Final project: Exploring the structure of correlation Forrest White, Jason Wei Joachim Edery, Kevin...

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

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fa

cts

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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

empirical

1 factor model"Implied"

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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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

3.5

density 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

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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

elat

ion

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

elat

ion

2222 ~~~~

~~~~)(

jjii

jiji

ij

rrrr

rrrr

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

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

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

1.5

2

2.5

3

3.5

4

4.5

5

Data

Density

Correlation distribution fitting

15min correlation

Gaussian 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

Density

1 factor correlation distribution fitting

15min 1 factor correlation

Gaussian fit

Student 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

Density

Correlation distribution fitting

25*15 correlation


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

ensity

1 factor correlation distribution fitting

25*15min 1 factor correlation


• empirical distribution : t-distribution fits better

• 1 factor model : normal distribution

• closer normal fit when time scale of returns increases

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

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

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• Interpretation

Consum

er D

iscre

tionar

y

Consum

er St

aples

Energy

Finan

cials

Health

Care

Industr

ials

Info

rmati

on Tec

hnology

Mate

rials

Teleco

mm

unicatio

n Serv

ices

Utiliti

es0

0.004

0.008

0.012

0.016

Comp 3

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

Factor modelSt

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• Results

Consumer d. Energy Materials

Mat

eria

ls

Ene

rgy

C

onsu

mer

d

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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 ?

Conclusion

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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

Q&A

• Thank you

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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

0.2

0.4

0.6

0.8

1

1.2


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


log(t)

log(F

2(t)

0.4 0.5 0.6 0.7 0.8 0.9 10

1

2

3

4

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6

7

8


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


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

0.4

0.6

0.8

1Spectrum for 12*15min returns

f()

Final project: Exploring the structure of correlation Forrest White, Jason Wei Joachim Edery, Kevin...

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Transcript of Final project: Exploring the structure of correlation Forrest White, Jason Wei Joachim Edery, Kevin...