Correlation between magnetic spin structure and the three ...
Final project: Exploring the structure of correlation
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Final project:Exploring the structure of
correlation
Forrest White, Jason WeiJoachim Edery, Kevin HsuYoan HassidMS&E 444 - 06/02/2010
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Stylized facts• Verification of empirical facts on correlation
• Data : 15min closing prices from Jan 2007 to Jan 2009 of the S&P 500
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Epps effect• empirical correlations virtually disappear at high
frequency
• trading asynchronous
• Epps effect observed
but data still
significant
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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"
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Memory effect and fractal analysisSt
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• 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 ~)(
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Memory effect and fractal analysisSt
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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
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Correlations vs absolute returns• Expect big correlation for extreme return
periods
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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
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Asymmetry in Correlations• Expect asymmetry for extreme negative return
periods vs extreme positive return periods
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• 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
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2222 ~~~~
~~~~)(
jjii
jijiij
rrrr
rrrr
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Beta vs Correlations• Stocks with the same betas show higher
correlation
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High Beta Mid Beta Low Beta
Low
Bet
a
Mid
Bet
a H
igh
Bet
a
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Factor modelSt
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• 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
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Distribution of correlationSt
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0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.5
1
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Data
Den
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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
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Data
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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
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2
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Data
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Correlation distribution fitting
25*15 correlationGaussian fitStudent fit
0.1 0.2 0.3 0.4 0.5 0.60
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DataD
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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
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One factor modelSt
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• 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
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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
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Factor modelSt
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• 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
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Factor modelSt
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• 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
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Factor modelSt
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• Results
Consumer d. Energy Materials
Mat
eria
ls
Ene
rgy
C
onsu
mer
d
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Styl
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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%
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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 ?
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ConclusionSt
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• 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
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Q&A• Thank you
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Memory effect and fractal analysisSt
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• 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
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• 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)
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1.5 2 2.5 3 3.5 4-0.6
-0.4
-0.2
0
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1
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H2=0.59 for 4 *15min returns
log(t)
log(
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1.5 2 2.5 3 3.5 4-0.4
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H2=0.74 for 16 *15min returns
log(t)
log(
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0.4 0.5 0.6 0.7 0.8 0.9 10
1
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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
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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
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• 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
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1Spectrum for 12*15min returns
f()