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Final project: Exploring the structure of correlation. Forrest White, Jason Wei Joachim Edery , Kevin Hsu Yoan Hassid MS&E 444 - 06/02/2010. MS&E 352 - 2/25/2010. Stylized facts. Conclusion. Verification of empirical facts on correlation - PowerPoint PPT Presentation
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Final project:Exploring the structure of
correlation
Forrest White, Jason WeiJoachim Edery, Kevin HsuYoan HassidMS&E 444 - 06/02/2010
Stylized facts• Verification of empirical facts on correlation
• Data : 15min closing prices from Jan 2007 to Jan 2009 of the S&P 500
Styl
ized
fa
cts
Fact
or m
odel
Copu
laCo
nclu
sion
2
Epps effect• empirical correlations virtually disappear at high
frequency
• trading asynchronous
• Epps effect observed
but data still
significant
Styl
ized
fa
cts
Fact
or m
odel
Copu
laCo
nclu
sion
3
0 5 10 15 20 25 300.32
0.34
0.36
0.38
0.4
0.42
0.44
0.46
0.48
0.5Epps Effect
empirical1 factor model"Implied"
Memory effect and fractal analysisSt
yliz
ed
fact
sFa
ctor
mod
elCo
pula
Conc
lusi
on
4
• Time series IC & AIC (instantaneous correlation)
• Average Instantaneous correlation :
• Detrented Fluctual Analysis :
• interpretation of H2 as Hurst exponent:
0.5<H2<1 : long-range memory
0<H2<0.5 : mean-reverting
H2 = 0.5 : no memory (Brownian motion)
1
1 1
)()()1(
2)(n
i
n
ijji tRtR
nntAIC
qHttAICf ~)(
Memory effect and fractal analysisSt
yliz
ed
fact
sFa
ctor
mod
elCo
pula
Conc
lusi
on
5
long-range memory for correlation on average
0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8-0.4
-0.2
0
0.2
0.4
0.6
0.8
1Spectrum for 12*15min returns
f()
1.5 2 2.5 3 3.5 4-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
H2=0.74 for 16 *15min returns
log(t)
log(
F 2(t)
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.20
0.5
1
1.5
2
2.5
3
3.5density of H2 for 16*15min returns
H2
behavior close to gaussian for pairwise
Multi-fractal behavior
Asymmetric shape
Correlations vs absolute returns• Expect big correlation for extreme return
periods
Styl
ized
fa
cts
Fact
or m
odel
Copu
laCo
nclu
sion
6
iii
jijiji
rrN
rrrrN
22
,2
1
1
)(
0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.050
0.1
0.2
0.3
0.4
0.5
0.6
0.7
correlation vs absolute returns
Absolute market return
Corr
elat
ion
Asymmetry in Correlations• Expect asymmetry for extreme negative return
periods vs extreme positive return periods
Styl
ized
fa
cts
Fact
or m
odel
Copu
laCo
nclu
sion
7
• Time period may be too short
-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.40.4
0.44
0.48
0.52
0.56
0.6
Asymmetry
realized negRealized pos
Returns
Corr
elat
ion
2222 ~~~~
~~~~)(
jjii
jijiij
rrrr
rrrr
Beta vs Correlations• Stocks with the same betas show higher
correlation
Styl
ized
fa
cts
Fact
or m
odel
Copu
laCo
nclu
sion
8
High Beta Mid Beta Low Beta
Low
Bet
a
Mid
Bet
a H
igh
Bet
a
Factor modelSt
yliz
ed
fact
sFa
ctor
mod
elCo
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Conc
lusi
on
9
• Compute the scores/loadings with a PCA
• Model values : Xi(t) ≈ βiV1(t)+ γiV2(t) + δiV3(t) …
• Correlation : ρij ≈ ρiV1 ρjV1 + ρiV2 ρjV2 +…
0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50
0.1
0.2
0.3
0.4
0.5
0.6
f(x) = 1.02930533057843 xR² = 0.998366917342644
Empirical correlation vs 1 factor model
Average empirical correlation
Mod
el
Distribution of correlationSt
yliz
ed
fact
sFa
ctor
mod
elCo
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on
10
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
Data
Den
sity
Correlation distribution fitting
15min correlationGaussian fitStudent fit
0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.550
1
2
3
4
5
6
7
Data
Den
sity
1 factor correlation distribution fitting
15min 1 factor correlationGaussian fitStudent fit
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.5
1
1.5
2
2.5
3
3.5
Data
Den
sity
Correlation distribution fitting
25*15 correlationGaussian fitStudent fit
0.1 0.2 0.3 0.4 0.5 0.60
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
DataD
ensi
ty
1 factor correlation distribution fitting
25*15min 1 factor correlationGaussian fitStudent fit
• empirical distribution : t-distribution fits better
• 1 factor model : normal distribution
• closer normal fit when time scale of returns increases
One factor modelSt
yliz
ed
fact
sFa
ctor
mod
elCo
pula
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11
• The one factor model works, on average!
• It tends to underestimate correlation for stocks of the same nature (sectors, betas…)
H-H M-M L-L0%5%
10%15%20%25%30%35%40%45%50%
Correl vs Betas
RealizedModel
Beta
Corr
elat
ion
0 0.01 0.02 0.03 0.04 0.05 0.060
0.10.20.30.40.50.60.70.80.9
Correl vs absolute returns
RealizedModel
max absolute returns
Corr
elat
ion
Factor modelSt
yliz
ed
fact
sFa
ctor
mod
elCo
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Conc
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on
12
• Interpretation
Consum
er D
iscre
tionar
y
Consum
er St
aples
Energy
Finan
cials
Health
Care
Industrial
s
Inform
ation T
echnolo
gy
Materia
ls
Teleco
mmunica
tion Se
rvice
s
Utiliti
es0
0.004
0.008
0.012
0.016
Comp 3
Factor modelSt
yliz
ed
fact
sFa
ctor
mod
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13
• Selection
1 2 3 4 5 6 7 8 9Health x x xUtilities x x xFinance x x x xconsumer d x x xconsumer s x x xindustrials x x xinfo tech x x x xmaterials x x x xtelecom x x xenergy x x x x x x
Factor modelSt
yliz
ed
fact
sFa
ctor
mod
elCo
pula
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lusi
on
14
• Results
Consumer d. Energy Materials
Mat
eria
ls
Ene
rgy
C
onsu
mer
d
Styl
ized
fa
cts
Fact
or m
odel
Copu
laCo
nclu
sion
15
Copula• Marginals + copula Joint distribution
• Sklar’s theorem, other properties
• Gaussian copula : )(),(),(),,( 111 cbacbaCP
• Easy but bad tail fitting
• Empirical ρ : 45%Optimal ρ : 60%
Styl
ized
fa
cts
Fact
or m
odel
Copu
laCo
nclu
sion
16
Copula
Market absolute log-returns ML Gaussian copula ML T copula df Relative difference< 0.30% (0-20% quantile) 40 45.5 10.4 13.8%< 0.58% (0-40% quantile 64 74.71 12.9 16.7%< 1.00% (0-60% quantile) 110 144 9.45 30.9%< 1.65% (0-80% quantile) 218 280 8.83 28.4%
all 792 974 4.34 23.0%
0.00% 0.50% 1.00% 1.50% 2.00% 2.50% 3.00% 3.50% 4.00%0
200
400
600
800
1000
1200Fitting with a Gaussian vs a T-copula
ML Gaussian copula
ML T copula
Max daily market returns
Max
imu
m L
ikel
ihoo
d
• Gaussian is ok for low returns
• T-distribution T-copula ?
ConclusionSt
yliz
ed
fact
sFa
ctor
mod
elCo
pula
Conc
lusi
on
17
• Some empirical facts in correlation can be captured with a low dimension model
• The Gaussian copula is very limited
• Trading strategies exist to take advantage of patterns
• Further studies
• Implied correlation vs historical correlation?
• Different time periods
• Higher frequencies
Q&A• Thank you
Styl
ized
fa
cts
Fact
or m
odel
Copu
laCo
nclu
sion
18
Memory effect and fractal analysisSt
yliz
ed
fact
sFa
ctor
mod
elCo
pula
Conc
lusi
on
19
• Time series IC & AIC (instantaneous correlation)
• normalized returns :
• Instantaneous correlation :
• Average Instantaneous correlation :
0 200 400 600 800 1000 1200-20
0
20
40AIC
0 200 400 600 800 1000 1200-10
0
10
20
30IC for JPM & PFE
Memory effect and fractal analysisSt
yliz
ed
fact
sFa
ctor
mod
elCo
pula
Conc
lusi
on
20
• Detrented Fluctual Analysis
• , with A=IC or A= AIC
• DFA functions :
• qth order of detrended function :
• power law behavior :
• interpretation of H2 as Hurst exponent:
0.5<H2<1 : long-range memory
0<H2<0.5 : anti-persistent
H2 = 0.5 : no memory (Brownian motion)
Memory effect and fractal analysisSt
yliz
ed
fact
sFa
ctor
mod
elCo
pula
Conc
lusi
on
21
1.5 2 2.5 3 3.5 4-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
H2=0.59 for 4 *15min returns
log(t)
log(
F 2(t)
1.5 2 2.5 3 3.5 4-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
H2=0.74 for 16 *15min returns
log(t)
log(
F 2(t)
0.4 0.5 0.6 0.7 0.8 0.9 10
1
2
3
4
5
6
7
8density of H2 for 4*15min returns
H2
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.20
0.5
1
1.5
2
2.5
3
3.5
density of H2 for 16*15min returns
H2
• long-range memory for correlation on average : persisent behavior, possible predictability
• behavior close to gaussian for pairwise correlation
Memory effect and fractal analysisSt
yliz
ed
fact
sFa
ctor
mod
elCo
pula
Conc
lusi
on
22
• Hq non constant : multifractality of signal
• Signal complex and turbulent with inhomogeneities in properties
• Spectrum of singularities :
• Asymmetry in spectrum => 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1Spectrum for 12*15min returns
f()
