = = =
+
. ..
·
M Bo
= =
