heavy tails and financial time series models
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Heavy Tails and Financial Time Series Models. Richard A. Davis Columbia University www.stat.columbia.edu/~rdavis Thomas Mikosch University of Copenhagen. Outline. Financial time series modeling General comments Characteristics of financial time series - PowerPoint PPT PresentationTRANSCRIPT
1SAMSI RISK WS 2007
Heavy TailsHeavy Tails and Financial and Financial Time Series ModelsTime Series Models
Richard A. Davis
Columbia University
www.stat.columbia.edu/~rdavis
Thomas MikoschUniversity of Copenhagen
2SAMSI RISK WS 2007
Outline
Financial time series modeling
General comments
Characteristics of financial time series
Examples (exchange rate, Merck, Amazon)
Multiplicative models for log-returns (GARCH, SV)
Regular variation
Multivariate case
Applications of regular variation
Stochastic recurrence equations (GARCH)
Stochastic volatility
Extremes and extremal index
Limit behavior of sample correlations
Wrap-up
3SAMSI RISK WS 2007
Financial Time Series Modeling
One possible goal: Develop models that capture essential features of financial data.
Strategy: Formulate families of models that at least exhibit these key characteristics. (e.g., GARCH and SV)
Linkage with goal: Do fitted models actually capture the desired characteristics of the real data?
Answer wrt to GARCH and SV models: Yes and no. Answer may depend on the features.
Stǎricǎ’s paper: “Is GARCH(1,1) Model as Good a Model as the Nobel Accolades Would Imply?”
Stǎricǎ’s paper discusses inadequacy of GARCH(1,1) model as a “data generating process” for the data.
4SAMSI RISK WS 2007
Financial Time Series Modeling (cont)
Goal of this talk: compare and contrast some of the features of GARCH and SV models.
• Regular-variation of finite dimensional distributions
• Extreme value behavior
• Sample ACF behavior
5SAMSI RISK WS 2007
Characteristics of financial time series
Define Xt = ln (Pt) - ln (Pt-1) (log returns)
• heavy tailed
P(|X1| > x) ~ RV(-),
• uncorrelated
near 0 for all lags h > 0
• |Xt| and Xt2 have slowly decaying autocorrelations
converge to 0 slowly as h increases.
• process exhibits ‘volatility clustering’.
)(ˆ hX
)(ˆ and )(ˆ 2|| hhXX
6SAMSI RISK WS 2007
day
log
retu
rns
(exc
hang
e ra
tes)
0 200 400 600 800
-20
24
lag
AC
F
0 10 20 30 40
0.0
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f squ
ares
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f abs
val
ues
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1.0
Example: Pound-Dollar Exchange Rates (Oct 1, 1981 – Jun 28, 1985; Koopman website)
7SAMSI RISK WS 2007
Example: Pound-Dollar Exchange Rates Hill’s estimate of alpha (Hill Horror plots-Resnick)
m
Hill
0 50 100 150
12
34
5
8SAMSI RISK WS 2007
Stǎricǎ Plots for Pound-Dollar Exchange Rates
15 realizations from GARCH model fitted to exchange rates + real exchange rate data. Which one is the real data?
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9SAMSI RISK WS 2007
Stǎricǎ Plots for Pound-Dollar Exchange Rates
ACF of the squares from the 15 realizations from the GARCH model on previous slide.
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LagA
CF
0 10 20 30 40
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12SAMSI RISK WS 2007
Example: Merck log(returns) (Jan 2, 2003 – April 28, 2006; 837 observations)
time
0 200 400 600 800
-0.3
-0.2
-0.1
0.0
0.1
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F
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s V
alu
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13SAMSI RISK WS 2007
Example: Merck log-returns Hill’s estimate of alpha (Hill Horror plots-Resnick)
m
Hill
0 50 100 150
12
34
5
15SAMSI RISK WS 2007
Example: Amazon-returns (May 16, 1997 – June 16, 2004)
time
log
re
turn
s
0 500 1000 1500
-1.0
-0.8
-0.6
-0.4
-0.2
0.0
0.2
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AC
F
0 10 20 30 40
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ua
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17SAMSI RISK WS 2007
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Stǎricǎ Plots for the Amazon Data
15 realizations from GARCH model fitted to Amazon + exchange rate data. Which one is the real data?
18SAMSI RISK WS 2007
Stǎricǎ Plots for Amazon
ACF of the squares from the 15 realizations from the GARCH model on previous slide.
Lag
AC
F
0 10 20 30 40
0.0
0.4
0.8
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AC
F
0 10 20 30 40
0.0
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19SAMSI RISK WS 2007
Multiplicative models for log(returns)
Basic model
Xt = ln (Pt) - ln (Pt-1) (log returns)
= t Zt ,
where
• {Zt} is IID with mean 0, variance 1 (if exists). (e.g. N(0,1) or a t-distribution with df.)
• {t} is the volatility process
• t and Zt are independent.
Properties:
• EXt = 0, Cov(Xt, Xt+h) = 0, h>0 (uncorrelated if Var(Xt) < )
• conditional heteroscedastic (condition on t).
24SAMSI RISK WS 2007
Two models for log(returns)-cont
Xt = t Zt (observation eqn in state-space formulation)
(i) GARCH(1,1) (General AutoRegressive Conditional Heteroscedastic – observation-driven specification):
(ii) Stochastic Volatility (parameter-driven specification):
)1,0(IID~}{ ,211
2110
2tt tt-t-tt Z σβ Xαα, σZX
Main question:
What intrinsic features in the data (if any) can be used to discriminate between these two models?
)N(0, IID~}{ , log log , 22110
2 ttttttt ZX
27SAMSI RISK WS 2007
Equivalence:
is a measure on RRm which satisfies for x > 0 and A bounded away
from 0,
(xA) = x(A).
Multivariate regular variation of X=(X1, . . . , Xm): There exists a
random vector Sm-1 such that
P(|X|> t x, X/|X| )/P(|X|>t) v xP( )
(v vague convergence on SSm-1, unit sphere in Rm) .
• P( ) is called the spectral measure
• is the index of X.
)()t |(|
)t (
vP
P
X
X)(
)t |(|
)t (
vP
P
X
X
Regular variation — multivariate case
28SAMSI RISK WS 2007
Examples:
1. If X1> 0 and X2 > 0 are iid RV(, then X= (X1, X2 ) is multivariate
regularly varying with index and spectral distribution
P( =(0,1) ) = P( =(1,0) ) =.5 (mass on axes).
Interpretation: Unlikely that X1 and X2 are very large at the same
time.
0 5 10 15 20
x_1
010
2030
40
x_2
Figure: plot of (Xt1,Xt2) for realization
of 10,000.
Regular variation — multivariate case (cont)
29SAMSI RISK WS 2007
2. If X1 = X2 > 0, then X= (X1, X2 ) is multivariate regularly varying
with index and spectral distribution
P( = (1/2, 1/2) ) = 1.
3. AR(1): Xt= .9 Xt-1 + Zt , {Zt}~IID symmetric stable (1.8)
sqrt(1.81), W.P. .9898
,W.P. .0102
-10 0 10 20 30
x_t
-10
010
2030
x_{t
+1}
Figure: scatter plot
of (Xt, Xt+1) for
realization of 10,000
Distr of
31SAMSI RISK WS 2007
Use vague convergence with Ac={y: cTy > 1}, i.e.,
where t-L(t) = P(|X| > t).
Linear combinations:
X ~RV() all linear combinations of X are regularly varying
),(w:)A()t |(|
)t (
)(
)tA ( T
cX
XcXc
c
P
P
tLt
P
i.e., there exist and slowly varying fcn L(.), s.t.
P(cTX> t)/(t-L(t)) w(c), exists for all real-valued c,
where
w(tc) = tw(c).
Ac),(w:)A(
)t |(|
)t (
)(
)tA ( T
cX
XcXc
c
P
P
tLt
P
Applications of multivariate regular variation (cont)
32SAMSI RISK WS 2007
Converse?
X ~RV() all linear combinations of X are regularly varying?
There exist and slowly varying fcn L(.), s.t.
(LC) P(cTX> t)/(t-L(t)) w(c), exists for all real-valued c.
Theorem (Basrak, Davis, Mikosch, `02). Let X be a random vector.
1. If X satisfies (LC) with non-integer, then X is RV().
2. If X > 0 satisfies (LC) for non-negative c and is non-integer,
then X is RV().
3. If X > 0 satisfies (LC) with an odd integer, then X is RV().
Applications of multivariate regular variation (cont)
33SAMSI RISK WS 2007
1. If X satisfies (LC) with non-integer, then X is RV().
2. If X > 0 satisfies (LC) for non-negative c and is non-integer, then X is RV().
3. If X > 0 satisfies (LC) with an odd integer, then X is RV().
Applications of multivariate regular variation (cont)
There exist and slowly varying fcn L(.), s.t.
(LC) P(cTX> t)/(t-L(t)) w(c), exists for all real-valued c.
Remarks:
• 1 cannot be extended to integer (Hult and Lindskog `05)
• 2 cannot be extended to integer (Hult and Lindskog `05)
• 3 can be extended to even integers (Lindskog et al. `07, under
review).
34SAMSI RISK WS 2007
1. Kesten (1973). Under general conditions, (LC) holds with L(t)=1
for stochastic recurrence equations of the form
Yt= At Yt-1+ Bt, (At , Bt) ~ IID,
At dd random matrices, Bt random d-vectors.
It follows that the distributions of Yt, and in fact all of the finite dim’l
distrs of Yt are regularly varying (no longer need to be non-even).
2. GARCH processes. Since squares of a GARCH process can be
embedded in a SRE, the finite dimensional distributions of a
GARCH are regularly varying.
Applications of theorem
35SAMSI RISK WS 2007
Example of ARCH(1): Xt=(01 X2t-1)1/2Zt, {Zt}~IID.
found by solving E| Z2| = 1.
.312 .577 1.00 1.57
8.00 4.00 2.00 1.00
Distr of
P( ) = E{||(B,Z)|| I(arg((B,Z)) )}/ E||(B,Z)||
where
P(B = 1) = P(B = -1) =.5
Examples
36SAMSI RISK WS 2007
Figures: plots of (Xt, Xt+1) and estimated distribution of for realization of 10,000.
Example of ARCH(1): 01=1, =2, Xt=(01 X2t-1)1/2Zt, {Zt}~IID
Examples (cont)
theta
-3 -2 -1 0 1 2 3
0.0
80
.10
0.1
20
.14
0.1
60
.18
x_1
x_2
-40 -20 0 20 40
-40
-20
02
04
0
39SAMSI RISK WS 2007
Is this process time-reversible?
Figures: plots of (Xt, Xt+1) and (Xt+1, Xt) imply non-reversibility.
Example of ARCH(1): 01=1, =2, Xt=(01 X2t-1)1/2Zt, {Zt}~IID
Examples (cont)
x_1
x_2
-40 -20 0 20 40
-40
-20
02
04
0
x_2
x_1
-40 -20 0 20 40
-40
-20
02
04
0
40SAMSI RISK WS 2007
x_1
x_2
-2000 0 2000
-20
00
02
00
0
Example: SV model Xt = t Zt
Suppose Zt ~ RV() and
Then Zn=(Z1,…,Zn)’ is regulary varying with index and so is
Xn= (X1,…,Xn)’ = diag(1,…, n) Zn
with spectral distribution concentrated on (1,0), (0, 1).
Figure: plot of
(Xt,Xt+1) for realization
of 10,000.
Examples (cont)
)N(0, IID~}{ , log log , 22110
2 ttttttt ZX )N(0, IID~}{ , log log , 22110
2 ttttttt ZX
41SAMSI RISK WS 2007
Example: SV model Xt = t Zt
Examples (cont)
theta
-3 -2 -1 0 1 2 3
0.1
20
.14
0.1
60
.18
0.2
0
x_1
x_2
-2000 0 2000
-20
00
02
00
0
x_2
x_1
-2000 0 2000
-20
00
02
00
0
• SV processes are time-reversible if log-volatility is Gaussian.
• Asymptotically time-reversible if log-volatility is nonGaussian
45SAMSI RISK WS 2007
Setup
Xt = t Zt , {Zt} ~ IID (0,1)
Xt is RV
Choose {bn} s.t. nP(Xt > bn) 1
Then }.exp{)( 11 xxXbP n
n
Then, with Mn= max{X1, . . . , Xn},
(i) GARCH:
is extremal index ( 0 < < 1).
(ii) SV model:
extremal index = 1 no clustering.
},exp{)( 1 xxMbP nn
},exp{)( 1 xxMbP nn
Extremes for GARCH and SV processes
46SAMSI RISK WS 2007
(i) GARCH:
(ii) SV model:
}exp{)( 1 xxMbP nn
}exp{)( 1 xxMbP nn
Remarks about extremal index.
(i) < 1 implies clustering of exceedances
(ii) Numerical example. Suppose c is a threshold such that
Then, if .5,
(iii) is the mean cluster size of exceedances.
(iv) Use to discriminate between GARCH and SV models.
(v) Even for the light-tailed SV model (i.e., {Zt} ~IID N(0,1), the extremal index is 1 (see Breidt and Davis `98 )
95.~)( 11 cXbP n
n
975.)95(.~)( 5.1 cMbP nn
Extremes for GARCH and SV processes (cont)
47SAMSI RISK WS 2007
0 20 40 60
time
010
2030
0 20 40 60
time
010
2030
** * * *** ***
Extremes for GARCH and SV processes (cont)
Absolute values of ARCH
48SAMSI RISK WS 2007
Extremes for GARCH and SV processes (cont)
time
0 10 20 30 40 50
01
23
45
* * * * * ** *
Absolute values of SV process
51SAMSI RISK WS 2007
Remark: Similar results hold for the sample ACF based on |Xt| and Xt
2.
,)/())(ˆ( ,,10,,1 mhhd
mhX VVh
.)0()(ˆ,,1
1,,1
/21mhhX
dmhX Vhn
.)0()(ˆ,,1
1,,1
2/1mhhX
dmhX Ghn
Summary of results for ACF of GARCH(p,q) and SV models
GARCH(p,q)
52SAMSI RISK WS 2007
.)0()(ˆ,,1
1,,1
2/1mhhX
dmhX Ghn
.)(ˆln/0
2
1
1h1/1
S
Shnn hd
X
Summary of results for ACF of GARCH(p,q) and SV models (cont)
SV Model
53SAMSI RISK WS 2007
Sample ACF for GARCH and SV Models (1000 reps)
-0.3
-0.1
0.1
0.3
(a) GARCH(1,1) Model, n=10000-0
.06
-0.0
20
.02
(b) SV Model, n=10000
54SAMSI RISK WS 2007
Sample ACF for Squares of GARCH (1000 reps)
(a) GARCH(1,1) Model, n=100000
.00
.20
.40
.60
.00
.20
.40
.6
b) GARCH(1,1) Model, n=100000
55SAMSI RISK WS 2007
Sample ACF for Squares of SV (1000 reps)0.
00.
010.
020.
030.
04
(d) SV Model, n=100000
0.0
0.0
50
.10
0.1
5
(c) SV Model, n=10000
56SAMSI RISK WS 2007
Example: Amazon-returns (May 16, 1997 – June 16, 2004)
time
log
re
turn
s
0 500 1000 1500
-1.0
-0.8
-0.6
-0.4
-0.2
0.0
0.2
Lag
AC
F o
f sq
ua
res
0 10 20 30 40
0.0
0.2
0.4
0.6
0.8
1.0
Lag
AC
F o
f a
bs
valu
es
0 10 20 30 40
0.0
0.2
0.4
0.6
0.8
1.0
57SAMSI RISK WS 2007
Amazon returns (GARCH model)
GARCH(1,1) model fit to Amazon returns:
0.000024931= .0385, = .957, Xt=(01 X2t-1)1/2Zt, {Zt}~IID
t(3.672)
Lag
AC
F a
bs
valu
es
0 10 20 30 40
0.0
0.2
0.4
0.6
0.8
1.0
Lag
AC
F o
f sq
ua
res
0 10 20 30 40
0.0
0.2
0.4
0.6
0.8
1.0
Simulation from GARCH(1,1) model
58SAMSI RISK WS 2007
Amazon returns (SV model)
Stochastic volatility model fit to Amazon returns:
Lag
AC
F o
f sq
ua
res
0 10 20 30 40
0.0
0.2
0.4
0.6
0.8
1.0
Lag
AC
F o
f a
bs
valu
es
0 10 20 30 40
0.0
0.2
0.4
0.6
0.8
1.0
59SAMSI RISK WS 2007
Wrap-up
• Regular variation is a flexible tool for modeling both dependence
and tail heaviness.
• Useful for establishing point process convergence of heavy-tailed
time series.
• Extremal index < 1 for GARCH and for SV.
• ACF has faster convergence for SV.