How do financial correlations grow? How do financial correlations - - PowerPoint PPT Presentation

How do financial correlations grow? How do financial correlations grow?

C.

• C. Borghesi

Borghesi (Paris), S. (Paris), S. Micciche Micciche’ (Palermo), MM (Trieste) ’ (Palermo), MM (Trieste)

• Financial correlations

Financial correlations

• Noise and structure

Noise and structure

• Dynamics: evolution and formation

Dynamics: evolution and formation

• How fast is information aggregated across stocks?

How fast is information aggregated across stocks?

• Early results: Market forms as an embryo

Early results: Market forms as an embryo

• Here: HF data from NYSE, LSE, PB

Here: HF data from NYSE, LSE, PB

• internal dynamics

internal dynamics →

→ market develops as a baby (

market develops as a baby (∆

∆t

t > > ∆

∆t

t0

0)

)

• Structure of overnight returns markedly different

Structure of overnight returns markedly different

• Summary of results

Summary of results

Financial correlations Financial correlations

xPFE xGE

T = window size t0 = initial time

∆t = time scale τ = time shift

The correlation matrix The correlation matrix

[ ]

[ ][ ] [ ] [ ]

N k i x t x x t x x t x x t x t T t C N i t x T x t t p t p t x

T t t t t k t k T t t t t i t i T t t t t k t k t i t i k i t T t t t i t i i i t i

, , 1 , , ) ( ) ( ) ( ) ( ) , , , ( , , 1 ) ( 1 , ) ( ) ( log ) (

2 2 ,

K K = − − − + − = ∆ = = ∆ − =

∑ ∑ ∑ ∑

+ = ∆ ∆ + = ∆ ∆ + = ∆ ∆ ∆ ∆ + = ∆ ∆ ∆

τ τ

Questions: Structure: i,k Noise dressing: T Evolution: t0 Efficiency τ Growth: ∆t

Log-returns: Pearson coefficient/sample correlation:

Noise or real correlations? Noise or real correlations?

• Eigenvalue

Eigenvalue distribution distribution and random matrix theory and random matrix theory

( (Laloux Laloux et al./ et al./ Gopikrishnan Gopikrishnan et al. …) et al. …)

• The bulk of

The bulk of eigenvalue eigenvalue distribution is dominated by distribution is dominated by sampling effects (noise) sampling effects (noise)

• One large

One large eigenvalue eigenvalue (market mode) (market mode)

• Few

Few eigenvalues eigenvalues with with “economic” meaning “economic” meaning

• “Localized” small

“Localized” small eigenvalues

Λ (N/T)1/2

eigenvalues

Structure Structure

( (∆

∆t

t= 1 day, N~ T~ 500) = 1 day, N~ T~ 500)

• Eigenvectors analysis

Eigenvectors analysis

( (Gopikrishnan Gopikrishnan et al. ) et al. )

• Minimal spanning trees

Minimal spanning trees

( (Mantegna Mantegna et al. ) et al. )

• There is significant non

There is significant non-

• trivial structure in C

trivial structure in C

Cluster structure Cluster structure

Ci,j = = 1 if i= j = Cs if si= sj = C0 else Idea: Model:

si = sector to which stock i belongs

) ( 1 ) ( ) ( ) (

2 2

t g a t g t a t x

i s i s s i i

i i i

ε η µ − − + + =

Maximize likelihood:

{ } ( )

∑

>

⎥ ⎦ ⎤ ⎢ ⎣ ⎡ − − − + − =

: 2 2

log 1 log 2 1 structure | data log

s

n s s s s s s s s

n n c n n n c P

cs ns ns

S&P500 clusters ~ sectors

Group: size/c/g/e 24 190.345795 0.431334049 -6.17717028 Oil & Gas ENE 710 Enron Corp. Natural Gas SLB 395 Schlumberger Ltd. Oil & Gas (Drilling & Equipment) RDC 395 Rowan Cos. Oil & Gas (Drilling & Equipment) HAL 395 Halliburton Co. Oil & Gas (Drilling & Equipment) BHI 395 Baker Hughes Oil & Gas (Drilling & Equipment) TX 390 Texaco Inc. Oil (International Integrated) RD 390 Royal Dutch Petroleum Oil (International Integrated) CHV 390 Chevron Corp. Oil (International Integrated) P 385 Phillips Petroleum Oil (Domestic Integrated) OXY 385 Occidental Petroleum Oil (Domestic Integrated) AHC 385 Amerada Hess Oil (Domestic Integrated) UCL 380 Unocal Corp. Oil & Gas (Exploration & Productn) KMG 380 Kerr-McGee Oil & Gas (Exploration & Productn) BR 380 Burlington Resources Oil & Gas (Exploration & Productn) XON 0 EXXON CORP SNT 0 SONAT INC PZL 0 PENNZOIL CO ORX 0 ORYX ENERGY CO MOB 0 MOBIL CORP LLX 0 LOUISIANA LAND HP 0 HELMERICH & PAYN DI 0 DRESSER INDUS ARC 0 ATL RICHFIELD CO AN 0 AMOCO CORP

Group: size/c/g/e 18 115.202408 0.465534151 -4.64141703 Computers AMAT 247 Applied Materials Equipment (Semiconductor) TXN 235 Texas Instruments Electronics (Semiconductors) NSM 235 National Semiconductor Electronics (Semiconductors) INTC 235 Intel Corp. Electronics (Semiconductors) AMD 235 Advanced Micro Devices Electronics (Semiconductors) SUNW 190 Sun Microsystems Computers (Hardware) IBM 190 International Bus. Machines Computers (Hardware) HWP 190 Hewlett-Packard Computers (Hardware) CPQ 190 COMPAQ Computer Computers (Hardware) AAPL 190 Apple Computer Computers (Hardware) ORCL 185 Oracle Corp. Computers (Software & Services) NOVL 185 Novell Inc. Computers (Software & Services) MSFT 185 Microsoft Corp. Computers (Software & Services) CA 185 Computer Associates Intl. Computers (Software & Services) MOT 180 Motorola Inc. Communications Equipment DIGI 0 DSC COMM CORP DEC 0 DIGITAL EQUIPMEN ACAD 0 AUTODESK INC Group: size/c/g/e 8 29.0933895 0.604280651 -2.01765895 SGP 285 Schering-Plough Health Care (Drugs-Major Pharmacs) PFE 285 Pfizer, Inc. Health Care (Drugs-Major Pharmacs) MRK 285 Merck & Co. Health Care (Drugs-Major Pharmacs) LLY 285 Lilly (Eli) & Co. Health Care (Drugs-Major Pharmacs) JNJ 280 Johnson & Johnson Health Care (Diversified) BMY 280 Bristol-Myers Squibb Health Care (Diversified) AHP 280 American Home Products Health Care (Diversified) ABT 280 Abbott Labs Health Care (Diversified)

Group: size/c/g/e 5 20.0271244 3.02181792 -4.17928696 PDG 265 Placer Dome Inc. Gold & Precious Metals Mining NEM 265 Newmont Mining Gold & Precious Metals Mining HM 265 Homestake Mining Gold & Precious Metals Mining ABX 265 Barrick Gold Corp. Gold & Precious Metals Mining ECO 0 ECHO BAY MINES Gold & Precious Metals Mining

Data from R.N. Mantegna

Hierarchical clustering of assets Hierarchical clustering of assets (N= 2000 NYSE 90 (N= 2000 NYSE 90-

• 98):

98):

“noise level”

• Statistically

significant

• Zipf’s law
• no orthogonality

Data from R.N. Mantegna

Note: totally unsupervised method Number of clusters not predefined!

Time evolution (t Time evolution (t 0

0): Persistence

): Persistence

( (Onnela Onnela et al. 2003, T~ 500 et al. 2003, T~ 500÷ ÷ 1500 days, 1500 days, δ

δt

t 0

0= 21 days)

= 21 days)

More than 80% of the links of the MST are conserved from one time window to the next The fraction of conserved links has algebraic decay with time lag between windows (no finite memory)

Time evolution (t Time evolution (t 0

0): Recurrence

): Recurrence

(MM ‘02) (MM ‘02)

Daily return assets

IBM, AAPL,HWP,…. GM,F,…. PDG, NEM,ABX,…. SGP,PFE,MRK,…

{

Computers,… cars,… gold/mining,… health,… day t

Do blue days, red days, … exist or is market activity following a continuum

Market states: Clustering days Market states: Clustering days

form this

• f

matrices

• f

class

• n the

likelihood maximize ' 1 ) ' ( ) ( 1 ) ' , ( : Idea

' ' 1

≡ ≠ = = = ⎪ ⎩ ⎪ ⎨ ⎧ = =

∑

= t t t t s N i i i

s s s s s t t g t x t x N t t C

Results:

• Market states do exist
• scale free frequency

distribution for most frequent clusters of days

• Identify market states

ω={sω,1,sω,2,…}

log likelihood/asset days

Conditional dynamics Dynamics after crashes

<r|ω> = average return in state ω

Non Non-

• equal time correlations (

equal time correlations (τ

τ)

)

(B. (B. Toth Toth + J. + J. Kertesz Kertesz 2005) 2005) C(bigtoday, smalltomorrow)

Correlation lag τ

→ market is getting more

and more efficient

Growth Growth

How does market structure forms as How does market structure forms as ∆

∆t

t grows? grows?

Early results Early results

The Epps effect The Epps effect

correlation grows with correlation grows with ∆

∆t

t

Information is aggregated faster today than in the past in bigger companies

(J. Kwapien, S. Drozdz, J. Speth 2003)

Market as an embryo: Market as an embryo:

growth and differentiation growth and differentiation

∆t= 1 day

∆t= 1/20 day ∆t= 1/10 day ∆t= 1/5 day ∆t= 1/2 day

( (Bonanno Bonanno et al. 2004, et al. 2004, Tumminello Tumminello et al. 2006) et al. 2006)

+ number of eigenvalues out of the noise band increases with ∆t

→ structure forms as ∆t increases

Correlated or interacting system? Correlated or interacting system?

→ → separate c separate center of mass motion

enter of mass motion and dynamics of relative coordinates and dynamics of relative coordinates

Data sets Data sets

• A: bare data

A: bare data

• B: without center of mass (mean return)

B: without center of mass (mean return)

• C: without market index (residues of one factor

C: without market index (residues of one factor model) model)

• D: without market mode (largest eigenvector)

D: without market mode (largest eigenvector)

• E: residues of the mean return

E: residues of the mean return

NYSE NYSE

100 most capitalized stocks of 2002 100 most capitalized stocks of 2002

Distribution of Ci,j

∆t = 5 min

15 min 30 min 65 min 195 min cl – op

• p – cl

cl – cl

Number of clusters Number of clusters

Note: Overnight!

Same intraday scale structure Same intraday scale structure

Cluster labels assets

A C B(/D) E

Cluster structure Cluster structure

A C B(/D) E 5 min 15 min 30 min 65 min 195 min cl – op 9:30-15:00

• vernight

Overlaps Overlaps

fraction* of pairs of assets in same cluster at scale fraction* of pairs of assets in same cluster at scale ∆

∆t

t which are in the same cluster at scale which are in the same cluster at scale ∆

∆t

t’ ’

A C E B(/D)

80% of the structure at daily scale is already formed at 5 min scale! But one has to look at relative coordinates!

* Weighted by log-likelihood

LSE ’92 most capitalized 2002 LSE ’92 most capitalized 2002

A B Time scale ∆t → Likelihood No. clusters Time scale ∆t

LSE structure LSE structure

Much stronger market mode

Sizeable fraction

• f daily structure

forms at 15 min

Paris bourse: Paris bourse:

75 most frequently traded assets in 2002 75 most frequently traded assets in 2002

• Tic time

Tic time vs vs real time real time

• Very high frequency (

Very high frequency (∆

∆t

t ~ 1 min) ~ 1 min)

• How does the market mode grows?

How does the market mode grows?

• How many eigenvectors are conserved?

How many eigenvectors are conserved?

Distribution of Distribution of C Ci,j

i,j

A A B B Tic time → Real time →

Structure Structure

Tic A Tic B Time A Time B 1 2 4 8 16 32 64 128 256 521 1024 Little structure for ∆t < 5 min

Overlaps Overlaps

Tic time Real time

A B B

Substantial rearrangement with ∆t Little structure ∆t < 5 min Structurally different overnight

Eigenvalue Eigenvalue distribution distribution and the market mode and the market mode

P(> λ)

λ

Market mode ~ log Market mode ~ log Dt Dt

Overlap of eigenvectors Overlap of eigenvectors

Largest Λ Second largest Λ Third largest Λ Smallest Λ Second smallest Λ Third smallest Λ

Conclusions Conclusions

• The formation of market structures is best revealed in

The formation of market structures is best revealed in relative coordinates relative coordinates

• Market index is not a good proxy for the center of mass

Market index is not a good proxy for the center of mass dynamics dynamics

• The structure of correlations forms at small time

The structure of correlations forms at small time-

• scales

scales (5 min for NYSE). (5 min for NYSE). It grows in strength keeping the same form It grows in strength keeping the same form

• Overnight returns have distinctly different structure than

Overnight returns have distinctly different structure than intraday returns intraday returns

• Market mode ~ log

Market mode ~ log ∆

∆t

t

• Both largest and

Both largest and smallest smallest eigenvalues eigenvalues have associated have associated eigenvectors which are conserved to some degree across eigenvectors which are conserved to some degree across scales scales

Thanks Thanks

www.ictp.trieste.it/~ marsili/ www.ictp.trieste.it/~ marsili/